Log24

Friday, December 1, 2017

The Architect and the Matrix

Filed under: Uncategorized — m759 @ 1:00 PM

In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —

Update from the morning of December 2 —

The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:

 .

See as well Tsimtsum in this journal.

Harold Bloom on tsimtsum as sublimation

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: Uncategorized — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —

.

From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Friday, November 24, 2017

The Matrix Meets the Grid

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

The Matrix —

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Saturday, October 14, 2017

In Principio:

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

Red October  continues …

See also Molloy in this  journal.

Related art  theory —

Geometry of the 4×4 Square 

Tuesday, September 5, 2017

Florence 2001

Filed under: Uncategorized — Tags: — m759 @ 4:44 AM

Or:  Coordinatization for Physicists

This post was suggested by the link on the word "coordinatized"
in the previous post.

I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.

Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.

Sunday, September 3, 2017

Broomsday Revisited

Filed under: Uncategorized — m759 @ 9:29 AM

Ivars Peterson in 2000 on a sort of conceptual art —

" Brill has tried out a variety of grid-scrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "

"Scrambled Grids," August 28, 2000

Or not.

If aesthetic sensibilities lead to a 23-cycle on a 4×6 grid, the results
may not be pretty —

From "Geometry of the 4×4 Square."

See a Log24 post, Noncontinuous Groups, on Broomsday 2009.

Saturday, September 2, 2017

A Touchstone

Filed under: Uncategorized — m759 @ 10:16 PM

From a paper by June Barrow-Green and Jeremy Gray on the history of geometry at Cambridge, 1863-1940

This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal Teichmuller Theory
," by S. Mochizuki.

* Specifically, Jacobi  and Kummer  (along with theta functions).
I do not know of any direct  connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.

Friday, July 14, 2017

March 26, 2006 (continued)

Filed under: Uncategorized — m759 @ 7:38 PM

4x4 array of Psychonauts images

The above image, posted here on March 26, 2006, was
suggested by this morning's post "Black Art" and by another
item from that date in 2006 —

Thursday, July 6, 2017

A Pleasing Situation

Filed under: Uncategorized — m759 @ 9:20 PM

The 4x4x4 cube is the natural setting
for the finite version of the Klein quadric
and the eight "heptads" discussed by
Conwell in 1910.

As R. Shaw remarked in 1995, 
"The situation is indeed quite pleasing."

Saturday, May 20, 2017

van Lint and Wilson Meet the Galois Tesseract*

Filed under: Uncategorized — m759 @ 12:12 AM

Click image to enlarge.

The above 35 projective lines, within a 4×4 array —


The above 15 projective planes, within a 4×4 array (in white) —

* See Galois Tesseract  in this journal.

Wednesday, April 26, 2017

A Tale Unfolded

Filed under: Uncategorized — Tags: , — m759 @ 2:00 AM

A sketch, adapted tonight from Girl Scouts of Palo Alto

From the April 14 noon post High Concept

From the April 14 3 AM post Hudson and Finite Geometry

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

From the April 24 evening post The Trials of Device

Pentagon with pentagram    

Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.

Monday, April 24, 2017

The Trials of Device

Filed under: Uncategorized — Tags: — m759 @ 3:28 PM

"A blank underlies the trials of device"
— Wallace Stevens, "An Ordinary Evening in New Haven" (1950)

A possible meaning for the phrase "the trials of device" —

See also Log24 posts mentioning a particular device, the pentagram .

For instance —

Related figures

Pentagon with pentagram    

Friday, April 14, 2017

Hudson and Finite Geometry

Filed under: Uncategorized — Tags: — m759 @ 3:00 AM

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element sets of black subsquares of a 4×4 square array 
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .

Hudson did not  view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Monday, April 3, 2017

Even Core

Filed under: Uncategorized — Tags: — m759 @ 9:16 PM

4x4x4 gray cube

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Monday, February 20, 2017

Mathematics and Narrative

Filed under: Uncategorized — Tags: , — m759 @ 2:40 PM

Mathematics —

Hudson's parametrization of the
4×4 square, published in 1905:

A later parametrization, from this date in 1986:

http://www.log24.com/log/pix11/110220-relativprob.jpg

A note from later in 1986 shows the equivalence of these
two parametrizations:

Narrative —

Posts tagged Memory-History-Geometry.

The mathematically challenged may prefer the narrative of the
Creation Matrix from the religion of the Transformers:

"According to religious legend, the core of the Matrix
was created from Solomus, the god of wisdom,
trapped in the form of a crystal by Mortilus, the god
of death. Following the defeat of Mortilus, Solomus
managed to transform his crystal prison into the Matrix—
a conduit for the energies of Primus, who had himself
transformed into the life-giving computer Vector Sigma."

Wednesday, January 4, 2017

A Drama of Many Forms

Filed under: Uncategorized — Tags: — m759 @ 1:24 PM

According to art historian Rosalind Krauss in 1979,
the grid's earliest employers

"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."

The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:

 

See instances of the above image.

Friday, November 25, 2016

Priority

Filed under: Uncategorized — Tags: , — m759 @ 12:00 AM

Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:

Hudson in 1905 —

Golomb in 1967 —

It is also likely that some figures illustrating Walsh functions  as
two-color square arrays were published prior to 1976.

Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

Friday, September 16, 2016

A Counting-Pattern

Filed under: Uncategorized — Tags: — m759 @ 10:48 AM

Wittgenstein, 1939

Dolgachev and Keum, 2002

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

For some related material, see posts tagged Priority.

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: Uncategorized — Tags: , — m759 @ 10:00 PM

The previous post discussed the parametrization of 
the 4×4 array as a vector 4-space over the 2-element 
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0  along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets —  were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator,"  what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click images for further details.

Monday, September 12, 2016

The Kummer Lattice

Filed under: Uncategorized — Tags: , — m759 @ 2:00 PM

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Thursday, July 28, 2016

The Giglmayr Foldings

Filed under: Uncategorized — m759 @ 1:44 PM

Giglmayr's transformations (a), (c), and (e) convert
his starting pattern

  1    2   5   6
  3    4   7   8
  9  10 13 14
11  12 15 16

to three length-16 sequences. Putting these resulting
sequences back into the 4×4 array in normal reading
order, we have

  1    2    3    4        1   2   4   3          1    4   2   3
  5    6    7    8        5   6   8   7          7    6   8   5 
  9  10  11  12      13 14 16 15       15 14 16 13
13  14  15  16       9  10 12 11        9  12 10 11

         (a)                         (c)                      (e)

Four length-16 basis vectors for a Galois 4-space consisting
of the origin and 15 weight-8 vectors over GF(2):

0 0 0 0       0 0 0 0       0 0 1 1       0 1 0 1
0 0 0 0       1 1 1 1       0 0 1 1       0 1 0 1 
1 1 1 1       0 0 0 0       0 0 1 1       0 1 0 1
1 1 1 1       1 1 1 1       0 0 1 1       0 1 0 1 .

(See "Finite Relativity" at finitegeometry.org/sc.)

The actions of Giglmayr's transformations on the above
four basis vectors indicate the transformations are part of
the affine group (of order 322,560) on the affine space
corresponding to the above vector space.

For a description of such transformations as "foldings,"
see a search for Zarin + Folded in this journal.

Tuesday, June 7, 2016

Art and Space…

Filed under: Uncategorized — m759 @ 6:00 AM

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title: 

A new graphic approach 
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24
"

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

Wednesday, May 25, 2016

Framework

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

Kummer and Dirac

Filed under: Uncategorized — Tags: , — m759 @ 11:00 AM

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam, 
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."

[4]

O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished


A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

Monday, May 2, 2016

Subjective Quality

Filed under: Uncategorized — m759 @ 6:01 AM

The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.

Siobhan Roberts recently wrote a book about the first of these
authors, Conway.  I just discovered that last fall she also had an
article about the second author, Sloane, published:

"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.

Meanwhile, in this  journal

Log24 on that same date, Oct. 22, 2015 —

Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective  quality, as opposed to the objective  quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here  last October.

Sunday, May 1, 2016

Sunday Appetizer from 1984

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —

“What would an organic Christian Sabbath look like today?”

The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —  

Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich
unter http://​www.​encyclopediaofma​th.​org/​index.​php/​
Cullinane_​diamond_​theorem
und http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."

The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the 
1988 first edition of Sphere Packings, Lattices and Groups
by J. H. Conway and N. J. A. Sloane —

A passage from the 1984 document —

Monday, April 25, 2016

Seven Seals

Filed under: Uncategorized — Tags: — m759 @ 11:00 PM

 An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —

"More poetically "

From Hermann Weyl's 1952 classic Symmetry

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

The seven seals from the previous post, with some context —

These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .

Friday, August 7, 2015

Parts

Filed under: Uncategorized — Tags: — m759 @ 2:19 AM

Spielerei  —

"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."

— Daniel Kehlmann, F: A Novel  (2014),
     translated from the German by
     Carol Brown Janeway

Nicht Spielerei  —

A figure from this journal at 2 AM ET
on Monday, August 3, 2015

Also on August 3 —

FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."

MANHATTAN — "Carol Brown Janeway, a Scottish-born
publishing executive, editor and award-winning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."

Related material —  Heisenberg on beauty, Munich, 1970                       

Monday, August 3, 2015

Text and Context*

Filed under: Uncategorized — Tags: — m759 @ 2:00 AM

"The ORCID organization offers an open and
independent registry intended to be the de facto  
standard for contributor identification in research
and academic publishing. On 16 October 2012,
ORCID launched its registry services and
started issuing user identifiers." — Wikipedia

This journal on the above date —

  

A more recent identifier —

Related material —

See also the recent posts Ein Kampf and Symplectic.

* Continued.

Wednesday, June 17, 2015

Slow Art, Continued

Filed under: Uncategorized — Tags: — m759 @ 10:01 AM

The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.

Example from mathematics:

  • Göpel tetrads as subsets of a 4×4 square in the classic
    1905 book Kummer's Quartic Surface  by R. W. H. T. Hudson.
    These subsets were constructed as helpful schematic diagrams,
    without any reference to the concept of finite  geometry they
    were later to embody.
     
  • Göpel tetrads (not then named as such), again as subsets of
    a 4×4 square, that form the 15 isotropic projective lines of the
    finite projective 3-space PG(3,2) in a note on finite geometry
    from 1986 —

    Göpel tetrads in an inscape, April 1986

  • Göpel tetrads as these figures of finite  geometry in a 1990
    foreword to the reissued 1905 book of Hudson:

IMAGE- Galois geometry in Wolf Barth's 1990 foreword to Hudson's 1905 'Kummer's Quartic Surface'

Click the Barth passage to see it with its surrounding text.

Related material:

Monday, June 15, 2015

Omega Matrix

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

See that phrase in this journal.

See also last night's post.

The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem.

Saturday, April 4, 2015

Harrowing of Hell (continued)

Filed under: Uncategorized — m759 @ 3:28 PM

Holy Saturday is, according to tradition, the day of 
the harrowing of Hell.

Notes:

The above passage on "Die Figuren der vier Modi
im Magischen Quadrat 
" should be read in the context of
a Log24 post from last year's Devil's Night (the night of
October 30-31).  The post, "Structure," indicates that, using
the transformations of the diamond theorem, the notorious
"magic" square of Albrecht Dürer may be transformed
into normal reading order.  That order is only one of
322,560 natural reading orders for any 4×4 array of
symbols. The above four "modi" describe another.

Monday, March 23, 2015

Gallucci’s Möbius Configuration

Filed under: Uncategorized — Tags: — m759 @ 12:05 PM

From H. S. M. Coxeter's 1950 paper
"Self-Dual Configurations and Regular Graphs," 
a 4×4 array and a more perspicuous rearrangement—

(Click image to enlarge.) 

The above rearrangement brings Coxeter's remarks into accord
with the webpage The Galois Tesseract.

Update of Thursday, March 26, 2015 —

For an explanation of Coxeter's Fig. 24, see Thursday's later
post titled "The Möbius Hypercube."

Wednesday, January 14, 2015

Serial Box

Filed under: Uncategorized — Tags: — m759 @ 1:20 PM

Enotes.com on Herman Wouk's 1985 novel Inside, Outside 

"The 'outside' of the title is the goyish world
into which David’s profession has drawn him;
the 'inside' is the warm life of his Russian-
Jewish family on which he, as narrator, reflects
in the course of the novel."

For a different sort of 'inside' life, see this morning's post
Gesamtkunstwerk , and Nathan Shields's Feb. 8, 2011,
tribute to a serial composer "In Memoriam, Milton Babbitt."
Some other context for Shields's musical remarks —

Doctor Faustus and Dürer Square.

For a more interesting contrast of inside with outside
that has nothing to do with ethnicity, see the Feb. 10,
2014, post Mystery Box III: Inside, Outside, about
the following box:

 .

Sunday, September 21, 2014

Sermon

Filed under: Uncategorized — m759 @ 11:00 AM

The previous post discussed the anatomy of the sum 9 + 6.

A different approach:  “A” and “The 6 spreads in A” below —

Tuesday, September 9, 2014

Smoke and Mirrors

Filed under: Uncategorized — Tags: , , — m759 @ 11:00 AM

This post is continued from a March 12, 2013, post titled
"Smoke and Mirrors" on art in Tromsø, Norway, and from
a June 22, 2014, post on the nineteenth-century 
mathematicians Rosenhain and Göpel.

The latter day was the day of death for 
mathematician Loren D. Olson, Harvard '64.

For some background on that June 22 post, see the tag 
Rosenhain and Göpel in this journal.

Some background on Olson, who taught at the
University of Tromsø, from the American Mathematical
Society yesterday:

Olson died not long after attending the 50th reunion of the
Harvard Class of 1964.

For another connection between that class (also my own) 
and Tromsø, see posts tagged "Elegantly Packaged."
This phrase was taken from today's (print) 
New York Times  review of a new play titled "Smoke."
The phrase refers here  to the following "package" for 
some mathematical objects that were named after 
Rosenhain and Göpel — a 4×4 array —

For the way these objects were packaged within the array
in 1905 by British mathematician R. W. H. T. Hudson, see
a page at finitegometry.org/sc. For the connection to the art 
in Tromsø mentioned above, see the diamond theorem.

Monday, August 4, 2014

A Wrinkle in Space

Filed under: Uncategorized — Tags: — m759 @ 10:30 AM

“There is  such a thing as a tesseract.” — Madeleine L’Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

Wednesday, May 21, 2014

The Tetrahedral Model of PG(3,2)

Filed under: Uncategorized — Tags: , — m759 @ 10:15 PM

The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).

The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.

Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square  model,
based on the 4×4 array I call the Galois tesseract  (at left below).

(Click to enlarge.)

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: Uncategorized — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis’s 35  4×6  1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this “by-hand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction,  not  by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.

* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Monday, February 10, 2014

Mystery Box III: Inside, Outside

Filed under: Uncategorized — Tags: , , , — m759 @ 2:28 PM

(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)

The Box

Inside the Box

Outside the Box

For the connection of the inside  notation to the outside  geometry,
see Desargues via Galois.

(For a related connection to curves  and surfaces  in the outside
geometry, see Hudson's classic Kummer's Quartic Surface  and
Rosenhain and Göpel Tetrads in PG(3,2).)

Saturday, January 18, 2014

The Triangle Relativity Problem

Filed under: Uncategorized — m759 @ 5:01 PM

A sequel to last night's post The 4×4 Relativity Problem —

IMAGE- Triangle Coordinatization

In other words, how should the triangle corresponding to
the above square be coordinatized ?

See also a post of July 8, 2012 — "Not Quite Obvious."

Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly  item published
in 1984.

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: Uncategorized — Tags: , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Thursday, January 16, 2014

Confession of a Sucker

Filed under: Uncategorized — m759 @ 12:00 PM

Today's 11 AM (ET) post was suggested by a New York Times
article, online yesterday, about art gallery owner Lisa Cooley.

A check of Cooley's website yields the image below,
related to Beckett's Molloy .

For the relevant passage from Molloy , click the following:

"I took advantage of being at the seaside
 to lay in a store of sucking-stones.
"

For posts on Molloy  in this journal, click Beckett + Molloy .

Cynthia Daignault, 2011:
The one I shall now describe, if I can…
Oil on linen, in 2 parts: 40 x 30 inches, 96 x 75 inches

Related art theory —

Geometry of the 4×4 Square 

Monday, October 28, 2013

Stella’s Goal

Filed under: Uncategorized — m759 @ 10:00 AM

The Whitney Museum of American Art has stated
that artist Frank Stella in 1959

"wanted to create work that was methodical,
intellectual and passionless."

Source: Whitney Museum, transcript of audio guide.

Related material:

A figure from this journal on July 13, 2003

and some properties of that figure.

Sunday, September 22, 2013

Incarnation, Part 2

Filed under: Uncategorized — Tags: , , — m759 @ 10:18 AM

From yesterday —

"…  a list of group theoretic invariants
and their geometric incarnation…"

David Lehavi on the Kummer 166 configuration in 2007

Related material —

IMAGE- 'This is not mathematics; this is theology.' - Paul Gordan

"The hint half guessed, the gift half understood, is Incarnation."

T. S. Eliot in Four Quartets

"This is not theology; this is mathematics."

— Steven H. Cullinane on  four quartets

To wit:


Click to enlarge.

Saturday, September 21, 2013

Geometric Incarnation

The  Kummer 166  configuration  is the configuration of sixteen
6-sets within a 4×4 square array of points in which each 6-set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.

See Configurations and Squares.

The Wikipedia article Kummer surface  uses a rather poetic
phrase* to describe the relationship of the 166 to a number
of other mathematical concepts — "geometric incarnation."

Geometric Incarnation in the Galois Tesseract

Related material from finitegeometry.org —

IMAGE- 4x4 Geometry: Rosenhain and Göpel Tetrads and the Kummer Configuration

* Apparently from David Lehavi on March 18, 2007, at Citizendium .

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags: , , , , — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Tuesday, July 16, 2013

Child Buyers

Filed under: Uncategorized — Tags: — m759 @ 10:00 PM

The title refers to a classic 1960 novel by John Hersey.

“How do you  get young people excited about space?”

— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)

Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society

“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”

This suggests a related image by Gauss, with a
remark on simplicity—

IMAGE- Complex Grid, by Gauss

Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).

Related material that does  feature the somewhat simpler 3×3 array
of squares, not  seen as part of an infinite array—

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

Click image for the original post.

For a purely mathematical view of the holy field, see Visualizing GL(2,p).

Wednesday, June 19, 2013

Ein Eck

Filed under: Uncategorized — Tags: , — m759 @ 9:29 PM

"Da hats ein Eck" —

"you've/she's (etc.) got problems there"

St. Galluskirche:

St. Gallus's Day, 2012:

Click image for a St. Gallus's Day post.

A related problem: 

Discuss the structure of the 4x4x4 "magic" cube
sent by Pierre de Fermat to Father Marin Mersenne
on April 1, 1640, in light of the above post.

Thursday, June 13, 2013

Gate

Filed under: Uncategorized — Tags: , — m759 @ 2:13 PM

"Eight is a Gate." — Mnemonic rhyme

Today's previous post, Window, showed a version
of the Chinese character for "field"—

This suggests a related image

The related image in turn suggests

Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.

Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.

Jan Krikke

As was the I Ching.  A related structure:

Tuesday, June 4, 2013

Cover Acts

Filed under: Uncategorized — m759 @ 11:00 AM

The Daily Princetonian  today:

IMAGE- 'How Jay White, a Neil Diamond cover act, duped Princeton'

A different cover act, discussed here  Saturday:

IMAGE- The diamond theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

See also, in this journal, the Galois tesseract and the Crosswicks Curse.

"There is  such a thing as a tesseract." — Crosswicks saying

Saturday, May 11, 2013

Core

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

"A core component in the construction
is a 3-dimensional vector space  over F."

—  Page 29 of "A twist in the M24 moonshine story," 
      by Anne Taormina and Katrin Wendland.
      (Submitted to the arXiv on 13 Mar 2013.)

Sunday, March 31, 2013

For Baker

Filed under: Uncategorized — m759 @ 8:00 PM

Baker, Principles of Geometry, Vol. IV  (1925), Title:

Baker, Principles of Geometry, Vol. IV  (1925), Frontispiece:

Baker's Vol. IV frontispiece shows "The Figure of fifteen lines 
and fifteen points, in space of four dimensions."

Another such figure in a vector space of four dimensions
over the two-element Galois field  GF(2):

(Some background grid parts were blanked by an image resizing process.)

Here the "lines" are actually planes  in the vector 4-space over GF(2),
but as planes through the origin  in that space, they are projective  lines .

For some background, see today's previous post and Inscapes.

Update of 9:15 PM March 31—

The following figure relates the above finite-geometry
inscape  incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.

Saturday, March 16, 2013

The Crosswicks Curse

Filed under: Uncategorized — Tags: — m759 @ 4:00 PM

Continues.

From the prologue to the new Joyce Carol Oates
novel Accursed

"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.

1905!—the very year of the Curse."

Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract  of Madeleine L'Engle.

The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —

"There is  such a thing as a tesseract."

A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also 
be viewed as a 4×4 array (with opposite edges
identified).

Meanwhile, back in 1905

For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).

See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.

Sunday, March 10, 2013

Galois Space

Filed under: Uncategorized — m759 @ 5:30 PM

(Continued)

The 16-point affine Galois space:

Further properties of this space:

In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.

Some closely related material:

  • Wolfgang Kühnel,
    "Minimal Triangulations of Kummer Varieties,"
    Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).

    For the first two pages, click here.

  • Jonathan Spreer and Wolfgang Kühnel,
    "Combinatorial Properties of the 3 Surface:
    Simplicial Blowups and Slicings,"
    preprint, 26 pages. (2009/10) (pdf).
    (Published in Experimental Math. 20,
    issue 2, 201–216 (2011).)

Thursday, March 7, 2013

Ten Years After

Filed under: Uncategorized — m759 @ 8:00 AM

Rock guitarist Alvin Lee, a founder of
the band Ten Years After , died
on March 6, 2013 (Michelangelo's
birthday). In his memory, a figure
from a post Ten Years Before —

Plato's reported motto for his Academy:
"Let no one ignorant of geometry enter."

For visual commentary by an artist ignorant
of geometry, see a work by Sol LeWitt.

For verbal commentary by an art critic  ignorant
of geometry, see a review of LeWitt by
Robert Hughes—

"A Beauty Really Bare" (TIME, Feb. 6, 2001).

See also Ten Years Group and Four Gods.

Tuesday, February 19, 2013

Configurations

Filed under: Uncategorized — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: Uncategorized — Tags: , — m759 @ 10:18 AM

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Wednesday, December 5, 2012

Arte Programmata*

Filed under: Uncategorized — m759 @ 9:30 PM

The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced 
according to strict rules, and were in this sense 
"programmed," but were drawn by hand.

Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not  in
Diamond Theory).

See Isaac Gierard's program at GitHub

https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde

As the suffix indicates, this program is in the
Processing Development Environment language.

It produces the following sketch:

IMAGE- Sketch programmed by Isaac Gierard to mimic some of the images of 'Diamond Theory' (© 1976 by Steven H. Cullinane).

The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the 
overall effect of the sketch is pleasing.

For some background for the program, see The ReCode Project.

It is good to learn that the Processing language is well-adapted to making the 
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory  in "Diamond Theory."

For some related remarks, see Theory (Sept. 30, 2012).

* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."

Tuesday, October 16, 2012

Cube Review

Filed under: Uncategorized — Tags: — m759 @ 3:00 PM

Last Wednesday's 11 PM post mentioned the
adjacency-isomorphism relating the 4-dimensional 
hypercube over the 2-element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.

A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).

In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6-dimensional hypercube over GF(2) 
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.

The above cube may be used to illustrate some properties
of the 64-point Galois 6-space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.

See

Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."

Sunday, July 29, 2012

The Galois Tesseract

Filed under: Uncategorized — Tags: — m759 @ 11:00 PM

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from  Chapter 11 of 
    Sphere Packings, Lattices and Groups , by John Horton
    Conway and N. J. A. Sloane, first published by Springer
    in 1988.)

The fact that any  4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Thursday, April 5, 2012

Meanwhile, back in 1950…

Filed under: Uncategorized — Tags: — m759 @ 10:30 AM

See also Solomon's Cube.

Tuesday, February 14, 2012

The Ninth Configuration

Filed under: Uncategorized — m759 @ 2:01 PM

The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title, 
that of a novel by the author of The Exorcist .

The Ninth Configuration 

The ninth* in a list of configurations—

"There is a (2d-1)d  configuration
  known as the Cox configuration."

MathWorld article on "Configuration"

For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6  ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc.  Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it 
relates to  γ6  , another form of the 4×4×4 Galois cube.

See also Solomon's Cube.

* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
  and then the ninth  would be a 153 and some related material would be Inscapes.

Wednesday, January 25, 2012

A Larger City

Filed under: Uncategorized — m759 @ 10:09 AM
ANY CHARACTER HERE
BOOKS OF THE TIMES

HOW IT ALL BEGAN
By Penelope Lively
229 pages. Viking. $26.95.

Review by Michiko Kakutani
in The New York Times ,
online Jan. 23, 2012

As a historian, Henry acknowledges that he has “a soft spot for what is known as the Cleopatra’s nose theory of history— the proposal that had the nose of Cleopatra been an inch longer, the fortunes of Rome would have been different.” It’s a bit of a reductio ad absurdum, he admits, but nonetheless “a reference to random causality that makes a lot of sense when we think about the erratic sequence of events that we call history.”

What Ms. Lively has done in this captivating volume is to use all her copious storytelling gifts to show how a similar kind of random causality rules individual lives, how one unlucky event can set off unexpected chain reactions, how the so-called butterfly effect— whereby the flapping of a tiny butterfly’s wings can supposedly lead to a huge storm elsewhere in the world— ripples through the ebb and flow of daily life.

Rhetorical question—

"Why walk when you can fly?"
— Mary Chapin Carpenter

Rhetorical answer—

Two excerpts from a webpage on random walks

A drunk man will find his way home,
but a drunk bird may get lost forever.

— Shizuo Kakutani

Now we move to a larger city

IMAGE- 'Now we move to a larger city...' illustrated by 4x4 grid with dots signifying extension of the grid

Tuesday, January 10, 2012

Defining Form

Filed under: Uncategorized — Tags: , — m759 @ 9:00 AM

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

Further detail, with a comparison to Dürer's magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011

 

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

Saturday, August 6, 2011

Correspondences

Filed under: Uncategorized — Tags: , — m759 @ 2:00 PM

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.

 

For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

Thursday, August 4, 2011

Midnight in Oslo

Filed under: Uncategorized — m759 @ 6:00 PM

For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth

(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)

http://www.log24.com/log/pix11B/110804-Pesic-PlatosDiamond.jpg

Plato's Diamond

The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—

Paradigms

"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate….

… Both in the Republic  and in the Sophist  there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno  that the square on the diagonal is double the original square in area."

– William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985

Plato's paradigm in the Meno

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Aspects of the paradigm change—

Monochrome figures to
   colored figures

Areas to
   transformations

Continuous transformations to
   non-continuous transformations

Euclidean geometry to
   finite geometry

Euclidean quantities to
   finite fields

The 24 patterns resulting from the paradigm change—

http://www.log24.com/log/pix11B/110805-The24.jpg

Each pattern has some ordinary or color-interchange symmetry.

This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem.

Related material: Plato's Diamond by Oslo artist Josefine Lyche.

Plato’s Ghost  evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”

— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics  (by Jeremy Gray, September 2008)

"Remember me to her."

— Closing words of the Algis Budrys novel Rogue Moon .

Background— Some posts in this journal related to Abel or to random thoughts from his birthday.

Friday, March 18, 2011

Defining Configurations*

Filed under: Uncategorized — m759 @ 7:00 PM

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

http://www.log24.com/log/pix11/110320-MoebiusKantorConfig500w.jpg

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

http://www.log24.com/log/pix11/110322-DolgachevIntro.gif

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Thursday, February 17, 2011

The Form, the Pattern

Filed under: Uncategorized — m759 @ 1:00 AM

"…  Only by the form, the pattern,     
Can words or music reach
The stillness…."

— T. S. Eliot,
Four Quartets

For further details, see Time Fold.

Saturday, January 22, 2011

High School Squares*

Filed under: Uncategorized — Tags: — m759 @ 1:20 AM

The following is from the weblog of a high school mathematics teacher—

http://www.log24.com/log/pix11/110121-LatinSquares4x4.jpg

This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory

http://www.log24.com/log/pix11/110122-DiamondTheoryCover.jpg

Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.

For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem.  The result is 24 structural diagrams—

    Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares

Some of the squares are structurally congruent under the group of 8 symmetries of the square.

This can be seen in the following regrouping—

   Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares, with Congruent Squares Adjacent

      (Image corrected on Jan. 25, 2011– "seven" replaced "eight.")

* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.

Saturday, June 19, 2010

Imago Creationis

Filed under: Uncategorized — Tags: , , , — m759 @ 6:00 PM

Image-- The Four-Diamond Tesseract

In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.

Four-Part Tesseract Divisions

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

The above figure shows how four-part partitions
of the 16 vertices  of a tesseract in an infinite
Euclidean  space are related to four-part partitions
of the 16 points  in a finite Galois  space

Euclidean spaces versus Galois spaces
in a larger context—

 


Infinite versus Finite

The central aim of Western religion —

"Each of us has something to offer the Creator...
the bridging of
                 masculine and feminine,
                      life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist  (1998)

The central aim of Western philosophy —

              Dualities of Pythagoras
              as reconstructed by Aristotle:
                 Limited     Unlimited
                     Odd     Even
                    Male     Female
                   Light      Dark
                Straight    Curved
                  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres  (1993)

Another picture related to philosophy and religion—

Jung's Four-Diamond Figure from Aion

http://www.log24.com/log/pix10A/100615-JungImago.gif

This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—

Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—

 

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science…  reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).

Notes:

  Paul Valéry, Oeuvres  (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—

… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.

* That is, uses multi-dimensional symbols beyond our grasp.

Related material:

Imago Creationis

A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).

http://www.log24.com/log/pix10A/100618-LeibnizMedaille.jpg

Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—

Frame of Reference

http://www.log24.com/log/pix10A/100619-ReferenceFrame.gif

The Diamond Theorem

http://www.log24.com/log/pix10A/100619-Dtheorem.gif

Some context by a British mathematician —

http://www.log24.com/log/pix10A/100619-Cameron.gif

Imago

by Wallace Stevens

Who can pick up the weight of Britain, 
Who can move the German load 
Or say to the French here is France again? 
Imago. Imago. Imago. 

It is nothing, no great thing, nor man 
Of ten brilliancies of battered gold 
And fortunate stone. It moves its parade 
Of motions in the mind and heart, 

A gorgeous fortitude. Medium man 
In February hears the imagination's hymns 
And sees its images, its motions 
And multitudes of motions 

And feels the imagination's mercies, 
In a season more than sun and south wind, 
Something returning from a deeper quarter, 
A glacier running through delirium, 

Making this heavy rock a place, 
Which is not of our lives composed . . . 
Lightly and lightly, O my land, 
Move lightly through the air again.

Wednesday, April 28, 2010

Eightfold Geometry

Filed under: Uncategorized — Tags: , — m759 @ 11:07 AM

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

Friday, November 6, 2009

Cowboy Song

Filed under: Uncategorized — m759 @ 9:00 AM

For a girl I saw once in USA TodayBlank spacer gif in 1995:

Part I:

Top of the online front page, NY Times this morning–

NY Times ad for film 'An Education'

Blank spacer gif

Blank spacer gif

Part II:

Sesame Street characters search for a lost image

Part III:

Adapted song lyrics from “Colorado Trail“–

Kendra was a pretty girl
God Almighty knows

Part IV:

Very like another girl 30 years earlier

Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: Uncategorized — Tags: — m759 @ 10:31 PM
For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

 
Background —

 
SAT
 
Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube



Part III:

From August 25th

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.
"

Wednesday, August 12, 2009

Wednesday August 12, 2009

Filed under: Uncategorized — m759 @ 12:00 PM
Exegesis

Text:

The Shining,
1977, page 162:

“A new headline, this one
 dated April 10….”

“The item on the next page
 was a mere squib, dated
 four months later….”

Exegesis:

April 10— Good Friday– See
The Paradise of Childhood.

Four months later– Aug. 10

“When he thought of the old man
  he could see him suddenly
  in a field in the spring,
  trying to move a gray boulder.”

Monday, June 22, 2009

Monday June 22, 2009

Filed under: Uncategorized — Tags: — m759 @ 4:00 AM

Text

Today’s birthday:
Kris Kristofferson

Kris Kristofferson in 'Heaven's Gate'

Heaven’s Gate

One year ago today
George Carlin died.

Online Etymology Dictionary

1369, “wording of anything written,” from O.Fr. texte, O.N.Fr. tixte (12c.), from M.L. textus “the Scriptures, text, treatise,” in L.L. “written account, content, characters used in a document,” from L. textus “style or texture of a work,” lit. “thing woven,” from pp. stem of texere “to weave,” from PIE base *tek- “make” (see texture).

“An ancient metaphor: thought is a thread, and the raconteur is a spinner of yarns– but the true storyteller, the poet, is a weaver. The scribes made this old and audible abstraction into a new and visible fact. After long practice, their work took on such an even, flexible texture that they called the written page a textus, which means cloth.” [Robert Bringhurst, “The Elements of Typographic Style”]

Text-book is from 1779.

The 4x4 square grid

“Discuss the geometry
underlying the above picture.”
Log24, June 11, 2009

Thursday, June 11, 2009

Thursday June 11, 2009

Filed under: Uncategorized — Tags: — m759 @ 7:11 PM

Geometry for Jews

(continued from Michelangelo's birthday, 2003)

The 4x4 square grid

"Discuss the geometry underlying the above picture."

Log24, March 6, 2003

Abstraction and the Holocaust  (Mark Godfrey, Yale University Press, 2007) describes one approach to such a discussion: Bochner "took a photograph of a new arrangement of blocks, cut it up, reprinted it as a negative, and arranged the four corners in every possible configuration using the serial principles of rotation and reversal to make Sixteen Isomorphs (Negative) of 1967, which he later illustrated alongside works by Donald Judd, Sol LeWitt and Eva Hesse in his Artforum article 'The Serial Attitude.' [December 1967, pp. 28-33]" Bochner's picture of "every possible configuration"–

Bochner's 'Sixteen Isomorphs' (or: 'Eight Isomorphs Short of a Load')

Compare with the 24 figures in Frame Tales
(Log24, Nov. 10, 2008) and in Theme and Variations.

Sunday, May 17, 2009

Sunday May 17, 2009

Filed under: Uncategorized — m759 @ 7:59 AM
Design Theory

Laura A. Smit, Calvin College, “Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams“–

“My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life.”

Hans Ludwig de Vries, “On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16),” Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–

“The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]….”

1. Biggs N.L., “T. P. Kirkman, Mathematician,” Bulletin of the London Mathematical Society 13, 97–120 (1981).

4. Frobenius G., “Über die Charaktere der mehrfach transitiven Gruppen,” Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).

Olli Pottonen, “Classification of Steiner Quadruple Systems” (Master’s thesis, Helsinki, 2005)–

“The concept of group actions is very useful in the study of isomorphisms of combinatorial structures.”

Olli Pottonen,  'Classification of Steiner Quadruple Systems'

“Simplify, simplify.”
Thoreau

Beauty is bound up
with symmetry.”
Weyl

Sixteen points in a 4x4 array

Pottonen’s thesis is
 dated Nov. 16, 2005.

For some remarks on
images and theology,
see Log24 on that date.

Click on the above image
 for some further details.

Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM
 
Hollywood Nihilism
Meets
Pantheistic Solipsism

Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
 to hear about our religion
… that we made up."

Tina Fey and Steve Martin at the 2009 Oscars

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:

… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer

 A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.


Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."

As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.

Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.

Heinlein:

"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
    I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."

Stevens:

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:

B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":

"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'….  Its subject is its speaker's sense of nothingness and his need to be cured of it."

This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism.

More positively…

Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space
(or the corresponding
5-dimensional projective space)

The 4x4x4 cube

over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."

Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.

Cara:

Philippe Cara on the Klein correspondence
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: Uncategorized — Tags: — m759 @ 8:30 AM
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."

Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Another possibility —
 

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
 

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: Uncategorized — Tags: — m759 @ 3:00 PM
Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

Lucia Joyce

 

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.

 

Log 24, Sept. 3, 2003:
 
Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year's entry on this date: 

 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

 

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and


FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles
.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.
 

Thursday, July 31, 2008

Thursday July 31, 2008

Filed under: Uncategorized — m759 @ 12:00 PM
Symmetry in Review

“Put bluntly, who is kidding whom?”

Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.

Good question.

Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —

 
Pavarotti takes a bow
Related material:

1. The performance of a work by
Richard Strauss,
Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008

2. Headline of a music review
in today’s New York Times:

Welcoming a Fresh Season of
Transformation and Death

3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:

Cover of 'Twelve Sporadic Groups'

4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”

5. Symmetry from Plato to
the Four-Color Conjecture

6. Geometry of the 4×4 Square

7. Yesterday’s entry,
Theories of Everything

Coda:

There is such a thing

Tesseract
     as a tesseract.

— Madeleine L’Engle

Cover of The New Yorker, April 12, 2004-- Roz Chast, Easter Eggs

For a profile of
L’Engle, click on
the Easter eggs.

Saturday, July 19, 2008

Saturday July 19, 2008

Filed under: Uncategorized — m759 @ 2:00 PM
Hard Core

(continued from yesterday)

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week’s New Yorker:

A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent ‘object’ in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure.”

Hermann Weyl on the hard core of objectivity:

“Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind.” (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)


Steven H. Cullinane on the symmetries of a 4×4 array of points:

A Structure-Endowed Entity

“A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way.”

— Hermann Weyl in Symmetry

Let us apply Weyl’s lesson to the following “structure-endowed entity.”

4x4 array of dots

What is the order of the resulting group of automorphisms?

The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
  called a “colorful tale”–

The Diamond 16 Puzzle

The Diamond 16 Puzzle

This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.

“You can make 322,560
pairs of patterns. Each
 pair pictures a different
symmetry of the underlying
16-point space.”

— Steven H. Cullinane,
July 17, 2008

For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings
.

Wednesday, June 18, 2008

Wednesday June 18, 2008

Filed under: Uncategorized — m759 @ 3:00 PM
CHANGE
 FEW CAN BELIEVE IN

What I Loved, a novel by Siri Hustvedt (New York, Macmillan, 2003), contains a paragraph on the marriage of a fictional artist named Wechsler–

Page 67 —

“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”

For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.

Related material:

Wechsler cubes

(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)

Wechsler blocks for psychological testing

These cubes are used to
make 3×3 patterns for
psychological testing.

Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–

Don Park at docuverse.com, Jan. 19, 2007:

“How to draw an Identicon

Designs from a web page on Identicons

A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners.

Positions and Rotations

For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.”

    

From a weblog by Scott Sherrill-Mix:

“… Don Park came up with the original idea for representing users with geometric shapes….”

Claire | 20-Dec-07 at 9:35 pm | Permalink

“This reminds me of a flash demo by Jarred Tarbell
http://www.levitated.net/daily/lev9block.html

ScottS-M | 21-Dec-07 at 12:59 am | Permalink

    

Jared Tarbell at levitated.net, May 15, 2002:

“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.

Designs from a web page by Jared Tarbell
Figure A. Four 9 block patterns,
arbitrarily assembled, show the
grid composition of the block.

Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.

The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.

Designs from a web page by Jared Tarbell

Figure B. The 16 possible shapes allowed
for each grid space. The 4 shapes allowed
in the center have bold numbers.”

   
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen.  See Block Designs in Art and Mathematics.

(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)

Monday, June 16, 2008

Monday June 16, 2008

Filed under: Uncategorized — m759 @ 9:00 PM
Bloomsday for Nash:
The Revelation Game

(American Mathematical Society Feb. 2008
review of Steven Brams’s Superior Beings:
If They Exist, How Would We Know?)

(pdf, 15 megabytes)

“Brams does not attempt to prove or disprove God. He uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….

In the ‘Revelation Game,’ for example, the Person (P) has two options:
1) P can believe in SB’s existence
2) P can not believe in SB’s existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself

Each player also has a primary and secondary goal. For the Person, the primary goal is to have his belief (or non-belief) confirmed by evidence (or lack thereof). The secondary goal is to ‘prefer to believe in SB’s existence.’ For the Supreme Being, the primary goal is to have P believe in His existence, while the secondary goal is to not reveal Himself. These goals allow us to rank all the outcomes for each player from best (4) to worst (1). We end up with a matrix as follows (the first number in the parentheses represents the SB’s ranking for that box; the second number represents P’s ranking):

Revelation Game payoff matrix

The question we must answer is: what is the Nash equilibrium in this case?”

Analogously:

Lotteries on
Bloomsday,
June 16,
2008
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
418

The Exorcist

No belief,
no revelation

064

4x4x4 cube summarizing geometry of the I Ching

Revelation
without belief

Evening
(Belief)
709

Human Conflict Number Five album by The 10,000 Maniacs

Belief without
revelation

198

(A Cheap
Epiphany)

Black disc from end of Ch. 17 of Ulysses

Belief and
revelation

The holy image

Black disc from end of Ch. 17 of Ulysses

denoting belief and revelation
may be interpreted as
a black hole or as a
symbol by James Joyce:

When?

Going to dark bed there was a square round Sinbad the Sailor roc’s auk’s egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.

Where?

Black disc from end of Ch. 17 in Ulysses

Ulysses, conclusion of Chapter 17

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: Uncategorized — Tags: , , , — m759 @ 8:00 AM
MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Tuesday, February 26, 2008

Tuesday February 26, 2008

Filed under: Uncategorized — m759 @ 8:00 PM

Eight is a Gate (continued)

Tom Stoppard, Jumpers:
“Heaven, how can I believe in Heaven?” she sings at the finale. “Just a lying rhyme for seven!”
“To begin at the beginning: Is God?…” [very long pause]

From “Space,” by Salomon Bochner

Makom. Our term “space” derives from the Latin, and is thus relatively late. The nearest to it among earlier terms in the West are the Hebrew makom and the Greek topos (τόπος). The literal meaning of these two terms is the same, namely “place,” and even the scope of connotations is virtually the same (Theol. Wörterbuch…, 1966). Either term denotes: area, region, province; the room occupied by a person or an object, or by a community of persons or arrangements of objects. But by first occurrences in extant sources, makom seems to be the earlier term and concept. Apparently, topos is attested for the first time in the early fifth century B.C., in plays of Aeschylus and fragments of Parmenides, and its meaning there is a rather literal one, even in Parmenides. Now, the Hebrew book Job is more or less contemporary with these Greek sources, but in chapter 16:18 occurs in a rather figurative sense:

O earth, cover not thou my blood, and let my cry have no place (makom).

Late antiquity was already debating whether this makom is meant to be a “hiding place” or a “resting place” (Dhorme, p. 217), and there have even been suggestions that it might have the logical meaning of “occasion,” “opportunity.” Long before it appears in Job, makom occurs in the very first chapter of Genesis, in:

And God said, Let the waters under the heaven be gathered together unto one place (makom) and the dry land appear, and it was so (Genesis 1:9).

This biblical account is more or less contemporary with Hesiod’s Theogony, but the makom of the biblical account has a cosmological nuance as no corresponding term in Hesiod. Elsewhere in Genesis (for instance, 22:3; 28:11; 28:19), makom usually refers to a place of cultic significance, where God might be worshipped, eventually if not immediately. Similarly, in the Arabic language, which however has been a written one only since the seventh century A.D., the term makām designates the place of a saint or of a holy tomb (Jammer, p. 27). In post-biblical Hebrew and Aramaic, in the first centuries A.D., makom became a theological synonym for God, as expressed in the Talmudic sayings: “He is the place of His world,” and “His world is His place” (Jammer, p. 26). Pagan Hellenism of the same era did not identify God with place, not noticeably so; except that the One (τὸ ἕν) of Plotinus (third century A.D.) was conceived as something very comprehensive (see for instance J. M. Rist, pp. 21-27) and thus may have been intended to subsume God and place, among other concepts. In the much older One of Parmenides (early fifth century B.C.), from which the Plotinian One ultimately descended, the theological aspect was only faintly discernible. But the spatial aspect was clearly visible, even emphasized (Diels, frag. 8, lines 42-49).

BIBLIOGRAPHY

Paul Dhorme, Le livre de Job (Paris, 1926).

H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed. (Berlin, 1938).

Max Jammer, Concepts of Space (Cambridge, Mass., 1954).

J. M. Rist, Plotinus: The Road to Reality (Cambridge, 1967).

Theologisches Wörterbuch zum Neuen Testament (1966), 8, 187-208, esp. 199ff.

— SALOMON BOCHNER

Related material: In the previous entry — “Father Clark seizes at one place (page eight)
upon the fact that….”

Father Clark’s reviewer (previous entry) called a remark by Father Clark “far fetched.”
This use of “place” by the reviewer is, one might say, “near fetched.”

Thursday, October 25, 2007

Thursday October 25, 2007

Filed under: Uncategorized — Tags: — m759 @ 9:19 AM

Something Anonymous

From this date–
Picasso's birthday–
five years ago:
 
"A work of art has an author
and yet,
when it is perfect,
it has something
which is
essentially anonymous about it."

Simone Weil, Gravity and Grace   

 
Michelangelo's birthday, 2003

4x4 square grid

Yesterday:

The color-analogy figures of Descartes

Nineteenth-century quilt design:

Tents of Armageddon quilt design

Related material:

Battlefield Geometry
 

Wednesday, September 12, 2007

Wednesday September 12, 2007

Filed under: Uncategorized — m759 @ 5:01 PM
Vector Logic

Geometry for Jews
(March 2003)
discussed the
following figure:

The 4x4 square

Some properties of
this figure were also
discussed last March
in my note
The Geometry of Logic.

I learned yesterday from Jonathan Westphal, a professor of philosophy at Idaho State University, that he and a colleague, Jim Hardy, have devised another geometric approach to logic: a system of arrow diagrams that illustrate classical propositional logic. The diagrams resemble those used to illustrate Euclidean vector spaces, and Westphal and Hardy call their approach “a vector system,” although it does not involve what a mathematician would regard as a vector space.
 
Westphal and Hardy, logic diagram with arrows
 
Journal of Logic and Computation
15(5) (October, 2005), pp. 751-765.
Related material:
 
(2) the quilt pattern
below (click for
the source) —
 
Quilt pattern Tents of Armageddon
 
and
(3) yesterday’s entry
 
“Christ! What are
patterns for?”
 

Monday, July 23, 2007

Monday July 23, 2007

Filed under: Uncategorized — Tags: , , , — m759 @ 8:00 AM
 
Daniel Radcliffe
is 18 today.
 
Daniel Radcliffe as Harry Potter
 

Greetings.

“The greatest sorcerer (writes Novalis memorably)
would be the one who bewitched himself to the point of
taking his own phantasmagorias for autonomous apparitions.
Would not this be true of us?”

Jorge Luis Borges, “Avatars of the Tortoise”

El mayor hechicero (escribe memorablemente Novalis)
sería el que se hechizara hasta el punto de
tomar sus propias fantasmagorías por apariciones autónomas.
¿No sería este nuestro caso?”

Jorge Luis Borges, “Los Avatares de la Tortuga

Autonomous Apparition
 
 

At Midsummer Noon:

 
“In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew from
a brief description in Waite’s
The Holy Kabbalah (1929) of
a supernatural cubic stone
on which was inscribed
‘the Divine Name.’”
 
The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.
 
Related material:
 
It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure

 

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit
,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

 
See also
 
as well as
Hofstadter on
his magnum opus:
 
“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”
 
Goedel Escher Bach cover

Hofstadter’s cover.

 
Here are three patterns,
“shadows” of a sort,
derived from a different
“central object”:
 
Faces of Solomon's Cube, related to Escher's 'Verbum'

Click on image for details.

Sunday, June 24, 2007

Sunday June 24, 2007

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM
Raiders of
the Lost Stone

(Continued from June 23)

Scott McLaren on
Charles Williams:
 
"In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew
from a brief description in Waite's
The Holy Kabbalah (1929)
of a supernatural cubic stone
on which was inscribed
'the Divine Name.'"

The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.

Related material:

The image “http://www.log24.com/log/pix07/070624-Cube.gif” cannot be displayed, because it contains errors.

Solomon's Cube,

Geometry of the 4x4x4 Cube,

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Friday, November 24, 2006

Friday November 24, 2006

Filed under: Uncategorized — m759 @ 1:06 PM
Galois’s Window:

Geometry
from Point
to Hyperspace


by Steven H. Cullinane

  Euclid is “the most famous
geometer ever known
and for good reason:
  for millennia it has been
his window
  that people first look through
when they view geometry.”

  Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace
,
by Leonard Mlodinow

“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
  concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”

— Paul Halmos in
I Want To Be a Mathematician

Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.

Each of these spaces
has infinitely many points.

Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:

+ 0 1
0 0 1
1 1 0
* 0 1
0 0 0
1 0 1
We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:
Galois affine spaces

From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call

Galois's Window

Galois’s Window.

The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.

For more on these cases, see

The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.

(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)

These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.

The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
  Elements of Finite Geometry.

Friday, May 12, 2006

Friday May 12, 2006

Filed under: Uncategorized — Tags: — m759 @ 3:00 AM
Tesseract

"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)

My reply–

Part I:

The image “http://www.log24.com/log/pix06A/WrinkleInTime1A.jpg” cannot be displayed, because it contains errors.

A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)

Part II:

Diamond Theory in 1937
and
Geometry of the 4×4 Square

Part III:

Catholic Schools Sermon

Conclusion:
 

"Wells and trees were dedicated to saints.  But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden.  Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Related material:

A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time.  For a related reflection on truth, stories, and values, see Saint's Day.  For a wider context, see the Log24 entries of February 1-15, 2003 and February 1-15, 2006.
 

Sunday, March 26, 2006

Sunday March 26, 2006

Filed under: Uncategorized — m759 @ 2:02 PM
‘Nauts

(continued from
Life of the Party, March 24)

Exhibit A —

From (presumably) a Princeton student
(see Activity, March 24):

The image “http://www.log24.com/log/pix06/060324-Activity.jpg” cannot be displayed, because it contains errors.

Exhibit B —

From today’s Sunday comics:

The image “http://www.log24.com/log/pix06/060326-Blondie2.gif” cannot be displayed, because it contains errors.

Exhibit C —

From a Smith student with the
same name as the Princeton student
(i.e., Dagwood’s “Twisterooni” twin):

The image “http://www.log24.com/log/pix06/060326-Smith.jpg” cannot be displayed, because it contains errors.

Related illustrations
(“Visual Stimuli“) from
the Smith student’s game —

The image “http://www.log24.com/log/pix06/060326-Psychonauts1.jpg” cannot be displayed, because it contains errors.

Literary Exercise:

Continuing the Smith student’s
Psychonauts theme,
compare and contrast
two novels dealing with
similar topics:

A Wrinkle in Time,
by the Christian author
Madeleine L’Engle,
and
Psychoshop,
by the secular authors
Alfred Bester and
Roger Zelazny.

Presumably the Princeton student
would prefer the Christian fantasy,
the Smith student the secular.

Those who prefer reality to fantasy —
not as numerous as one might think —
may examine what both 4×4 arrays
illustrated above have in common:
their structure.

Both Princeton and Smith might benefit
from an application of Plato’s dictum:

The image “http://www.log24.com/theory/images/motto2.jpg” cannot be displayed, because it contains errors.

Sunday, January 15, 2006

Sunday January 15, 2006

Filed under: Uncategorized — Tags: , — m759 @ 7:59 AM

Inscape

My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S6] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S6 contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S6 by a subgroup acting triply transitively on the 6 letters as PGL2(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

The image “http://www.log24.com/theory/images/Cremona-Richmond.gif” cannot be displayed, because it contains errors.

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

The image “http://www.log24.com/theory/images/Inscape.gif” cannot be displayed, because it contains errors.

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

Monday, January 9, 2006

Monday January 9, 2006

Filed under: Uncategorized — m759 @ 5:01 AM
Cornerstone

“In 1782, the Swiss mathematician Leonhard Euler posed a problem whose mathematical content at the time seemed about as much as that of a parlor puzzle. 178 years passed before a complete solution was found; not only did it inspire a wealth of mathematics, it is now a cornerstone of modern design theory.”

— Dean G. Hoffman, Auburn U.,
    July 2001 Rutgers talk

Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):

The image “http://www.log24.com/log/pix06/060109-TarryProof.gif” cannot be displayed, because it contains errors.

Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).

Saturday, June 4, 2005

Saturday June 4, 2005

Filed under: Uncategorized — m759 @ 7:00 PM
  Drama of the Diagonal
  
   The 4×4 Square:
  French Perspectives

Earendil_Silmarils:
The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.
  
   Les Anamorphoses:
 
   The image “http://www.log24.com/log/pix05A/050604-DesertSquare.jpg” cannot be displayed, because it contains errors.
 
  “Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d’horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4).”
   _______________________________
  
  Serge Mehl,
   Perspective &
  Géométrie Projective:
  
   “… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s’opposait à la géométrie
   euclidienne: élémentaire
  
  La géométrie projective, certes supérieure
   car assez ardue, permet d’établir
   de façon élégante des résultats de
   la géométrie élémentaire.”
  
  Similarly…
  
  Finite projective geometry
  (in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
  quilt-pattern symmetry
  and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.
  
  Other Related Material…
  
   from algebra rather than
   geometry, and from a German
   rather than from the French:  

This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

 Weyl also says that the profound branch
of mathematics known as Galois theory

   “… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
    finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory.”
  — Weyl, Symmetry,
   Princeton U. Press, 1952
  
   Metaphor and Algebra…  

“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.” 

   — attributed, in varying forms, to
   Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see  

  “Symmetry invariance
  in a diamond ring,”

  A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

  
More on Max Black…

“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”

— Paul Thompson, University College, Oxford,
    The Nature and Role of Intuition
     in Mathematical Epistemology

  A New Slant…  

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known.  The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.”  The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

Friday, May 27, 2005

Friday May 27, 2005

Filed under: Uncategorized — m759 @ 12:25 PM
Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares
(ps),
a discussion of a 4×4 array
of square linear designs.
  Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle:  the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16.  The first group is the most obvious.  The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
  Being and Nothingness,
  Philosophical Library, 1956
  [reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
 

  The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1. on Page 51:
“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being.  Man is ‘a being of distances.'”
2. on Page 154:
“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities.  Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3. on Page 155:
“… end of the desires.  But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the  glass-in-the-midst-of-the-world.”
4. on Page 158:
“…  it is in time that my possibilities appear on the horizon of the world which they make mine.  If, then, human reality is itself apprehended as temporal….”
5. on Page 180:
“… else time is an illusion and chronology disguises a strictly logical order of  deducibility.  If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6. on Page 186:
“…  It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7. on Page 332:
“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon.  It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8. on Page 359:
“… eyes as objects which manifest the look.  The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9. on Page 392:
“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.  on Page 502:
“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”
11.  on Page 506:
“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.  on Page 661:
“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.  on Page 754:
“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself.  The pen and the pipe, the clothing, the desk, the house– are myself.  The totality of my possessions reflects the totality of my being.  I am what I have.  It is I myself which I touch in this cup, in this trinket.  This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

The image “http://www.log24.com/log/pix05/050527-CipraLogo.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050527-CIPRAview.jpg” cannot be displayed, because it contains errors.
 
From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

Friday, May 6, 2005

Friday May 6, 2005

Filed under: Uncategorized — Tags: — m759 @ 7:28 PM

Fugues

"To improvise an eight-part fugue
is really beyond human capability."

— Douglas R. Hofstadter,
Gödel, Escher, Bach

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Order of a projective
 automorphism group:
168

"There are possibilities of
contrapuntal arrangement
of subject-matter."

— T. S. Eliot, quoted in
Origins of Form in Four Quartets.

The image “http://www.log24.com/theory/images/Grid4x4A.gif” cannot be displayed, because it contains errors.

Order of a projective
 automorphism group:
20,160

Monday, January 24, 2005

Monday January 24, 2005

Filed under: Uncategorized — Tags: — m759 @ 2:45 PM

Old School Tie

From a review of A Beautiful Mind:

"We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "

The image “http://www.log24.com/log/pix05/050124-Tie.gif” cannot be displayed, because it contains errors.

"Three readings of diamond and box
have been extremely influential."

Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke

"Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections….

… diamonds and boxes are upper and lower adjoints of Galois connections…."

— "Modal Kleene Algebra
and Applications: A Survey"
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
 
See also
Galois Correspondence

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

and Log24.net, May 20, 2004:

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

Saturday, January 1, 2005

Saturday January 1, 2005

Filed under: Uncategorized — m759 @ 8:08 AM

Metamorphosis

This illustration was added yesterday
to Geometry of the 4×4 Square.

Friday, November 19, 2004

Friday November 19, 2004

Filed under: Uncategorized — m759 @ 11:00 PM

From Tate to Plato

In honor of Allen Tate‘s birthday (today)
and of the MoMA re-opening (tomorrow)

“For Allen Tate the concept of tension was the most useful formal tool at the critic’s disposal, as irony and paradox were for Brooks. The principle of tension sustains the whole structure of meaning, and, as Tate declares in Tension in Poetry (1938), he derives it from lopping the prefixes off the logical terms extension and intension (which define the abstract and denotative aspect of the poetic language and, respectively, the concrete and connotative one). The meaning of the poem is ‘the full organized body of all the extension and intension that we can find in it.’  There is an infinite line between extreme extension and extreme intension and the readers select the meaning at the point they wish along that line, according to their personal drives, interests or approaches. Thus the Platonist will tend to stay near the extension end, for he is more interested in deriving an abstraction of the object into a universal….”

— from Form, Structure, and Structurality,
   by Radu Surdulescu

“Eliot, in a conception comparable to Wallace Stevens’ ‘Anecdote of the Jar,’ has suggested how art conquers time:

        Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

F. O. Matthiessen
   in The Achievement of T.S. Eliot,
   Oxford University Press, 1958

From Writing Chinese Characters:

“It is practical to think of a character centered within an imaginary square grid…. The grid can… be… subdivided, usually to 9 or 16 squares….”

The image “http://www.log24.com/log/pix04B/041119-ZhongGuo.jpg” cannot be displayed, because it contains errors.

These “Chinese jars”
(as opposed to their contents)
are as follows:

The image “http://www.log24.com/log/pix04B/041119-Grids.gif” cannot be displayed, because it contains errors.

Various previous Log24.net entries have
dealt with the 3×3 “form” or “pattern”
(to use the terms of T. S. Eliot).

For the 4×4 form, see Poetry’s Bones
and Geometry of the 4×4 Square.

Saturday, June 5, 2004

Saturday June 5, 2004

Filed under: Uncategorized — m759 @ 11:11 AM
A Form,
 continued…

Some cognitive uses
of the 3×3 square
are discussed in

From Lullus to Cognitive Semantics:
The Evolution of a Theory of Semantic Fields

by Wolfgang Wildgen and in

Another Page in the Foundation of Semiotics:
A Book Review of On the Composition of Images, Signs & Ideas, by Giordano Bruno…
by Mihai Nadin

“We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn’t merely sensational, that doesn’t get its message across in 10 seconds, that isn’t falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game.”

Robert Hughes, speech of June 2, 2004

Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder.

For a meditation on the related 4×4 square grid as “art that holds time,” see Time Fold.

Thursday, May 20, 2004

Thursday May 20, 2004

Filed under: Uncategorized — m759 @ 7:00 AM

Parable

“A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’), a term used to designate the geometric application we call a ‘parabola.’….  The basic parables are extended similes or metaphors.”

http://religion.rutgers.edu/nt/
    primer/parable.html

“If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals.  Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind.”

Sharon Begley, “The Puzzle of Genius,” Newsweek magazine, June 28, 1993, p. 50

“The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem ‘Pied Beauty’ has to do with ‘creation.’ “

Speaking in Parables, Ch. 2, by Sallie McFague

“The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels….  According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ‘see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”

— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

For further details, see

Speaking in Parables:
A Study in Metaphor and Theology

by Sallie McFague

Fortress Press, Philadelphia, 1975

Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7

“Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra.”

— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962

For metaphor and algebra combined, see

“Symmetry invariance in a diamond ring,” A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.

Sunday, April 25, 2004

Sunday April 25, 2004

Filed under: Uncategorized — m759 @ 3:31 PM

Small World

Added a note to 4×4 Geometry:

The 4×4 square model  lets us visualize the projective space PG(3,2) as well as the affine space AG(4,2).  For tetrahedral and circular models of PG(3,2), see the work of Burkard Polster.  The following is from an advertisement of a talk by Polster on PG(3,2).

The Smallest Perfect Universe

“After a short introduction to finite geometries, I’ll take you on a… guided tour of the smallest perfect universe — a complex universe of breathtaking abstract beauty, consisting of only 15 points, 35 lines and 15 planes — a space whose overall design incorporates and improves many of the standard features of the three-dimensional Euclidean space we live in….

Among mathematicians our perfect universe is known as PG(3,2) — the smallest three-dimensional projective space. It plays an important role in many core mathematical disciplines such as combinatorics, group theory, and geometry.”

— Burkard Polster, May 2001

Thursday, April 22, 2004

Thursday April 22, 2004

Filed under: Uncategorized — Tags: — m759 @ 10:07 PM

Minimalism

"It's become our form of modern classicism."

— Nancy Spector in 
   the New York Times of April 23, 2004

Part I: Aesthetics

In honor of the current Guggenheim exhibition, "Singular Forms" — A quotation from the Guggenheim's own website

"Minimalism refers to painting or sculpture

  1. made with an extreme economy of means
  2. and reduced to the essentials of geometric abstraction….
  3. Minimalist art is generally characterized by precise, hard-edged, unitary geometric forms….
  4. mathematically regular compositions, often based on a grid….
  5. the reduction to pure self-referential form, emptied of all external references….
  6. In Minimal art what is important is the phenomenological basis of the viewer’s experience, how he or she perceives the internal relationships among the parts of the work and of the parts to the whole….
  7. The repetition of forms in Minimalist sculpture serves to emphasize the subtle differences in the perception of those forms in space and time as the spectator’s viewpoint shifts in time and space."

Discuss these seven points
in relation to the following:

 
Form,
by S. H. Cullinane

Logos and Logic

Mark Rothko's reference
to geometry as a "swamp"
and his talk of "the idea" in art

Michael Kimmelman's
remarks on ideas in art 

Notes on ideas and art

Geometry
of the 4×4 square

The Grid of Time

ART WARS:
Judgment Day
(2003, 10/07)

Part II: Theology

Today's previous entry, "Skylark," concluded with an invocation of the Lord.   Of course, the Lord one expects may not be the Lord that appears.


 John Barth on minimalism:

"… the idea that, in art at least, less is more.

It is an idea surely as old, as enduringly attractive and as ubiquitous as its opposite. In the beginning was the Word: only later came the Bible, not to mention the three-decker Victorian novel. The oracle at Delphi did not say, 'Exhaustive analysis and comprehension of one's own psyche may be prerequisite to an understanding of one's behavior and of the world at large'; it said, 'Know thyself.' Such inherently minimalist genres as oracles (from the Delphic shrine of Apollo to the modern fortune cookie), proverbs, maxims, aphorisms, epigrams, pensees, mottoes, slogans and quips are popular in every human century and culture–especially in oral cultures and subcultures, where mnemonic staying power has high priority–and many specimens of them are self-reflexive or self-demonstrative: minimalism about minimalism. 'Brevity is the soul of wit.' "


Another form of the oracle at Delphi, in minimalist prose that might make Hemingway proud:

"He would think about Bert.  Bert was an interesting man.  Bert had said something about the way a gambler wants to lose.  That did not make sense.  Anyway, he did not want to think about it.  It was dark now, but the air was still hot.  He realized that he was sweating, forced himself to slow down the walking.  Some children were playing a game with a ball, in the street, hitting it against the side of a building.  He wanted to see Sarah.

When he came in, she was reading a book, a tumbler of dark whiskey beside her on the end table.  She did not seem to see him and he sat down before he spoke, looking at her and, at first, hardly seeing her.  The room was hot; she had opened the windows, but the air was still.  The street noises from outside seemed almost to be in the room with them, as if the shifting of gears were being done in the closet, the children playing in the bathroom.  The only light in the room was from the lamp over the couch where she was reading.

He looked at her face.  She was very drunk.  Her eyes were swollen, pink at the corners.  'What's the book,' he said, trying to make his voice conversational.  But it sounded loud in the room, and hard.

She blinked up at him, smiled sleepily, and said nothing.

'What's the book?'  His voice had an edge now.

'Oh,' she said.  'It's Kierkegaard.  Soren Kierkegaard.' She pushed her legs out straight on the couch, stretching her feet.  Her skirt fell back a few inches from her knees.  He looked away.

'What's that?' he said.

'Well, I don't exactly know, myself."  Her voice was soft and thick.

He turned his face away from her again, not knowing what he was angry with.  'What does that mean, you don't know, yourself?'

She blinked at him.  'It means, Eddie, that I don't exactly know what the book is about.  Somebody told me to read it once, and that's what I'm doing.  Reading it.'

He looked at her, tried to grin at her — the old, meaningless, automatic grin, the grin that made everbody like him — but he could not.  'That's great,' he said, and it came out with more irritation than he had intended.

She closed the book, tucked it beside her on the couch.  She folded her arms around her, hugging herself, smiling at him.  'I guess this isn't your night, Eddie.  Why don't we have a drink?'

'No.'  He did not like that, did not want her being nice to him, forgiving.  Nor did he want a drink.

Her smile, her drunk, amused smile, did not change.  'Then let's talk about something else,' she said.  'What about that case you have?  What's in it?'  Her voice was not prying, only friendly, 'Pencils?'

'That's it,' he said.  'Pencils.'

She raised her eyebrows slightly.  Her voice seemed thick.  'What's in it, Eddie?'

'Figure it out yourself.'  He tossed the case on the couch."

— Walter Tevis, The Hustler, 1959,
    Chapter 11


See, too, the invocation of Apollo in

A Mass for Lucero, as well as 

GENERAL AUDIENCE OF JOHN PAUL II
Wednesday 15 January 2003
:

"The invocation of the Lord is relentless…."

and

JOURNAL ENTRY OF S. H. CULLINANE
Wednesday 15 January 2003
:

Karl Cullinane —
"I will fear no evil, for I am the
meanest son of a bitch in the valley."

Tuesday, January 6, 2004

Tuesday January 6, 2004

Filed under: Uncategorized — m759 @ 10:10 PM

720 in the Book

Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.

An article titled On Mathematical Imagination concludes by looking forward to

“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”

Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.

Hmm.

The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:

Abel’s Proof:  An Essay
on the Sources and Meaning
of Mathematical Unsolvability

by Peter Pesic,
MIT Press, 2003

From a review:

“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….

Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations.  The reader is left with little clarity on this sequel to the story….”

— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244

Here, it seems, is my epiphany:

“Elliptic modular functions suffice to solve all polynomial equations.”


Incidental Remarks
on Synchronicity,
Part I

Those who seek a star
on this Feast of the Epiphany
may click here.


Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.

Just how such equations can be solved is a less familiar story.  I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.

The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions.  Modular functions are also distantly related, via the topic of “moonshine” and via the  “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.


Incidental Remarks
on Synchronicity,
Part II

There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.


Here is what I was able to find on the Web about Pesic’s claim:

From Wolfram Research:

From Solving the Quintic —

“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”

From Siegel Theta Function —

“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”

From Polynomial

“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable.  Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron.  Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”

Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.

King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.

Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999. 

From Angel Zhivkov,

Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:

“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker:  in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function….  Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist.  This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions.  In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”  

— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants


Incidental Remarks
on Synchronicity,
Part III

From Music for Dunne’s Wake:

Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”

— Carrie Fisher,
Postcards from the Edge

     

720 in  
the Book”

and
Paradise

“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.

Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.

For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see

Log24, June 2003.

For the relevance of the time
of this entry, 10:10, see

  1. the reference to Paradise
    on the “postcard” above, and
  2. Storyline (10/10, 2003).

Related recreational reading:

Labyrinth



The Shining

Shining Forth

Wednesday, November 12, 2003

Wednesday November 12, 2003

Filed under: Uncategorized — Tags: — m759 @ 9:58 AM

The Silver Table

“And suddenly all was changed.  I saw a great assembly of gigantic forms all motionless, all in deepest silence, standing forever about a little silver table and looking upon it.  And on the table there were little figures like chessmen who went to and fro doing this and that.  And I knew that each chessman was the idolum or puppet representative of some one of the great presences that stood by.  And the acts and motions of each chessman were a moving portrait, a mimicry or pantomine, which delineated the inmost nature of his giant master.  And these chessmen are men and women as they appear to themselves and to one another in this world.  And the silver table is Time.  And those who stand and watch are the immortal souls of those same men and women.  Then vertigo and terror seized me and, clutching at my Teacher, I said, ‘Is that the truth?….’ ”

— C.S. Lewis, The Great Divorce, final chapter

Follow-up to the previous four entries:

St. Art Carney, whom we may imagine to be a passenger on the heavenly bus in The Great Divorce, died on Sunday, Nov. 9, 2003.

The entry for that date (Weyl’s birthday) asks for the order of the automorphism group of a 4×4 array.  For a generalization to an 8×8 array — i.e., a chessboard — see

Geometry of the I Ching.

Audrey Meadows, said to have been the youngest daughter of her family, was born in Wuchang, China.

Tui: The Youngest Daughter

“Tui means to ‘give joy.’  Tui leads the common folk and with joy they forget their toil and even their fear of death. She is sometimes also called a sorceress because of her association with the gathering yin energy of approaching winter.  She is a symbol of the West and autumn, the place and time of death.”

Paraphrase of Book III, Commentaries of Wilhelm/Baynes.

Tuesday, November 11, 2003

Tuesday November 11, 2003

Filed under: Uncategorized — Tags: — m759 @ 11:11 AM

11:11

“Why do we remember the past
but not the future?”

— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

The flow of imaginary time is at right angles to that of ordinary time.“Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”

Is Time Quantized?

Yes.

Maybe.

We don’t really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional.  Then the following picture,

from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:

“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….

I sometimes wonder if that is
what Krishna meant—
Among other things—or one way
of putting the same thing:
That the future is a faded song,
a Royal Rose or a lavender spray
Of wistful regret for those who are
not yet here to regret,
Pressed between yellow leaves
of a book that has never been opened.
And the way up is the way down,
the way forward is the way back.”

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

Sunday, November 9, 2003

Sunday November 9, 2003

Filed under: Uncategorized — m759 @ 5:00 PM

For Hermann Weyl’s Birthday:

A Structure-Endowed Entity

“A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way.”

— Hermann Weyl in Symmetry

Exercise:  Apply Weyl’s lesson to the following “structure-endowed entity.”

4x4 array of dots

What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array’s structure you choose to examine.  It could be in the hundreds, or in the hundreds of thousands.)

Tuesday, September 16, 2003

Tuesday September 16, 2003

Filed under: Uncategorized — m759 @ 2:56 PM

The Form, the Pattern

“…the sort of organization that Eliot later called musical, in his lecture ‘The Music of Poetry’, delivered in 1942, just as he was completing Four Quartets: ‘The use of recurrent themes is as natural to poetry as to music,’ Eliot says:

There are possibilities for verse which bear some analogy to the development of a theme by different groups of instruments [‘different voices’, we might say]; there are possibilities of transitions in a poem comparable to the different movements of a symphony or a quartet; there are possibilities of contrapuntal arrangement of subject-matter.”

— Louis L. Martz, from
Origins of Form in Four Quartets,
in Words in Time: New Essays on Eliot’s Four Quartets, ed. Edward Lobb, University of Michigan Press, 1993

“…  Only by the form, the pattern,     
Can words or music reach
The stillness….”

— T. S. Eliot,
Four Quartets

Four Quartets

For a discussion of the above
form, or pattern, click here.

Wednesday, September 3, 2003

Wednesday September 3, 2003

Filed under: Uncategorized — Tags: — m759 @ 3:00 PM

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date: 

 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and

The Golden Theorem.

Sunday, July 13, 2003

Sunday July 13, 2003

Filed under: Uncategorized — Tags: — m759 @ 5:09 PM

ART WARS, 5:09

The Word in the Desert

For Harrison Ford in the desert.
(See previous entry.)

    Words strain,
Crack and sometimes break,
    under the burden,
Under the tension, slip, slide, perish,
Will not stay still. Shrieking voices
Scolding, mocking, or merely chattering,
Always assail them.
    The Word in the desert
Is most attacked by voices of temptation,
The crying shadow in the funeral dance,
The loud lament of
    the disconsolate chimera.

— T. S. Eliot, Four Quartets

The link to the word "devilish" in the last entry leads to one of my previous journal entries, "A Mass for Lucero," that deals with the devilishness of postmodern philosophy.  To hammer this point home, here is an attack on college English departments that begins as follows:

"William Faulkner's Snopes trilogy, which recounts the generation-long rise of the drily loathsome Flem Snopes from clerk in a country store to bank president in Jefferson, Mississippi, teems with analogies to what has happened to English departments over the past thirty years."

For more, see

The Word in the Desert,
by Glenn C. Arbery
.

See also the link on the word "contemptible," applied to Jacques Derrida, in my Logos and Logic page.

This leads to an National Review essay on Derrida,

The Philosopher as King,
by Mark Goldblatt

A reader's comment on my previous entry suggests the film "Scotland, PA" as viewing related to the Derrida/Macbeth link there.

I prefer the following notice of a 7-11 death, that of a powerful art museum curator who would have been well cast as Lady Macbeth:

Die Fahne Hoch,
Frank Stella,
1959


Dorothy Miller,
MOMA curator,

died at 99 on
July 11, 2003
.

From the Whitney Museum site:

"Max Anderson: When artist Frank Stella first showed this painting at The Museum of Modern Art in 1959, people were baffled by its austerity. Stella responded, 'What you see is what you see. Painting to me is a brush in a bucket and you put it on a surface. There is no other reality for me than that.' He wanted to create work that was methodical, intellectual, and passionless. To some, it seemed to be nothing more than a repudiation of everything that had come before—a rational system devoid of pleasure and personality. But other viewers saw that the black paintings generated an aura of mystery and solemnity.

The title of this work, Die Fahne Hoch, literally means 'The banner raised.'  It comes from the marching anthem of the Nazi youth organization. Stella pointed out that the proportions of this canvas are much the same as the large flags displayed by the Nazis.

But the content of the work makes no reference to anything outside of the painting itself. The pattern was deduced from the shape of the canvas—the width of the black bands is determined by the width of the stretcher bars. The white lines that separate the broad bands of black are created by the narrow areas of unpainted canvas. Stella's black paintings greatly influenced the development of Minimalism in the 1960s."

From Play It As It Lays:

   She took his hand and held it.  "Why are you here."
   "Because you and I, we know something.  Because we've been out there where nothing is.  Because I wanted—you know why."
   "Lie down here," she said after a while.  "Just go to sleep."
   When he lay down beside her the Seconal capsules rolled on the sheet.  In the bar across the road somebody punched King of the Road on the jukebox again, and there was an argument outside, and the sound of a bottle breaking.  Maria held onto BZ's hand.
   "Listen to that," he said.  "Try to think about having enough left to break a bottle over it."
   "It would be very pretty," Maria said.  "Go to sleep."

I smoke old stogies I have found…    

Cigar Aficionado on artist Frank Stella:

" 'Frank actually makes the moment. He captures it and helps to define it.'

This was certainly true of Stella's 1958 New York debut. Fresh out of Princeton, he came to New York and rented a former jeweler's shop on Eldridge Street on the Lower East Side. He began using ordinary house paint to paint symmetrical black stripes on canvas. Called the Black Paintings, they are credited with paving the way for the minimal art movement of the 1960s. By the fall of 1959, Dorothy Miller of The Museum of Modern Art had chosen four of the austere pictures for inclusion in a show called Sixteen Americans."

For an even more austere picture, see

Geometry for Jews:

For more on art, Derrida, and devilishness, see Deborah Solomon's essay in the New York Times Magazine of Sunday, June 27, 1999:

 How to Succeed in Art.

"Blame Derrida and
his fellow French theorists…."

See, too, my site

Art Wars: Geometry as Conceptual Art

For those who prefer a more traditional meditation, I recommend

Ecce Lignum Crucis

("Behold the Wood of the Cross")

THE WORD IN THE DESERT

For more on the word "road" in the desert, see my "Dead Poet" entry of Epiphany 2003 (Tao means road) as well as the following scholarly bibliography of road-related cultural artifacts (a surprising number of which involve Harrison Ford):

A Bibliography of Road Materials

Wednesday, March 12, 2003

Wednesday March 12, 2003

Filed under: Uncategorized — m759 @ 2:03 AM

Daimon Theory

Today is allegedly the anniversary of the canonization, in 1622, of two rather important members of the Society of Jesus (Jesuits):

Ignatius Loyola
  Click here for Loyola’s legacy of strategic intelligence.

Francis Xavier
  Click here for Xavier’s legacy of strategic stupidity.

We can thank (or blame) a Jesuit (Gerard Manley Hopkins) for the poetic phrase “immortal diamond.”  He may have been influenced by Plato, who has Socrates using a diamond figure in an argument for the immortality of the soul.  Confusingly, Socrates also talked about his “daimon” (pronounced dye-moan).  Combining these similar-sounding concepts, we have Doctor Stephen A. Diamond writing about daimons — a choice of author and topic that neatly combines the strategic intelligence of Loyola with the strategic stupidity of Xavier.

The cover illustration is perhaps not of Dr. Diamond himself.

A link between diamond theory and daimon theory is furnished by the charitable legacy of the non-practicing Jew Walter Annenberg.

For Annenberg and diamond theory, see this site on the elementary geometry of quilt blocks, which credits the Annenberg Foundation for support.

For Annenberg and daimon theory, see this site on Socrates, which has a similar Annenberg support credit.

Advanced disciples of Annenberg can learn much from the Perseus site about daimon theory. Let us pray that Abrahamic religious bigotry does not stand in their way.  Less advanced disciples of Annenberg may find fulfillment in teaching children the beauty of elementary 4×4 quilt-block symmetry.  Let us pray that academic bigotry does not prevent these same children, when they have grown older, from learning the deeper, and more difficult, beauties of diamond theory.

 
Daimon Theory

 
Diamond Theory

Thursday, March 6, 2003

Thursday March 6, 2003

Filed under: Uncategorized — Tags: , — m759 @ 2:35 AM

ART WARS:

Geometry for Jews

Today is Michelangelo's birthday.

Those who prefer the Sistine Chapel to the Rothko Chapel may invite their Jewish friends to answer the following essay question:

Discuss the geometry underlying the above picture.  How is this geometry related to the work of Jewish artist Sol LeWitt? How is it related to the work of Aryan artist Ernst Witt?  How is it related to the Griess "Monster" sporadic simple group whose elements number 

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000?

Some background:

Thursday, December 5, 2002

Thursday December 5, 2002

Filed under: Geometry — Tags: , — m759 @ 3:17 AM

Sacerdotal Jargon

From the website

Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:

Paper:  clf-alg/good9601
From:  David M. Goodmanson
Address:  2725 68th Avenue S.E., Mercer Island, Washington 98040

Title:  A graphical representation of the Dirac Algebra

Abstract:  The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra.  The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….

Published:  Am. J. Phys. 64, 870-880 (1996)


The following is a picture of K6, the complete graph on six points.  It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.


From
"The Relations between Poetry and Painting,"
by Wallace Stevens:

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: Uncategorized — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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