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Monday, July 16, 2012

Mapping Problem continued

Filed under: General,Geometry — Tags: Figurate Geometry, Springer — m759 @ 2:56 am

Another approach to the square-to-triangle
mapping problem (see also previous post)—

IMAGE- Triangular analogs of the hyperplanes in the square model of PG(3,2)

For the square model referred to in the above picture, see (for instance)

Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.

This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.

Update of 9:35 AM ET July 16, 2012:

Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —

— share the following 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.)

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Sunday, January 8, 2012

Big Apple

Filed under: General,Geometry — Tags: 2x4 — m759 @ 12:00 pm

“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”

A 2x4 array of squares

This suggests that the above structure
be viewed as illustrating not eight parts
but rather 8! = 40,320 parts.

"The Cardinal seemed a little preoccupied today."

The New Yorker , May 13, 2002

Saturday, September 3, 2011

The Galois Tesseract (continued)

Filed under: General,Geometry — Tags: Fields, Galois's Space, Geometry, Natural Diagram, Octad, Octad Generator, Priority — m759 @ 1:00 pm

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M₂₄, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Update of Sept. 4— This post is now a page at finitegeometry.org.

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Thursday, September 1, 2011

How It Works

Filed under: General,Geometry — Tags: Design Theory, Geometry, Priority, September Morn — m759 @ 11:00 am

“Design is how it works.” — Steven Jobs (See Symmetry and Design.)

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010—

“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”

– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)

See also “Do you like apples?“

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Thursday, August 25, 2011

Design

Filed under: General,Geometry — Tags: Geometry — m759 @ 11:07 pm

"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)

Today's American Mathematical Society home page—

Some related material—

The above Rowley paragraph in context (click to enlarge)—

"We employ Curtis's MOG …
both as our main descriptive device and
also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122

And the MOG incorporates the
Geometry of the 4×4 Square.

For this geometry's relation to "design"
in the graphic-arts sense, see
Block Designs in Art and Mathematics.

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Wednesday, August 24, 2011

Symmetry

Filed under: General,Geometry — m759 @ 11:07 pm

An article from cnet.com tonight —

For Jobs, design is about more than aesthetics

By: Jay Greene August 24, 2011 7:21 PM PDT

… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.

Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod, Jobs laid out his vision for product design.

''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''

Related material: Open, Sesame Street (Aug. 19) continues… Brought to you by the number 24—

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

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Saturday, August 20, 2011

Castle Rock

Filed under: General,Geometry — Tags: 2x4 — m759 @ 6:29 pm

Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)

"The metaphor for metamorphosis…" —Endgame

Related material:

"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."

— Scholarly paper on "Correlative Cosmologies"

"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.

The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….

'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."

— Alastair Macaulay in today's New York Times

SOCRATES: Let us turn off the road a little….

— Libretto for Mark Morris's 'Socrates'

See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:

A 2x4 array of squares

The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S₈
acting on the above eight-part figure.

See also The Moore Correspondence from last year
on today's date, August 20.

For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—

"At the still point, there the dance is."
— Four Quartets

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Saturday, August 6, 2011

Correspondences

Filed under: General,Geometry — Tags: Four Color, four-set, Geometry, Natural Diagram, Octad, Octad Generator, Springer — 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”—

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—

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 .

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Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: Gordon Baker, Modal Diamond Box — m759 @ 9:29 am

Philosophical Investigations (1953)—

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest thing there is
(Tractatus Logico-Philosophicus No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach sein. Sie ist vor aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh. No. 5.5563.)

See also—

Related language in Łukasiewicz (1937)—

* Updates of 9:29 PM ET July 10, 2011—

A mnemonic from a course titled “Galois Connections and Modal Logics“—

“Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and (diamond) for possibly. Then, for example, diamond P can be read as
‘it is possibly the case that P .'”

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

“The diamond ⋄ symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬⋄¬. The dual symbol □ was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich ‘possible.’”

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

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Wednesday, July 6, 2011

Nordstrom-Robinson Automorphisms

Filed under: General,Geometry — Tags: Coding Theory, Fields, Geometry, Octad, Octad Generator, Springer — m759 @ 1:01 am

A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—

"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."

— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22

A statement by Bierbrauer from 2004 has an error that doubles the above figure—

The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order |M24| = 24 × 23 × 22 × 21 × 20 × 48 in its 5-transitive action on the 24 coordinates. As M24 is transitive on octads, the stabilizer of an octad has order |M24|/759 [=322,560]. The stabilizer of NR has index 8 in this group. It follows that NR admits an automorphism group of order |M24| / (759 × 8 ) = [?] 16 × 7! [=80,640]. This is a huge symmetry group. Its structure can be inferred from the embedding in G as well. The automorphism group of NR is a semidirect product of an elementary abelian group of order 16 and the alternating group A₇.

— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A₇-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170

The error is corrected (though not detected) later in the same 2004 paper—

In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).

For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—

Click to enlarge:

For some context, see the group of order 322,560 in Geometry of the 4×4 Square.

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Sunday, June 19, 2011

Abracadabra (continued)

Filed under: General,Geometry — Tags: Octad, Octad Generator — m759 @ 12:00 am

Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.

A search tonight for related material on the Web yielded…

(Click to enlarge.)

   Weblog post by Steve Richards titled "The Search for Invariants:
   The Diamond Theory of Truth, the Miracle Octad Generator
   and Metalibrarianship." The artwork is by Steven H. Cullinane.
   Richards has omitted Cullinane's name and retitled the artwork.

The author of the post is an artist who seems to be interested in the occult.

His post continues with photos of pages, some from my own work (as above), some not.

My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.

The artist's post concludes with the following (note also the beginning of the preceding post)—

"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.

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Sunday, June 5, 2011

Edifice Complex

Filed under: General,Geometry — Tags: Geometry, Kummerhenge, Octad, Octad Generator — m759 @ 7:00 pm

"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."

— Wallace Stevens, "To an Old Philosopher in Rome"

The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…

"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology

The Kummer 16₆ Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)

IMAGE-- 16_6 configuration from '2-Transitive Symmetric Designs,' by William M. Kantor (AMS Transactions, 1969)

For some background, see Configurations and Squares.

For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)

Related material— The Schwartz Notes of June 1.

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Wednesday, June 1, 2011

The Schwartz Notes

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator, Priority, The Omega Matrix — m759 @ 2:00 pm

A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)

The same search limited to results from only the past month yielded,
in addition, the following—

This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.

Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.

See Schwartz’s page 12, page 13, and page 14.

Compare to the web pages from today’s earlier search.

There are no references at the end of the Schwartz document,
but there is this at the beginning—

These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.

It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.

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Wednesday, March 2, 2011

Labyrinth of the Line

Filed under: General,Geometry — Tags: Finite Relativity, Octad, Octad Generator — m759 @ 11:24 am

“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—

Related material —

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.

A search in the latter for miracle octad is updated below.

This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

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Tuesday, November 16, 2010

Meanwhile, back in 1953…

Filed under: General — Tags: The Horcrux Narrative, Unholy Crux — m759 @ 9:29 am

"I open at the close" — The Resurrection Stone

Related material — Click on images for details —

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Monday, October 25, 2010

The Embedding*

Filed under: General,Geometry — Tags: Intake Manifold, Octad — m759 @ 4:04 pm

A New York Times "The Stone" post from yesterday (5:15 PM, by John Allen Paulos) was titled—

Stories vs. Statistics

Thursday, September 9, 2010

Building a Mystery

Filed under: General,Geometry — Tags: Waymark Prize — m759 @ 2:02 pm

Notes on Mathematics and Narrative, continued

Patrick Blackburn, meet Gideon Summerfield…

From a summary of a politically correct 1995 feminist detective novel about quilts, A Piece of Justice—

The story deals with “one Gideon Summerfield, deceased.” Summerfield, a former tutor at (the fictional) St. Agatha’s College, Cambridge University, “is about to become the recipient of the Waymark prize. This prize is awarded in Mathematics and has the same prestige as the Nobel. Summerfield had a rather lackluster career at St. Agatha’s, with the exception of one remarkable result that he obtained. It is for this result that he is being awarded the prize, albeit posthumously.” Someone is apparently trying to prevent a biography of Summerfield from being published.

The following page contains a critical part of the solution to the mystery:

The image “http://www.log24.com/log/pix06B/PieceOfJustice138.gif” cannot be displayed, because it contains errors.

Compare and contrast with an episode from the resume of a real Gideon Summerfield—

Head of Strategy, Designer City (May 1999 — January 2002)

Secured Web agency business from new and existing clients with compelling digital media strategies and oversaw delivery of creative, production and technical teams…. Clients included… Greenfingers and Lord of the Dance .

For material related to Greenfingers and Lord of the Dance , see Castle Kennedy Gardens at Wicker Man Locations.

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Friday, August 20, 2010

The Moore Correspondence

Filed under: General,Geometry — m759 @ 5:01 pm

There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A₈ with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M₂₄.

A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.

Edge says that

It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….

Excerpts from the Edge paper—

Excerpts from the Moore paper—

Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439

* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72

** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.

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Thursday, August 5, 2010

Eightgate

Filed under: General,Geometry — Tags: 2x4 — m759 @ 2:02 pm

"Eight is a gate."
— This journal, December 2002

Tralfamadorian Structure
in Slaughterhouse-Five

includes the following passage:

A 2x4 array of squares

This suggests that the above structure be viewed
as illustrating not eight parts but rather
8! = 40,320 parts.

Saturday, July 24, 2010

Playing with Blocks

Filed under: General,Geometry — Tags: Geometry, Lexicon Omega, Octad, Octad Generator, Springer, SymOmega, The Building-Block Metaphor — m759 @ 12:00 pm

"Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry."

— Finite geometry page at the Centre for the Mathematics of
Symmetry and Computation at the University of Western Australia
(Alice Devillers, John Bamberg, Gordon Royle)

For such symmetries, see Robert A. WIlson's recent book The Finite Simple Groups.

The finite simple groups are often described as the "building blocks" of finite group theory.

At least some of these building blocks have their own building blocks. See Non-Euclidean Blocks.

For instance, a set of 24 such blocks (or, more simply, 24 unit squares) appears in the Miracle Octad Generator (MOG) of R.T. Curtis, used in the study of the finite simple group M₂₄.

(The octads of the MOG illustrate yet another sort of mathematical blocks— those of a block design.)

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Thursday, June 24, 2010

Midsummer Noon

Filed under: General,Geometry — Tags: Galois Window, Midsummer Geometry, The Four-Point Plane — m759 @ 12:00 pm

Geometry Simplified

Image-- The Three-Point Line: A Finite Projective Space
(a projective space)

The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

(an affine space)

The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

For further details, see Galois Geometry.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —

Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

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Saturday, June 19, 2010

Imago Creationis

Filed under: General,Geometry — Tags: Aion Art, Cubehenge, Eden, Galois's Space, Imago, Self, Stevens Rock, Valéry — m759 @ 6:00 pm

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—

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—

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:

O 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).

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

Frame of Reference

The Diamond Theorem—

Some context by a British mathematician —

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.

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Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: Aion Art, Change Logos, Eden, Self, Strange Change, Valéry — m759 @ 11:07 am

Recommended— an online book—

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

See in particular

Valéry and the Discourse On His Method.

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:

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

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

Compare Jung's image in Aion of the Self as a four-diamond figure:

and Cullinane's purely geometric four-diamond figure:

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation "Theme and Variations" in Oslo in 2009. Lyche titled this part of her installation "The Smallest Perfect Universe," a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry's notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

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Saturday, May 15, 2010

Mathematics and Narrative continued…

Filed under: General,Geometry — Tags: Octad — m759 @ 4:16 pm

Step Two

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

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Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: Folklore, four-set, Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 11:07 am

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.

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Wednesday, October 14, 2009

Wednesday October 14, 2009

Filed under: General,Geometry — Tags: Geometry, Natural Diagram — m759 @ 9:29 am

Singer 7-Cycles

Click on images for details.

The 1985 Cullinane version gives some algebraic background for the 1987 Curtis version.

The Singer referred to above is James Singer. See his "A Theorem in Finite Projective Geometry and Some Applications to Number Theory," Transactions of the American Mathematical Society 43 (1938), 377-385.For other singers, see Art Wars and today's obituaries.

Some background: the Log24 entry of this date seven years ago, and the entries preceding it on Las Vegas and painted ponies.

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Thursday, August 6, 2009

Thursday August 6, 2009

Filed under: General,Geometry — Tags: Octad, Octad Generator, on090806 — m759 @ 1:44 pm

A Fisher of Men

Cover, Schulberg's novelization of 'Waterfront,' Bantam paperback

Update: The above image was added
at about 11 AM ET Aug. 8, 2009.

From a webpage of the First United Methodist Church of Bloomington, Indiana–

Dr. Joe Emerson, April 24, 2005–

"The Ultimate Test"

— Text: I Peter 2:1-9

Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"– was based on a series of newspaper articles by Malcolm Johnson.)

"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando. That’s the scary part of growing old when you see what he looked like then and when he grew old.) It is based on a book by Budd Schulberg."

Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–

"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River. A priest is called to give last rites after they drag him out."

New York Times today

Dr. Emerson flunks the test.

Dr. Emerson's sermon is, as noted above (Text: I Peter 2:1-9), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.

My own remarks on the date of Dr. Emerson's sermon—

The 4x6 array used in the Miracle Octad Generator of R. T. Curtis

Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.

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Wednesday, May 20, 2009

Wednesday May 20, 2009

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator — m759 @ 4:00 pm

From Quilt Blocks to the
Mathieu Group M₂₄

_{Diamonds
(a traditional

quilt block):

_Octads:}

Click on illustrations for details.

The connection:

The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)

The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in–

"A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.

Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."

"A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.

Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."

For the connection of S(5,8,24) with the Mathieu group M₂₄, see the references in The Miracle Octad Generator.

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Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: General,Geometry — Tags: Geometry, May 19 Gestalt, Octad, Octad Generator — m759 @ 7:20 pm

Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M₂₄."

— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

— William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.

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Saturday, April 4, 2009

Saturday April 4, 2009

Filed under: General,Geometry — Tags: Autist Artist, Geometry — m759 @ 7:01 pm

Steiner Systems

"Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns). The child-master, like his adult counterpart, is able to visualize in an instantaneous yet preternaturally confident way how the thing should look several moves hence. He sees the logical, the necessary harmonic and melodic argument as it arises out of an initial key relation or the preliminary fragments of a theme. He knows the order, the appropriate dimension, of the sum or geometric figure before he has performed the intervening steps. He announces mate in six because the victorious end position, the maximally efficient configuration of his pieces on the board, lies somehow 'out there' in graphic, inexplicably clear sight of his mind…."

"… in some autistic enchantment, pure as one of Bach's inverted canons or Euler's formula for polyhedra."

— George Steiner, "A Death of Kings," in The New Yorker, issue dated Sept. 7, 1968

Related material:

A correspondence underlying
the Steiner system S(5,8,24)–

The Steiner here is
Jakob, not George.

See "Pope to Pray on
Autism Sunday 2009."
See also Log24 on that
Sunday– February 8:

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Sunday, February 15, 2009

Sunday February 15, 2009

Filed under: General,Geometry — Tags: 2x4, Facets — m759 @ 11:00 am

The Pediment
of Appearance

From April 28, 2008:

Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
an epiphany.

A 2x4 array of squares

One approach to
the epiphany:

"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein

See 4/28/08 for examples
of such transformations.

Related material:

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, pp. 117-118:

"… his point of origin is external nature, the fount to which we come seeking inspiration for our fictions. We come, many of Stevens's poems suggest, as initiates, ritualistically celebrating the place through which we will travel to achieve fictive shape. Stevens's 'real' is a bountiful place, continually giving forth life, continually changing. It is fertile enough to meet any imagination, as florid and as multifaceted as the tropical flora about which the poet often writes. It therefore naturally lends itself to rituals of spring rebirth, summer fruition, and fall harvest. But in Stevens's fictive world, these rituals are symbols: they acknowledge the real and thereby enable the initiate to pass beyond it into the realms of his fictions.

Two counter rituals help to explain the function of celebration as Stevens envisions it. The first occurs 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. For in 'Notes Toward a Supreme Fiction' he tells us that

... the first idea was not to shape the clouds
In imitation. The clouds preceded us.      

There was a muddy centre before we breathed.
There was a myth before the myth began,
Venerable and articulate and complete.      

From this the poem springs: that we live in a place
That is not our own and, much more, not ourselves
And hard it is in spite of blazoned days.      

We are the mimics.

                                (Collected Poems, 383-84)

Believing that they are the life and not the mimics thereof, the world and not its fiction-forming imitators, these young men cannot find the savage transparence for which they are looking. In its place they find the pediment, a scowling rock that, far from being life's source, is symbol of the human delusion that there exists a 'form alone,' apart from 'chains of circumstance.'

A far more productive ritual occurs in 'Sunday Morning.'…."

For transformations of a more
specifically religious nature,
see the remarks on
Richard Strauss,
"Death and Transfiguration,"
(Tod und Verklärung, Opus 24)
in Mathematics and Metaphor
on July 31, 2008, and the entries
of August 3, 2008, related to the
death of Alexander Solzhenitsyn.

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Tuesday, January 6, 2009

Tuesday January 6, 2009

Filed under: General,Geometry — Tags: Aion Art, Eightfold Cube, Octad, Octad Generator — m759 @ 12:00 am

Archetypes, Synchronicity,
and Dyson on Jung

The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson's 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung's theory of archetypes:

"… we do not need to accept Jung’s theory as true in order to find it illuminating."

The same is true of Jung's remarks on synchronicity.

For example —

Yesterday's entry, "A Wealth of Algebraic Structure," lists two articles– each, as it happens, related to Jung's four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:

R. T. Curtis's 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.

Curtis's 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.

On these dates, the entries in this journal discussed…

Oct. 24:
Cube Space, 1984-2003

Material related to that entry:

Dec. 19:
Art and Religion: Inside the White Cube

That entry discusses a book by Mark C. Taylor:

The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).

In Chapter 3, "Sutures of Structures," Taylor asks —

"What, then, is a frame, and what is frame work?"

One possible answer —

Hermann Weyl on the relativity problem in the context of the 4×4 "frame of reference" found in the above Cambridge University Press articles.

"Examples are the stained-glass
windows of knowledge."
— Vladimir Nabokov

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Monday, January 5, 2009

Monday January 5, 2009

Filed under: General,Geometry — Tags: Fields, Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 9:00 pm

A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

(Received July 20 1987)

— Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):

A new combinatorial approach to M₂₄, by R. T. Curtis. Abstract:

“In this paper, we define M₂₄ from scratch as the subgroup of S₂₄ preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”

(Received June 15 1974)

— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

* For instance:

Click for details.

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Saturday, December 27, 2008

Saturday December 27, 2008

Filed under: General — Tags: Octad — m759 @ 9:00 pm

keen

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Saturday, December 13, 2008

Saturday December 13, 2008

Filed under: General — m759 @ 1:06 pm

The Shining
of Dec. 13

continued from
Dec. 13, 2003

“There is a place for a hint
somewhere of a big agent
to complete the picture.”

— Notes for an unfinished novel,
The Last Tycoon,
by F. Scott Fitzgerald

Internet Movie Database
Filmography:William Grady

The Good Earth (1937)
casting: Chinese extras
(uncredited)

A Place for a Hint:

http://www.log24.com/log/pix08A/081213-Tea2.jpg

(From the book Tangram)

See also
yesterday’s entries
as well as…

Serpent’s Eyes Shine,
Alice’s Tea Party,
Janet’s Tea Party,
Hollywood Memory,
and
Hope of Heaven.

“… it’s going to be
accomplished in steps,
this establishment of
the Talented
in the scheme of things.”

— Anne McCaffrey

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Monday, November 24, 2008

Monday November 24, 2008

Filed under: General,Geometry — Tags: Geometric Theology, Octad, Octad Generator — m759 @ 12:00 pm

Frame Tale

Click on image for details.

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Sunday, October 12, 2008

Sunday October 12, 2008

Filed under: General,Geometry — Tags: 2x4 — m759 @ 2:22 am

“Elegant”

— Today’s New York Times
review of the Very Rev.
Francis Bowes Sayre Jr.

Related material:

Log24 entries from
the anniversary this
year of Sayre’s birth
and from the date
of his death:

A link from the former
suggests the following
graphic meditation–

(Click on figure for details.)

A link from the latter
suggests another
graphic meditation–

(Click on figure for details.)

Although less specifically
American than the late
Reverend, who was
born in the White House,
hence perhaps irrelevant
to his political views,
these figures are not
without relevance to
his religion, which is
more about metanoia
than about paranoia.

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Thursday, July 31, 2008

Thursday July 31, 2008

Filed under: General,Geometry — Tags: Four Color, Octad, Octad Generator — 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 —

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

as a tesseract.“

— Madeleine L’Engle

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

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Monday, April 28, 2008

Monday April 28, 2008

Filed under: General,Geometry — Tags: 2x4, Brick Space, four-set — m759 @ 7:00 am

Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
an epiphany.

A 2x4 array of squares

One approach to
the epiphany:

"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein

The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:

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

For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model."

For a related structure–
not rectangle but cube–
see Epiphany 2008.

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Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General,Geometry — Tags: Lichtung, Pieces of April — m759 @ 5:01 am

A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into

n(n -1) … (n – k + 1)
____________________

m(m – 1) … (m – k + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

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Thursday, March 6, 2008

Thursday March 6, 2008

Filed under: General,Geometry — Tags: Diadema, Four Color, March Fourth Death — m759 @ 12:00 pm

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

"The historical road
from the Platonic solids
to the finite simple groups
is well known."

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. "the king of geometry."

Picture from a 1980 article
in a journal of
The Mathematical Association
of America:

Coxeter as King of Geometry (from an MAA journal, 1980)

Wednesday, January 16, 2008

Wednesday January 16, 2008

Filed under: General,Geometry — Tags: Chess Problem, Cube Moves, Cube School, Eightfold Cube, Knight Move — m759 @ 12:25 pm

Knight Moves:
Geometry of the
Eightfold Cube

Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.

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Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: Innermost Kernel, Local/Global, Octad, Strange Change, Structural Logic — m759 @ 12:07 pm

Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

—Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A₈ on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
— Physics Today

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

— The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

"Pauli linked this symbolism
with the concept of automorphism."

— The Innermost Kernel
(previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them? A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic (4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
as many symmetries as possible."

"There is such a thing
as a tesseract."

— Madeleine L'Engle

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Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: Beadgame Space, Geometry, Glasperlenspiel, Octad, Octad Generator — m759 @ 5:00 pm

Space-Time

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”

The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.

Part II: A Corresponding Finite Model

The Klein quadric also occurs in a finite model of projective 5-space. See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail. This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads. These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M₂₄.

Related material:

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group AGL(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

See

Finite Geometry of
the Square and Cube

A 4x4x4 cube

and

Geometry of the I Ching.

“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game. Elder Brother laughed. ‘Go ahead and try,’ he exclaimed. ‘You’ll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”

— Hermann Hesse, The Glass Bead Game,

translated by Richard and Clara Winston

Comments (2)

Thursday, May 24, 2007

Thursday May 24, 2007

Filed under: General,Geometry — m759 @ 4:00 am

Day 24

“The miraculous enters….“

Discuss.

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Monday, April 30, 2007

Monday April 30, 2007

Filed under: General,Geometry — Tags: Structural Logic, Structure and Mutability — m759 @ 6:24 pm

Structure and Logic

The phrase “structural logic” in yesterday’s entry was applied to Bach’s cello suites. It may equally well be applied to geometry. In particular:

“The aim of this thesis is to classify certain structures which are, from a certain point of view, as homogeneous as possible, that is which have as many symmetries as possible.”

— Alice Devillers, “Classification of Some Homogeneous and Ultrahomogeneous Structures,” Ph.D. thesis, Université Libre de Bruxelles, academic year 2001-2002

Related material:

New models of some small finite spaces

In Devillers’s words, the above spaces with 8 and 16 points are among those structures that have “as many symmetries as possible.” For more details on what this means, see Devillers’s thesis and Finite Geometry of the Square and Cube.

The above models for the corresponding projective spaces may be regarded as illustrating the phrase “structural logic.”

For a possible application of the 16-point space’s “many symmetries” to logic proper, see The Geometry of Logic.

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Wednesday, February 28, 2007

Wednesday February 28, 2007

Filed under: General,Geometry — Tags: Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 7:59 am

Elements
of Geometry

The title of Euclid’s Elements is, in Greek, Stoicheia.

From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –

Stoicheia

“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….

Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–

stix, gen. stichos, a row, a line of soldiers
stichos, a row, a line; distich, a couplet
steicho, estichon, to march in order, step by step; to mount
stoichos, a row, a file; stoichein, to march in a line

In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….

Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”

Example:

The image “http://www.log24.com/log/pix07/070228-MOG.gif” cannot be displayed, because it contains errors.

The Miracle Octad Generator of R. T. Curtis

For the geometry of these stoicheia, see
The Smallest Perfect Universe and
Finite Geometry of the Square and Cube.

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Tuesday, February 27, 2007

Tuesday February 27, 2007

Filed under: General,Geometry — Tags: Brick Space — m759 @ 7:59 am

Continued from 2/06:

The Poetics of Space

Log24 yesterday:

“Imprimatur.
+John Cardinal Farley,
Archbishop of New York”

Tom Hanks as Robert Langdon in The Da Vinci Code

Tom Hanks as Robert Langdon
in “The Da Vinci Code”

“… and by ‘+’ I mean
artistic vision.”

New York State Lottery
yesterday, Feb. 26, 2007:

Mid-day 206
Evening 888

For more on the artistic
significance of 206,
see 2/06.

For more on the artistic
significance of 888, see
St. Bonaventure on the
Trinity at math16.com.

A trinity:

Click on picture for further details.

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Sunday, December 10, 2006

Sunday December 10, 2006

Filed under: General,Geometry — Tags: Bullshit Studies, Waymark Prize — m759 @ 12:00 pm

On This Date

“… in 1896 Alfred Nobel,
the inventor of dynamite and
founder of the Nobel prizes,
died in San Remo, Italy,
at age 63.”

— “Today in History,”
by The Associated Press

… And the Nobel Prize
for Bullshit goes to…

David Titcher,

author and co-producer of
The Librarian: Quest for the Spear.

First Runner-up —

A Piece of Justice.

From a summary of the novel:

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Wednesday, December 6, 2006

Wednesday December 6, 2006

Filed under: General,Geometry — Tags: Waymark Prize — m759 @ 3:15 am

Mathematical Imagery

From the current
American Mathematical Society
“Mathematical Imagery” page:

From today’s New York Times:

“Rosie Lee Tompkins, a renowned African-American quiltmaker whose use of dazzling color and vivid geometric forms made her work internationally acclaimed despite her vehement efforts to remain completely unknown, was found dead on Friday at her home in Richmond, Calif. She was 70.” —Margalit Fox, NY Times 12/6/06

Tompkins was found dead
on December 1, 2006.

From Log24 on that date:

The image “http://www.log24.com/log/pix06B/061201-DayWithout.jpg” cannot be displayed, because it contains errors.

That entry contained an excerpt from
Tom Wolfe’s The Painted Word—

“What I saw before me was the critic-in-chief of The New York Times saying: In looking at a painting today, ‘to lack a persuasive theory is to lack something crucial.’ I read it again. It didn’t say ‘something helpful’ or ‘enriching’ or even ‘extremely valuable.’ No, the word was crucial….”

Related material:

Diamond Theory

and a politically correct
1995 feminist detective novel
about quilts,

A Piece of Justice.

From a summary of the novel:

The following page contains
a critical part of the solution
to the mystery:

The image “http://www.log24.com/log/pix06B/PieceOfJustice138.gif” cannot be displayed, because it contains errors.

Meanwhile, back in real life…

It is said that the late Ms. Tompkins
liked to work while listening to the
soundtrack of “Saturday Night Fever.”

“It’s just your jive talkin’
you’re telling me lies, yeah
Jive talkin’
you wear a disguise
Jive talkin’
so misunderstood, yeah
Jive talkin’
You really no good”

These lyrics may also serve
to summarize reviews
of Diamond Theory written
in the summer of 2005.

For further details, see
Mathematics and Narrative.

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Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator, Sundown Ending — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

— Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

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Saturday, July 29, 2006

Saturday July 29, 2006

Filed under: General,Geometry — Tags: Big Rock, Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 2:02 pm

Big Rock

Thanks to Ars Mathematica, a link to everything2.com:

“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”

Another example:

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis. See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:

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

The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts. (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.

“The rock cannot be broken.
It is the truth.”

— Wallace Stevens,
“Credences of Summer”

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Friday, June 16, 2006

Friday June 16, 2006

Filed under: General,Geometry — Tags: Quilt Geometry, Sherlock Narratives, Waymark Prize — m759 @ 9:00 am

For Bloomsday 2006:

Hero of His Own Story

"The philosophic college should spare a detective for me."

— Stephen Hero. Epigraph to Chapter 2, "Dedalus and the
Beauty Maze," in Joyce and Aquinas, by William T. Noon, S. J.,
Yale University Press, 1957 (in the Yale paperback edition of
1963, page 18)

"Dorothy Sayers makes a great deal of sense when she points out
in her highly instructive and readable book The Mind of the Maker
that 'to complain that man measures God by his own measure is
a waste of time; man measures everything by his own experience;
he has no other yardstick.'"

— William T. Noon, S. J., Joyce and Aquinas (in the Yale paperback
edition of 1963, page 106)

Related material:

Dorothy Sayers and Jill Paton Walsh
Jill Paton Walsh's detective novel A Piece of Justice (1995):
"The mathematics of tilings and quilting play background
roles in this mystery in which a graduate student attempts
to write a biography of the (fictitious) mathematician
Gideon Summerfield. Summerfield is about to posthumously
receive the prestigious (and, I should point out, also fictitious)
Waymark Prize in mathematics…but it soon becomes clear
that someone with evil intentions does not want the student's
book to be published!By all accounts this is a well written mystery…
the second by the author with college nurse Imogen Quy playing
the role of the detective."— Mathematical Fiction by Alex Kasman,
College of Charleston

Quilt Geometry, by Steven H. Cullinane

AD PULCHRITUDINEM TRIA REQUIRUNTUR:
INTEGRITAS, CONSONANTIA, CLARITAS.

— St. Thomas Aquinas

Comments (1)

Sunday, May 7, 2006

Sunday May 7, 2006

Filed under: General,Geometry — Tags: Natural Diagram, Octad, Octad Generator — m759 @ 3:00 am

Bagombo Snuff Box

(in memory of
Burt Kerr Todd)

“Well, it may be the devil
or it may be the Lord
But you’re gonna have to
serve somebody.”

— “Bob Dylan”
(pseudonym of Robert Zimmerman),
quoted by “Bob Stewart”
on July 18, 2005

“Bob Stewart” may or may not be the same person as “crankbuster,” author of the “Rectangular Array Theorem” or “RAT.” This “theorem” is intended as a parody of the “Miracle Octad Generator,” or “MOG,” of R. T. Curtis. (See the Usenet group sci.math, “Steven Cullinane is a Crank,” July 2005, messages 51-60.)

“Crankbuster” has registered at Math Forum as a teacher in Sri Lanka (formerly Ceylon). For a tall tale involving Ceylon, see the short story “Bagombo Snuff Box” in the book of the same title by Kurt Vonnegut, who has at times embodied– like Martin Gardner and “crankbuster“– “der Geist, der stets verneint.”

Here is my own version (given the alleged Ceylon background of “crankbuster”) of a Bagombo snuff box:

Related material:

Log24 entries of
April 16-30, 2005,

and the 5 Log24 entries
ending on Friday,
April 28, 2006.

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Monday, January 23, 2006

Monday January 23, 2006

Filed under: General,Geometry — Tags: Free-Floating Signs, Springer — m759 @ 6:00 pm

In Defense of Hilbert
(On His Birthday)

Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:

“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”

Harris adds in a footnote:

“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”

If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”

For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):

[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”

Some of Zach’s references:

[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].

[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990

From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:

“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”

[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….

[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.

[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.

From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:

“What Hilbert wants to secure is not the truth, but the consistency of the old analysis. This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics. In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one. The statements become meaningless figures built up from signs. Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess. Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs. One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess. And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae. By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game. The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules. Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color. Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game. Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge. What is meant to be recognized is that a contradiction can never occur as an end formula of a proof. Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration. One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color. In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game. It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together. Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”

Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:

“Hilbert’s program… received its share of criticism. Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’ But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time? I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous. It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”

— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)

Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:

“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”

One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136. The quotation in the English translation given there:

“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”

Corfield says of this quotation that he’d love to be told the original German. He should consult the above references cited by Richard Zach.

For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.

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Wednesday, December 14, 2005

Wednesday December 14, 2005

Filed under: General — Tags: in1954 — m759 @ 1:00 am

From Here
to Eternity

For Loomis Dean

See also
For Rita Moreno
on Her Birthday
(Dec. 11, 2005)

Los Angeles Times

Tuesday, Dec. 13, 2005

OBITUARIES

The image “http://www.log24.com/log/pix05B/051214-LoomisDean.jpg” cannot be displayed, because it contains errors.

LOOMIS DEAN
After many years at Life magazine,
he continued to find steady work
as a freelancer and as a still
photographer on film sets.
(Dean Family)

Loomis Dean, 88;
Life Magazine Photographer
Known for Pictures of
Celebrities and Royalty

By Jon Thurber, Times Staff Writer

Loomis Dean, a Life magazine photographer who made memorable pictures of the royalty of both Europe and Hollywood, has died. He was 88.

Dean died Wednesday [December 7, 2005] at Sonoma Valley Hospital in Sonoma, Calif., of complications from a stroke, according to his son, Christopher.

In a photographic career spanning six decades, Dean's leading images included shirtless Hollywood mogul Darryl F. Zanuck trying a one-handed chin-up on a trapeze bar, the ocean liner Andrea Doria listing in the Atlantic and writer Ernest Hemingway in Spain the year before he committed suicide. One of his most memorable photographs for Life was of cosmopolitan British playwright and composer Noel Coward in the unlikely setting of the Nevada desert.

Dean shot 52 covers for Life, either as a freelance photographer or during his two stretches as a staffer with the magazine, 1947-61 and 1966-69. After leaving the magazine, Dean found steady freelance work in magazines and as a still photographer on film sets, including several of the early James Bond movies starring Sean Connery.

Born in Monticello, Fla., Dean was the son of a grocer and a schoolteacher.

When the Dean family's business failed during the Depression, they moved to Sarasota, Fla., where Dean's father worked as a curator and guide at the John and Mable Ringling Museum of Art.

Dean studied engineering at the University of Florida but became fascinated with photography after watching a friend develop film in a darkroom. He went off to what is now the Rochester Institute of Technology, which was known for its photography school.

After earning his degree, Dean went to work for the Ringling circus as a junior press agent and, according to his son, cultivated a side job photographing Ringling's vast array of performers and workers.

He worked briefly as one of Parade magazine's first photographers but left after receiving an Army Air Forces commission during World War II. During the war, he worked in aerial reconnaissance in the Pacific and was along on a number of air raids over Japan.

His first assignment for Life in 1946 took him back to the circus: His photograph of clown Lou Jacobs with a giraffe looking over his shoulder made the magazine's cover and earned Dean a staff job.

In the era before television, Life magazine photographers had some of the most glamorous work in journalism. Life assigned him to cover Hollywood. In 1954, the magazine published one of his most memorable photos, the shot of Coward dressed for a night on the town in New York but standing alone in the stark Nevada desert.

Dean had the idea of asking Coward, who was then doing a summer engagement at the Desert Inn in Las Vegas, to pose in the desert to illustrate his song "Mad Dogs and Englishmen Go Out in the Midday Sun."

As Dean recalled in an interview with John Loengard for the book "Life Photographers: What They Saw," Coward wasn't about to partake of the midday sun. "Oh, dear boy, I don't get up until 4 o'clock in the afternoon," Dean recalled him saying.

But Dean pressed on anyway. As he related to Loengard, he rented a Cadillac limousine and filled the back seat with a tub loaded with liquor, tonic and ice cubes — and Coward.

The temperature that day reached 119 as Coward relaxed in his underwear during the drive to a spot about 15 miles from Las Vegas. According to Dean, Coward's dresser helped him into his tuxedo, resulting in the image of the elegant Coward with a cigarette holder in his mouth against his shadow on the dry lake bed.

"Splendid! Splendid! What an idea! If we only had a piano," Coward said of the shoot before hopping back in the car and stripping down to his underwear for the ride back to Las Vegas.

In 1956, Life assigned Dean to Paris. While sailing to Europe on the Ile de France, he was awakened with the news that the Andrea Doria had collided with another liner, the Stockholm.

The accident occurred close enough to Dean's liner that survivors were being brought aboard.

His photographs of the shaken voyagers and the sinking Andrea Doria were some of the first on the accident published in a U.S. magazine.

During his years in Europe, Dean photographed communist riots and fashion shows in Paris, royal weddings throughout Europe and noted authors including James Jones and William S. Burroughs.

He spent three weeks with Hemingway in Spain in 1960 for an assignment on bullfighting. In 1989, Dean published "Hemingway's Spain," about his experiences with the great writer.

In 1965, Dean won first prize in a Vatican photography contest for a picture of Pope Paul VI. The prize included an audience with the pope and $750. According to his son, it was Dean's favorite honor.

In addition to his son, he is survived by a daughter, Deborah, and two grandsons.

Instead of flowers, donations may be made to the American Child Photographer's Charity Guild (www.acpcg.com) or the Make-A-Wish Foundation.

Related material:
The Big Time

(Log 24, July 29, 2003):

A Story That Works

"There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles —
- that minds should really touch, or
- that the silent universe should speak, tell minds a story, or (perhaps the same thing)
- that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
and lastly, there is lonely, story-telling, wonder-questing, mortal me."

— Fritz Leiber in "The Button Molder"

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Wednesday, November 30, 2005

Wednesday November 30, 2005

Filed under: General,Geometry — Tags: Octad, Octad Generator — m759 @ 1:00 am

For St. Andrew’s Day

“The miraculous enters…. When we investigate these problems, some fantastic things happen….”

— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)

The remarkable Mathieu group M₂₄, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.

A picture of the Miracle Octad Generator, with my comments, is available online.

Cartoon by S.Harris

Related material:
Mathematics and Narrative.

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Friday, November 25, 2005

Friday November 25, 2005

Filed under: General,Geometry — m759 @ 9:00 pm

Holy Geometry

What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p. 10 in the Schilpp book below) that so impressed the young Albert Einstein?

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.”

(“Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am Anfang eines Schuljahres in die Hand bekam. Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die– obwohl an sich keineswegs evident– doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien. Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich.”)

— Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

From a website by Hans-Josef Küpper:

“Today it cannot be said with certainty which book is Einstein’s ‘holy geometry book.’ There are three different titles that come into question:

Theodor Spieker, 1890
Lehrbuch der ebenen Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.

Heinrich Borchert Lübsen, 1870
Ausführliches Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Selbstunterricht. Mit Rücksicht auf die Zwecke des praktischen Lebens.

Adolf Sickenberger, 1888
Leitfaden der elementaren Mathematik.

Young Albert Einstein owned all of these three books. The book by T. Spieker was given to him by Max Talmud (later: Talmey), a Jewish medic. The book by H. B. Lübsen was from the library of his uncle Jakob Einstein and the one of A. Sickenberger was from his parents.”

Küpper does not state clearly his source for the geometry-book information.

According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

    “Throughout Einstein’s five and a half years at the Luitpold Gymnasium, he was taught mathematics from one or another edition of the separately published parts of Sickenberger’s Textbook of Elementary Mathematics. When it first appeared in 1888 the book constituted a major contribution to reform pedagogy. Sickenberger based his book on twenty years of experience that in his view necessarily took precedence over ‘theoretical doubts and systematic scruples.’ At the same time Sickenberger made much use of the recent pedagogical literature, especially that published in the pages of Immanuel Carl Volkmar Hoffmann’s Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, the leading pedagogical mathematics journal of the day. Following in the tradition of the reform movement, he sought to present everything in the simplest, most intuitive way possible. He opposed introducing scientific rigour and higher approaches in an elementary text. He emphasised that he would follow neither the synthesis of Euclidean geometry nor the so-called analytical-genetic approach. He opted for a great deal of freedom in the form of presentation because he believed that a textbook was no more than a crutch for oral instruction. The spoken word, in Sickenberger’s view, could infuse life into the dead forms of the printed text. Too often, he insisted in the preface to his text, mathematics was seen and valued ‘as the pure science of reason.’ In reality, he continued, mathematics was also ‘an essential tool for daily work.’ In view of the practical dimension of mathematics Sickenberger sought most of all to present basic propositions clearly rather than to arrive at formal conciseness.   Numerous examples took the place of long, complicated, and boring generalities. In addition to the usual rules of arithmetic Sickenberger introduced diophantine equations. To solve three linear, homogeneous, first-order equations with three unknowns he specified determinants and determinant algebra. Then he went on to quadratic equations and logarithms. In the second part of his book, Sickenberger treated plane geometry.
     According to a biography of Einstein written by his step-son-in-law, Rudolf Kayser– one that the theoretical physicist described as ‘duly accurate’– when he was twelve years old Einstein fell into possession of the ‘small geometry book’ used in the Luitpold Gymnasium before this subject was formally presented to him. Einstein corroborated Kayser’s passage in autobiographical notes of 1949, when he described how at the age of twelve ‘a little book dealing with Euclidean plane geometry’ came into his hands ‘at the beginning of a school year.’ The ‘lucidity and certainty’ of plane geometry according to this ‘holy geometry booklet’ made, Einstein wrote, ‘an indescribable impression on me.’ Einstein saw here what he found in other texts that he enjoyed: it was ‘not too particular’ in logical rigour but ‘made up for this by permitting the main thoughts to stand out clearly and synoptically.’ Upon working his way through this text, Einstein was then presented with one of the many editions of Theodor Spieker’s geometry by Max Talmey, a medical student at the University of Munich who dined with the Einsteins and who was young Einstein’s friend when Einstein was between the ages of ten and fifteen. We can only infer from Einstein’s retrospective judgment that the first geometry book exerted an impact greater than that produced by Spieker’s treatment, by the popular science expositions of Aaron Bernstein and Ludwig Büchner also given to him by Talmey, or by the texts of Heinrich Borchert Lübsen from which Einstein had by the age of fourteen taught himself differential and integral calculus.
     Which text constituted the ‘holy geometry booklet’? In his will Einstein gave ‘all his books’ to his long-time secretary Helen Dukas. Present in this collection are three bearing the signature ‘J Einstein’: a logarithmic and trigonometric handbook, a textbook on analysis, and an introduction to infinitesimal calculus. The signature is that of Einstein’s father’s brother Jakob, a business partner and member of Einstein’s household in Ulm and Munich. He presented the books to his nephew Albert. A fourth book in Miss Dukas’s collection, which does not bear Jakob Einstein’s name, is the second part of a textbook on geometry, a work of astronomer Eduard Heis’s which was rewritten after his death by the Cologne schoolteacher Thomas Joseph Eschweiler. Without offering reasons for his choice Banesh Hoffmann has recently identified Heis and Eschweiler’s text as the geometry book that made such an impression on Einstein. Yet, assuming that Kayser’s unambiguous reporting is correct, it is far more likely that the geometrical part of Sickenberger’s text was what Einstein referred to in his autobiographical notes. Sickenberger’s exposition was published seven years after that of Heis and Eschweiler, and unlike the latter it appeared with a Munich press. Because it was used in the Luitpold Gymnasium, copies would have been readily available to Uncle Jakob or to whoever first acquainted Einstein with Euclidean geometry.”

What might be the modern version of a “holy geometry book”?

I suggest the following,

first published in 1940:

The image “http://www.log24.com/log/pix05B/BasicGeometry.gif” cannot be displayed, because it contains errors.

Click on picture for details.

Comments (3)

Sunday, May 22, 2005

Sunday May 22, 2005

Filed under: General — Tags: Labyrinth for Penelope — m759 @ 12:25 pm

The Shining
of Friday the 13th

From Margalit Fox in today’s New York Times:

“Eddie Barclay, who for three decades after World War II was arguably the most powerful music mogul in Europe and inarguably the most flamboyant, died on [Friday] May 13 in Paris. He was 84….

… Mr. Barclay was best known for three things: popularizing American jazz in France in the postwar years; keeping the traditional French chanson alive into the age of rock ‘n’ roll; and presiding over parties so lavish that they were considered just the tiniest bit excessive even by the standards of the French Riviera….

Among the guests at some of his glittering parties… Jack Nicholson….”

The image “http://www.log24.com/log/pix05/050522-Jack.jpg” cannot be displayed, because it contains errors.

Related material:

“Joyce’s confidant in Zurich in 1918, Frank Budgen, luckily for us described the process of writing Ulysses…. ‘Not Bloom, not Stephen is here the principal personage, but Dublin itself… All towns are labyrinths…’ While working… Joyce bought a game called Labyrinth, which he played every evening for a time with his daughter, Lucia. From this game he cataloged the six main errors of judgment into which one might fall in seeking a way out of a maze.”

— quoted by Bruce Graham from The Creators by Daniel Boorstin

“We’ll always have Paris.”

— An Invariant Feast, Log24, Sept. 6, 2004

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Wednesday, April 6, 2005

Wednesday April 6, 2005

Filed under: General — m759 @ 12:00 pm

Paratext:

A Birthday Gift for Barry Levinson
(born April 6, or maybe June 2, 1942)

The following excerpts from page 162*
in three different books
with Catholic backgrounds†
may or may not prove useful
to a film director.

Locution:

Narrative Form

162

satires, forgeries, fakes
parody
illocutionary stance
documentary novel
pseudofactual fiction
authorial reading
history of the book

Illocution:

Pocket Catholic Dictionary

162

wisdom (sapientia)
understanding (intellectus)
knowledge (scientia)
fortitude or courage (fortitudo)
counsel (consilium)
piety or love (pietas), and
fear of the Lord (timor Domini)

Perlocution:

The Nick Tosches Reader

162

The image “http://www.log24.com/log/pix05/050406-Tosches2.jpg” cannot be displayed, because it contains errors.

Never play the pizza man
for a fool.

† The seven items in the list from the

Pocket Catholic Dictionary are from the
definition of “Gifts of the Holy Spirit.”

* The page number 162 may be regarded,
in honor of the late Saul Bellow
(see previous entry), as
Humboldt’s Gift.

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Saturday, January 1, 2005

Saturday January 1, 2005

Filed under: General,Geometry — m759 @ 8:08 am

Metamorphosis

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

Comments (2)

Wednesday, December 22, 2004

Wednesday December 22, 2004

Filed under: General,Geometry — m759 @ 7:59 am

Geometry Update

Added a new section,
“How the MOG works,” to
Geometry of the 4×4 Square.

Comments (1)

Sunday, November 21, 2004

Sunday November 21, 2004

Filed under: General — Tags: on041121 — m759 @ 3:00 pm

Trinity and Counterpoint

Today's Roman Catholic meditation is from Gerry Adams, leader of Sinn Fein, the political arm of the Irish Republican Army:

"I certainly regret what happened and I make no bones about that," Adams said on the 30th anniversary of pub bombings that killed 21 on Nov. 21, 1974, in Birmingham, England.

Those who care what Roman Catholics think of the Trinity may read the remarks of St. Bonaventure at math16.com.

That site also offers a less holy but more intelligible trinity based on the irrefutable fact that 3 x 8 = 24 and on a remarkable counterpoint between group actions on a 4×2 array and group actions on a 4×4 array.

For a Protestant view of this trinity, see a website at the University of Birmingham in England.

That site's home page links to Birmingham's City Evangelical Church.

Comments (1)

Friday, October 1, 2004

Friday October 1, 2004

Filed under: General — m759 @ 4:07 pm

Ig Nobel
Geography Prize

Background from David Brooks (NYT Magazine, Aug. 29, 2004):

“The war on Islamic extremism….
This has been miscast as a war on terror, but terror is just the means our enemies use. In reality, we’re fighting a war against a specific brand of Islamic extremism, a loose federation of ideologues who seek to dominate the Middle East and return it to the days of the caliphate.

We are in the beginning of this war, where we were against Bolshevism around 1905 or Fascism in the early 1930’s, with enemies that will continue to gain strength, thanks to the demographic bulge in the Middle East….”

On last night’s Ig Nobel ceremony at Harvard:

“Coming a week before their more noble cousins are announced in Stockholm, the awards have become a keenly contested, globally reported event.”

— The Guardian, Oct. 1

From the Bush-Kerry debate last night:

“Kerry: The president just talked about Iraq as a center of the war on terror. Iraq was not even close to the center of the war on terror before the president invaded it.”

As David Brooks has pointed out, the so-called war on terror is actually a war on an Islamic extremism that seeks to dominate the Middle East. The United States government may deny, and Kerry may decry, the strategy of building permanent bases at the very center of the Middle East, Iraq (shown above), but such a strategy seems not without merit if the long-term strategic interests of the United States are placed ahead of political considerations.

Kerry’s apparent ignorance of such a strategy’s merits entitles him to this year’s Ig Nobel prize in geography– the above map.

Comments (1)

Tuesday, September 2, 2003

Tuesday September 2, 2003

Filed under: General,Geometry — Tags: Fields, Figaro's Passover — m759 @ 1:11 pm

One Ring to Rule Them All

In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.

Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:

“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of beta X (the Stone-Cech compactification of X) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’ Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”

— Leonard Gillman: An Interview

This clears up a question I asked earlier in this journal….

Wednesday, May 14, 2003

Common Sense

On the mathematician Kolmogorov:

“It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.”

— Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

the discussion of the Gelfand representation theorem on p. 397, “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich,” by Pierre Cartier, Bulletin of the American Mathematical Society, Vol. 38 (2001) No. 4, pp. 389-408,
a remark on Gelfand’s work on page 467 of the above AMS Bulletin issue,
V. S. Varadarajan‘s discussion of the “Hilbert-Gel’fand principle” on page 11 of “The Concept of a Supermanifold” and the “Gel’fand Principle” on page 11 of “What Is the Geometry of Physical Space?,” and
the excellent 1963 textbook Introduction to Topology and Modern Analysis, by George F. Simmons, chapters 13 and 14.

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses. Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case. A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR 22 (1939), 11-15. This is cited by Gillman and Jerison in the classic Rings of Continuous Functions.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area. See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”).

Any other references to Kolmogorov’s work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone. I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

A response by Richard Cudney:

“In regard to your entry, it is largely correct. The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works. So, I suppose my entry was unfair to Gelfand. You’re right, the issue of credit is a bit touchy since Gelfand was his student. In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student. At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them.

And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst. In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A. That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up.

This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras. In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique. That is to say, this whole vein of Gelfand’s work started in this joint paper.

Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge. Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third).

As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin. The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.”

Posted 5/14/2003 at 8:44 PM by rcudney

Comments (1)

Sunday, June 15, 2003

Sunday June 15, 2003

Filed under: General,Geometry — Tags: Geometric Theology, Seventh Seal, Trinity Knot — m759 @ 3:00 pm

Readings for Trinity Sunday

Triune knot:
Problems in Combinatorial Group Theory, 7 and 8, in light of the remark in Section 8.3 of Lattice Polygons and the Number 12
Cardinal Newman:
Sermon 24
Simon Nickerson:
24=8×3.

For more on the structure
discussed by Nickerson, see

Raiders of the Lost Matrix:

For theology in general, see

Jews Telling Stories.

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Thursday, May 15, 2003

Thursday May 15, 2003

Filed under: General — m759 @ 1:23 pm

Birthday Present

Today is the birthday of Emile Mathieu.
Here is a present.

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Wednesday, May 14, 2003

Wednesday May 14, 2003

Filed under: General,Geometry — Tags: Figaro's Passover — m759 @ 2:00 pm

Common Sense

On the mathematician Kolmogorov:

— Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

the discussion of the Gelfand representation theorem on p. 397, “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich,” by Pierre Cartier, Bulletin of the American Mathematical Society, Vol. 38 (2001) No. 4, pp. 389-408,
a remark on Gelfand’s work on page 467 of the above AMS Bulletin issue,
V. S. Varadarajan‘s discussion of the “Hilbert-Gel’fand principle” on page 11 of “The Concept of a Supermanifold” and the “Gel’fand Principle” on page 11 of “What Is the Geometry of Physical Space?,” and
the excellent 1963 textbook Introduction to Topology and Modern Analysis, by George F. Simmons, chapters 13 and 14.

Any other references to Kolmogorov’s work in this area would be of interest.

Comments (1)

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — 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