Log24

Saturday, July 8, 2017

Desargues and Galois in Japan

Filed under: General,Geometry — m759 @ 1:00 AM

Related material now available online —

A less business-oriented sort of virtual reality —

Link to 'Desargues via Galois' in Japan

For example, "A very important configuration is obtained by
taking the plane section of a complete space five-point." 
(Veblen and Young, 1910, p. 39)—

'Desargues via Galois' in Japan (via Pinterest)

Tuesday, August 6, 2013

Desargues via Galois

Filed under: General,Geometry — Tags: , — m759 @ 5:12 PM

The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.

IMAGE- The large Desargues configuration in light of Galois geometry

Update of Aug. 7, 2013:  See also an expanded PDF version.

Sunday, November 22, 2020

The Galois-Fano Plane

Filed under: General — Tags: — m759 @ 9:52 PM

A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —

Related material — The Eightfold Cube.

Update at 10:51 PM ET the same day —

Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .

DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.

The arXiv versions

Wednesday, May 2, 2018

Galois’s Space

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 PM

(A sequel to Foster's Space and Sawyer's Space)

See posts now tagged Galois's Space.

Sunday, November 19, 2017

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

This is a sequel to yesterday's post Cube Space Continued.

Saturday, May 20, 2017

van Lint and Wilson Meet the Galois Tesseract*

Filed under: General,Geometry — Tags: — m759 @ 12:12 AM

Click image to enlarge.

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


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

* See Galois Tesseract  in this journal.

Sunday, August 14, 2016

The Boole-Galois Games

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM

Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.

Tuesday, May 31, 2016

Galois Space —

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

A very brief introduction:

Seven is Heaven...

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.

For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

Thursday, March 20, 2014

Classical Galois

Filed under: General,Geometry — Tags: , , — m759 @ 12:26 PM

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Click image for more details.

To enlarge image, click here.

Friday, April 19, 2013

The Large Desargues Configuration

Filed under: General,Geometry — Tags: — m759 @ 9:25 AM

Desargues' theorem according to a standard textbook:

"If two triangles are perspective from a point
they are perspective from a line."

The converse, from the same book:

"If two triangles are perspective from a line
they are perspective from a point."

Desargues' theorem according to Wikipedia
combines the above statements:

"Two triangles are in perspective axially  [i.e., from a line]
if and only if they are in perspective centrally  [i.e., from a point]."

A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.

A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.

This large  Desargues configuration involves a third triangle,
needed for the proof   (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large  configuration is the
frontispiece to Volume I (Foundations)  of Baker's 6-volume
Principles of Geometry .

Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads  (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).

The third triangle, within the larger configuration,
is pictured below.

IMAGE- The proof of the converse of Desargues' theorem involves a third triangle.

Saturday, April 13, 2013

Veblen and Young in Light of Galois

Filed under: General,Geometry — m759 @ 1:00 AM

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:

IMAGE- Veblen and Young,1910 Desargues illustration, with 2013 Galois-geometry version

See also the original Veblen-Young figure in context.

Monday, April 1, 2013

Desargues via Rosenhain

Filed under: General,Geometry — Tags: , — m759 @ 6:00 PM

Background: Rosenhain and Göpel Tetrads in PG(3,2)

Introduction:

The Large Desargues Configuration

Added by Steven H. Cullinane on Friday, April 19, 2013

Desargues' theorem according to a standard textbook:

"If two triangles are perspective from a point
they are perspective from a line."

The converse, from the same book:

"If two triangles are perspective from a line
they are perspective from a point."

Desargues' theorem according to Wikipedia 
combines the above statements:

"Two triangles are in perspective axially  [i.e., from a line]
if and only if they are in perspective centrally  [i.e., from a point]."

A figure often used to illustrate the theorem, 
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.

A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line 
and 4 lines on each point.

This large  Desargues configuration involves a third triangle,
needed for the proof   (though not the statement ) of the 
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large  configuration is the
frontispiece to Volume I (Foundations)  of Baker's 6-volume
Principles of Geometry .

Point-line incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely 
by 20 Rosenhain tetrads  (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).

The third triangle, within the larger configuration,
is pictured below.

IMAGE- The proof of the converse of Desargues' theorem involves a third triangle.

 

 

A connection discovered today (April 1, 2013)—

(Click to enlarge the image below.)

Update of April 18, 2013

Note that  Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for 
further details.

(End of April 18, 2013 update.)

Update of April 14, 2013

See Baker's Proof (Edited for the Web) for a detailed explanation 
of the above picture of Baker's Desargues-theorem frontispiece.

(End of April 14, 2013 update.)

Update of April 12, 2013

A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:

IMAGE- Desargues' theorem with three triangles, and Galois-geometry version

(End of update of April 12, 2013)

Update of April 13, 2013

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
IMAGE- Veblen and Young 1910 Desargues illustration, with 2013 Galois-geometry version

See also the original Veblen-Young figure in context.

(End of update of April 13, 2013)

Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection.  For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.

For the recent  context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.

For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.

In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):

Volume I of Baker's Principles  
has a cover closely related to 
the Rosenhain tetrads in PG(3,2)
Volume IV of Baker's Principles 
has a cover closely related to
the Göpel tetrads in PG(3,2) 
Foundations
(click to enlarge)

 

 

 

Higher Geometry
(click to enlarge)

 

 

 

 

Sunday, March 10, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

(Continued)

The 16-point affine Galois space:

Further properties of this space:

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

Some closely related material:

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

    For the first two pages, click here.

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

Monday, March 4, 2013

Occupy Galois Space

Filed under: General,Geometry — Tags: — m759 @ 3:00 AM

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Thursday, February 21, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 6:00 PM

(Continued)

The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon

Illustrations—

(Click to enlarge.)

Sunday, July 29, 2012

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

(Continued)

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

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

— share the same vector-space structure:

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

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

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

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

Source:

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

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 PM

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
[Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Tuesday, July 10, 2012

Euclid vs. Galois

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM

(Continued)

Euclidean square and triangle

Galois square and triangle

Background—

This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.

Saturday, September 3, 2011

The Galois Tesseract (continued)

Filed under: General,Geometry — Tags: , , — 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—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

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

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

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:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

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

Thursday, September 1, 2011

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 7:11 PM

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

IMAGE- Review of Conway and Sloane's 'Sphere Packings...' by Rota

Friday, September 17, 2010

The Galois Window

Filed under: General,Geometry — Tags: , , — m759 @ 5:01 AM

Yesterday’s excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.

That approach will appeal to few mathematicians, so here is another.

Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace  is a book by Leonard Mlodinow published in 2002.

More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design  (published on September 7, 2010).

A review of Mlodinow’s book on geometry—

“This is a shallow book on deep matters, about which the author knows next to nothing.”
— Robert P. Langlands, Notices of the American Mathematical Society,  May 2002

The Langlands remark is an apt introduction to Mlodinow’s more recent work.

It also applies to Martin Gardner’s comments on Galois in 2007 and, posthumously, in 2010.

For the latter, see a Google search done this morning—

http://www.log24.com/log/pix10B/100917-GardnerGalois.jpg

Here, for future reference, is a copy of the current Google cache of this journal’s “paged=4” page.

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron’s web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

See also, in this  journal, Window and Window, continued (July 5 and 6, 2010).

Gardner scoffs at the importance of Galois’s last letter —

Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers.”
Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society  in March 1899 and February 1909.

Thursday, April 2, 2020

Pattern

Filed under: General — m759 @ 7:59 PM

See also Plan 9 from Yale
and Galois Desargues.

Monday, February 22, 2021

Design Theory

Filed under: General — Tags: , , , , — m759 @ 10:59 PM

A related image —

Related design theory in mathematics

http://m759.net/wordpress/?p=9221

Wednesday, December 16, 2020

Kramer’s Cross

Filed under: General — Tags: — m759 @ 12:21 AM

See Crucial Kramer and Galois Window.

Sunday, December 6, 2020

“Binary Coordinates”

Filed under: General — Tags: — m759 @ 3:09 PM

The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false  and  true , to the absence
or presence  of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .

Related material from the Web —

Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —

A related anonymous change to Wikipedia today —

The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.

Sunday, November 15, 2020

Map Methods

Filed under: General — Tags: , — m759 @ 2:04 PM

See also Priority (November 25, 2016).

Friday, September 11, 2020

Kauffman on Algebra

Filed under: General — Tags: , — m759 @ 11:07 PM

Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry.  See other posts now tagged Boole vs. Galois.

Detail, 8/14/2016 Google image search for 'Galois Boole'

See also “A Four-Color Epic” (April 16, 2020).

Friday, February 21, 2020

Frozen

Filed under: General — Tags: — m759 @ 12:30 PM

Tuesday, August 13, 2019

Putting the Structure  in Structuralism

Filed under: General — Tags: , , — m759 @ 8:34 PM

The Matrix of Lévi-Strauss —

(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée 
, No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)

The structure  of the matrix of Lévi-Strauss —

Illustration from Diamond Theory , by Steven H. Cullinane (1976).

The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.

Sunday, June 16, 2019

Master Plan from Outer Space

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

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Tuesday, October 23, 2018

Plan 9 from Inner Space

Filed under: G-Notes,General,Geometry — m759 @ 9:57 AM

Click the image for some context.

Sunday, September 9, 2018

Plan 9 Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:00 AM

"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.

Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."

— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics
(May, 1997), pp. 183-196. Published by: Springer.

Stable URL: https://www.jstor.org/stable/20117627.

Related figures —

Note the 3×3 subsquare containing the triangles ABC, etc.

"That in which space itself is contained" — Wallace Stevens

Sunday, April 29, 2018

Amusement

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

From the online New York Times  this afternoon:

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

From this  journal this morning:

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to 
make a fancy amusement-park ride
.
There had to be more—"

Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"The sensation of folding space . . . ."

Or unfolding:

Click the above unfolded space for some background.

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Sunday, March 4, 2018

The Square Inch Space: A Brief History

Filed under: General,Geometry — Tags: , — m759 @ 11:21 AM

1955  ("Blackboard Jungle") —

1976 —

2009 —

2016 —

 Some small Galois spaces (the Cullinane models)

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • “The digital revolution based on the 2 symbols (0,1)”
  • “The algebra of George Boole”
  • “Binary codes”
  • “Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Thursday, January 25, 2018

Beware of Analogical Extension

Filed under: General,Geometry — Tags: — m759 @ 11:29 AM

"By an archetype  I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."

— Max Black in Models and Metaphors 
    (Cornell, 1962, p. 241)

"Others … spoke of 'ultimate frames of reference' …."
Ibid.

A "frame of reference" for the concept  four quartets

A less reputable analogical extension  of the same
frame of reference

Madeleine L'Engle in A Swiftly Tilting Planet :

"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."

See also the phrase Galois tesseract .

Friday, September 29, 2017

Principles Before Personalities*

Filed under: General,Geometry — m759 @ 12:00 PM

(Some Remarks for Science Addicts)

Principles —

IMAGE- The large Desargues configuration in light of Galois geometry

Personalities —

* See "Tradition Twelve."

Saturday, September 23, 2017

The Turn of the Frame

Filed under: General,Geometry — Tags: , , — m759 @ 2:19 AM

"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."

— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies  No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.

See also the previous post and The Galois Tesseract.

Friday, September 15, 2017

Space Art

Filed under: General,Geometry — Tags: — m759 @ 2:05 PM

Silas in "Equals" (2015) —

Ever since we were kids it's been drilled into us that 
Our purpose is to explore the universe, you know.
Outer space is where we'll find 
…  the answers to why we're here and 
…  and where we come from.

Related material — 

'The Art of Space Art' in The Paris Review, Sept. 14, 2017

See also Galois Space  in this  journal.

Sunday, August 27, 2017

Black Well

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.

Related material —

The Galois Tesseract and, more generally, Binary Coordinate Systems.

Friday, August 11, 2017

Symmetry’s Lifeboat

Filed under: General,Geometry — Tags: , , — m759 @ 9:16 PM

A post suggested by the word tzimtzum  (see Wednesday)
or tsimtsum  (see this morning) —

Lifeboat from the Tsimtsum  in Life of Pi  —

Another sort of tsimtsum, contracting infinite space to a finite space —

IMAGE- Desargues's theorem in light of Galois geometry

Tuesday, July 11, 2017

A Date at the Death Cafe

Filed under: General,Geometry — Tags: , — m759 @ 8:48 PM

The New York TImes  reports this evening that
"Jon Underwood, Founder of Death Cafe Movement,"
died suddenly at 44 on June 27. 

This  journal on that date linked to a post titled "The Mystic Hexastigm."

A related remark on the complete 6-point   from Sunday, April 28, 2013

(See, in Veblen and Young's 1910 Vol. I, exercise 11,
page 53: "A plane section of a 6-point in space can  
be considered as 3 triangles perspective in pairs
from 3 collinear points with corresponding sides
meeting in 3 collinear points." This is the large  
Desargues configuration. See Classical Geometry
in Light of Galois Geometry
.)

This  post was suggested, in part, by the philosophical ruminations
of Rosalind Krauss in her 2011 book Under Blue Cup . See 
Sunday's post  Perspective and Its Transections . (Any resemblance
to Freud's title Civilization and Its Discontents  is purely coincidental.)

Sunday, July 9, 2017

Perspective and Its Transections

Filed under: General,Geometry — m759 @ 5:27 PM

The title phrase is from Rosalind Krauss (Under Blue Cup , 2011) —

Another way of looking at the title phrase —

"A very important configuration is obtained by
taking the plane section of a complete space five-point." 
(Veblen and Young, 1910, p. 39) —

'Desargues via Galois' in Japan (via Pinterest) 

For some context, see Desargues + Galois in this journal.

Monday, June 26, 2017

Four Dots

Analogies — “A : B  ::  C : D”  may be read  “A is to B  as  C is to D.”

Gian-Carlo Rota on Heidegger…

“… The universal as  is given various names in Heidegger’s writings….

The discovery of the universal as  is Heidegger’s contribution to philosophy….

The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”

— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
See also Four Dots in this journal.

Some context:  McLuhan + Analogy.

Saturday, June 3, 2017

Expanding the Spielraum (Continued*)

Filed under: General,Geometry — Tags: — m759 @ 1:13 PM

Or:  The Square

"What we do may be small, but it has
 a certain character of permanence."
— G. H. Hardy

* See Expanding the Spielraum in this journal.

Tuesday, May 23, 2017

Pursued by a Biplane

Filed under: General,Geometry — Tags: — m759 @ 9:41 PM

The Galois Tesseract as a biplane —

Cary Grant in 'North by Northwest'

Saturday, May 20, 2017

The Ludicrous Extreme

Filed under: General,Geometry — Tags: — m759 @ 1:04 AM

From a review of the 2016 film "Arrival"

"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."

Jordan Brower in the Los Angeles Review of Books

Further on in the review —

"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."

For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —

van Lint and Wilson Meet the Galois Tesseract.

The death process of van Lint occurred on Sept. 28, 2004.

See this journal on that date

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Friday, April 28, 2017

A Generation Lost in Space

Filed under: General,Geometry — m759 @ 2:00 PM

The title is from Don McLean's classic "American Pie."

A Finite Projective Space —

A Non-Finite Projective Space —

Thursday, April 27, 2017

Partner, Anchor, Decompose

Filed under: General,Geometry — Tags: — m759 @ 12:31 PM

See also a figure from 2 AM ET April 26 

" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "

Greg Gutfeld, September 2014

Thursday, April 20, 2017

Stone Logic

Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

See also “Romancing the Omega” —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Related mathematics — Guitart in this journal —

From 'Moving Logic, from Boole to Galois,' by René Guitart, 2005

See also Weyl + Palermo in this journal —

http://www.log24.com/log/pix11B/110922-TriquetrumCube.jpg

Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM

This post’s title is from the tags of the previous post

 

The title’s “shift” is in the combined concepts of

Space and Number

From Finite Jest (May 27, 2012):

IMAGE- History of Mathematics in a Nutshell

The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —

Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange

io9 , July 29, 2016

” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego Comic-Con . . . .

‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “

[Typo now corrected, except in a comment.]

Wednesday, October 5, 2016

Sources

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

From a Google image search yesterday

Sources (left to right, top to bottom) —

Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)

Wednesday, August 24, 2016

Core Statements

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

"That in which space itself is contained" — Wallace Stevens

An image by Steven H. Cullinane from April 1, 2013:

The large Desargues configuration of Euclidean 3-space can be 
mapped canonically to the 4×4 square of Galois geometry —

'Desargues via Rosenhain'- April 1, 2013- The large Desargues configuration mapped canonically to the 4x4 square

On an Auckland University of Technology thesis by Kate Cullinane —
On Kate Cullinane's book 'Sample Copy' - 'The core statement of this work...'
The thesis reportedly won an Art Directors Club award on April 5, 2013.

Tuesday, August 16, 2016

Midnight Narrative

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —


Boulez and Boole      and

Cross and Boolean lattice.

The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Saturday, June 18, 2016

Midnight in Herald Square

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 AM

In memory of New Yorker  artist Anatol Kovarsky,
who reportedly died at 97 on June 1.

Note the Santa, a figure associated with Macy's at Herald Square.

See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.

A note related both to Galois space and to
the "Herald Square"-tagged posts —

"There is  such a thing as a length-16 sequence."
— Saying adapted from a young-adult novel.

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface
.

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Sunday, May 8, 2016

The Three Solomons

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:00 PM

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick’s Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word “conceptual”
in his approach to “conceptual art.”

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

THE SQUARE AND THE CUBE
by Sol LeWitt

“The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.”

Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)”

See also the Cullinane models of some small Galois spaces

Some small Galois spaces (the Cullinane models)

Friday, May 6, 2016

Review

Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

 Some small Galois spaces (the Cullinane models)

Thursday, April 14, 2016

One Funeral at a Time

Filed under: General,Geometry — Tags: — m759 @ 1:37 PM

On this date in 2005, mathematician Saunders Mac Lane died at 95.

Related material —

Max Planck quotations:

Mac Lane on Boolean algebra:

Mac Lane’s summary chart (note the absence of Galois geometry ):

I disagree with Mac Lane’s assertion that “the finite models of
Boolean algebra are dull.”  See Boole vs. Galois in this journal.

Wednesday, January 13, 2016

Geometry for Jews

Filed under: General,Geometry — Tags: — m759 @ 7:45 AM

(Continued from previous episodes)

'Games Played by Boole and Galois'

Boole and Galois also figure in the mathematics of space
i.e. , geometry.  See Boole + Galois in this journal.

Related material, according to Jung’s notion of synchronicity —

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: , — m759 @ 3:15 PM

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.

A Google search today for Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.

Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978

Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981

An illustration from the Lechner paper above —

“There is  such a thing as harmonic analysis of switching functions.”

— Saying adapted from a young-adult novel

Monday, December 28, 2015

ART WARS Continues

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

Combining two headlines from this morning’s
New York Times  and Washington Post , we have

Deceptively Simple Geometries
on a Bold Scale

     Voilà —

Click image for details.

More generally, see
Boole vs. Galois.

Friday, December 25, 2015

Dark Symbol

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

Related material:

The previous post (Bright Symbol) and
a post from Wednesday,
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.

“An analogy between mathematics and religion is apposite.”

— Harvard Magazine  review by Avner Ash of
Mathematics without Apologies
(Princeton University Press, January 18, 2015)

Wednesday, December 23, 2015

Splitting Apart

Filed under: General,Geometry — Tags: , — m759 @ 1:01 PM

Bleecker Street logo —

Click image for some background.

Related remarks on mathematics:

Boole vs. Galois

Sunday, December 13, 2015

The Monster as Big as the Ritz

Filed under: General,Geometry — Tags: , , — m759 @ 11:30 AM

“The colorful story of this undertaking begins with a bang.”

— Martin Gardner on the death of Évariste Galois

Tuesday, December 1, 2015

Pascal’s Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 12:01 AM

See a search for "large Desargues configuration" in this journal.

The 6 Jan. 2015 preprint "Danzer's Configuration Revisited," 
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the 
interesting context of Pascal's Hexagrammum Mysticum .

They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .

This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.

The duality was, as they note, previously described in 1898.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Related material on six-set geometry from a more recent source —

Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage

Monday, November 2, 2015

The Devil’s Offer

Filed under: General,Geometry — Tags: , — m759 @ 11:09 AM

This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space.  From the latter —

” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “

George Boole in image posted on All Souls' Day 2015

Colorful Story

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."

— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575

Or gold letters.

From a search for Seagram in this  journal —

Seagram VO ad, image posted on All Souls's Day 2015

A Seagram 'colorful tale'

"The colorful story of this undertaking begins with a bang."

— Martin Gardner on the death of Évariste Galois

Saturday, October 31, 2015

Raiders of the Lost Crucible

Filed under: General,Geometry — Tags: , — m759 @ 10:15 AM

Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —

Paraconsistent Logic

“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”

This  journal on the date Friday, April 5, 2013 —

The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .

For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching  enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching  studies is,
I maintain, not Boolean algebra  but rather Galois geometry.

See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.

Saturday, October 24, 2015

Two Views of Finite Space

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

The following slides are from lectures on “Advanced Boolean Algebra” —

The small Boolean  spaces above correspond exactly to some small
Galois  spaces. These two names indicate approaches to the spaces
via Boolean algebra  and via Galois geometry .

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

” ‘All you need to do is give me your soul:  give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “

Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977.  That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra  than about Galois geometry .

For later remarks on diamond theory, see finitegeometry.org/sc.

Wednesday, October 21, 2015

Algebra and Space

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space —

"The Quality Without a Name" (Log24, August 26, 2015).

Sunday, September 6, 2015

Elementally, My Dear Watson

Filed under: General,Geometry — Tags: — m759 @ 9:45 AM

Sarah Larson in the online New Yorker  on Sept. 3, 2015,
discussed Google’s new parent company, “Alphabet”—

“… Alphabet takes our most elementally wonderful
general-use word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm.”

Emma Watson in “The Bling Ring”

This journal, also on September 3 —

Thursday, September 3, 2015

Rings of August

Filed under: Uncategorized — m759 @ 7:20 AM

For the title, see posts from August 2007
tagged Gyges.

Related theological remarks:

Boolean  spaces (old)  vs. Galois  spaces (new)  in 
The Quality Without a Name. . . .

* Actually, Sarah, that would be “phonemes.”

Friday, September 4, 2015

Space Program

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Galois via Boole

(Courtesy of Intel)

Thursday, September 3, 2015

Rings of August

Filed under: General,Geometry — Tags: , — m759 @ 7:20 AM

For the title, see posts from August 2007 tagged Gyges.

Related theological remarks:

Boolean  spaces (old) vs. Galois  spaces  (new) in
The Quality Without a Name
(a post from August 26, 2015) and the

Related literature:  A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.

Wednesday, August 26, 2015

“The Quality Without a Name”

Filed under: General,Geometry — Tags: , , — m759 @ 8:00 AM

The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander’s book
The Timeless Way of Building  (Oxford University Press, 1979).

A quote from the publisher:

“Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself.”

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of “Boolean space”:

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the “unfolding,” as Christopher Alexander would have it.

These “Boolean” spaces of 1, 2, 4, 8, and 16 points
are also Galois  spaces.  See the diamond theorem —

Friday, August 14, 2015

Discrete Space

Filed under: General,Geometry — Tags: — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Tuesday, June 9, 2015

Colorful Song

Filed under: General,Geometry — Tags: — m759 @ 8:40 PM

For geeks* —

Domain, Domain on the Range , "

where Domain = the Galois tesseract  and
Range = the four-element Galois field.

This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.

* A term from the 1947 film "Nightmare Alley."

Sunday, May 31, 2015

Foundations

Filed under: General,Geometry — m759 @ 6:00 AM

IMAGE- Desargues's theorem in light of Galois geometry

To enlarge image, click here.

Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: , — m759 @ 12:31 AM

The incidences of points and planes in the
Möbius 8 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.* 
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —

Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his 
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots.  The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points  (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.

Here a plane  (represented by a dotless intersection) contains
the four points  that are represented in the square array as lying
in the same row or same column as the plane. 

The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube. 

In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.

Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in 
Coxeter's labels above may be viewed as naming the positions 
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.  In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

*  "Self-Dual Configurations and Regular Graphs," 
    Bulletin of the American Mathematical Society,
    Vol. 56 (1950), pp. 413-455

The subscripts' usual 1-2-3-4 order is reversed as a reminder
    that such a vector may be viewed as labeling a binary number 
    from 0  through 15, or alternately as labeling a polynomial in
    the 16-element Galois field GF(24).  See the Log24 post
     Vector Addition in a Finite Field (Jan. 5, 2013).

Monday, January 5, 2015

Gitterkrieg*

Filed under: General,Geometry — Tags: , — m759 @ 2:00 PM
 

Wednesday, March 13, 2013

Blackboard Jungle

Filed under: Uncategorized — m759 @ 8:00 AM 

From a review in the April 2013 issue of
Notices of the American Mathematical Society

"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."

From the book under review—

"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor 
of the place."

— Lockhart, Paul (2009-04-01). 
A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form 
 (p. 92).
Bellevue Literary Press. Kindle Edition. 

Related material: Blackboard Jungle in this journal.

See also Galois Space and Solomon's Mines.

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:

I showed my masterpiece to the
grown-ups, and asked them whether
the drawing frightened them.

But they answered: 'Why should
anyone be frightened by a hat?'"

The Little Prince

* For the title, see Plato Thanks the Academy (Jan. 3).

Monday, December 29, 2014

Dodecahedron Model of PG(2,5)

Filed under: General,Geometry — Tags: , , — m759 @ 2:28 PM

Recent posts tagged Sagan Dodecahedron 
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.  

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: General,Geometry — Tags: , , — m759 @ 2:23 PM

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Wednesday, December 3, 2014

Pyramid Dance

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).

My response —

Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid"

and remarks from a Log24 post of August 14, 2013 :

Norway dance (as interpreted by an American)

IMAGE- 'The geometry of the dance' is that of a tetrahedron, according to Peter Pesic

I prefer a different, Norwegian, interpretation of "the dance of four."

Related material:
The clash between square and tetrahedral versions of PG(3,2).

See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —

(Click image below to enlarge.)

Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :

From On Art and Magic (May 5, 2011) —

http://www.log24.com/log/pix11A/110505-ThemeAndVariations-Hofstadter.jpg

http://www.log24.com/log/pix11A/110505-BlockDesignTheory.jpg

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows  symbol—
Two blocks short of  a design.

 

(Updated at about 7 PM ET on Dec. 3.)

Sunday, November 30, 2014

Two Physical Models of the Fano Plane

Filed under: General,Geometry — Tags: , — m759 @ 1:23 AM

The Regular Tetrahedron

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains 
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three 
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book pp. 16-17.)

The Cube

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the 
other two members.

(This is the eightfold cube  discussed at finitegeometry.org.)

Wednesday, November 26, 2014

A Tetrahedral Fano-Plane Model

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

Update of Nov. 30, 2014 —

It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by 
Burkard Polster (Springer, 1998).

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" 
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

Class Act

Filed under: General,Geometry — Tags: — m759 @ 7:18 AM

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie  I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge, 
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science 
, 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…


… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled.  So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge.  It’s been a rich life.  I’m grateful. 
 
Steve
 

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

Monday, September 22, 2014

Space

Filed under: General,Geometry — Tags: — m759 @ 11:17 AM

Review of an image from a post of May 6, 2009:

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Sunday, September 21, 2014

Uncommon Noncore

Filed under: General,Geometry — m759 @ 10:30 AM

This post was suggested by Greg Gutfeld’s Sept. 4 remarks on Common Core math.

Problem: What is 9 + 6 ?

Here are two approaches suggested by illustrations of Desargues’s theorem.

Solution 1:

9 + 6 = 10 + 5,
as in Common Core (or, more simply, as in common sense), and
10 + 5 = 5 + 10 = 15 as in Veblen and Young:

Solution 2:

In the figure below,
9 + 6 = no. of  V’s + no. of  A’s + no. of C’s =
no. of nonempty squares = 16 – 1 = 15.
(Illustration from Feb. 10, 2014.)

The silly educationists’ “partner, anchor, decompose” jargon
discussed by Gutfeld was their attempt to explain “9 + 6 = 10 + 5.”

As he said of the jargon, “That’s not math, that’s the plot from ‘Silence of the Lambs.'”

Or from Richard, Frank, and Marcus in last night’s “Intruders”
(BBC America, 10 PM).

Sunday, September 14, 2014

Sensibility

Filed under: General,Geometry — Tags: , — m759 @ 9:26 AM

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Sunday, August 31, 2014

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc.  on
15 June 1974).  Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history: 

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

[Rewritten for clarity on Sept. 3, 2014.]

Sunday, July 20, 2014

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

Paradigms of Geometry:
Continuous and Discrete

The discovery of the incommensurability of a square’s
side with its diagonal contrasted a well-known discrete 
length (the side) with a new continuous  length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous  configuration at
left is embodied in the discrete  unit cells of the square at right.

IMAGE- Concepts of Space: The Large Desargues Configuration, the Related 4x4 Square, and the 4x4x4 Cube

See Desargues via Galois (August 6, 2013).

Sunday, July 6, 2014

Sticks and Stones

Filed under: General,Geometry — Tags: — m759 @ 6:29 AM

The title is from this morning’s previous post.

From a theater review in that post—

… “all flying edges and angles, a perpetually moving and hungry soul”

… “a formidably centered presence, the still counterpoint”

A more abstract perspective:

IMAGE- Concepts of Space

See also Desargues via Galois (August 6, 2013).

Sunday, June 8, 2014

Vide

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Tuesday, May 20, 2014

Play

Filed under: General,Geometry — m759 @ 7:47 PM

From a recreational-mathematics weblog yesterday:

“This appears to be the arts section of the post,
so I’ll leave Martin Probert’s page on
The Survival, Origin and Mathematics of String Figures
here. I’ll be back to pick it up at the end. Maybe it’d like
to play with Steven H. Cullinane’s pages on the
Finite Geometry of the Square and Cube.”

I doubt they would play well together.

Perhaps the offensive linking of  the purely recreational topic
of string figures to my own work was suggested by the
string figures’ resemblance to figures of projective geometry.

A pairing I prefer:  Desargues and Galois —

IMAGE- Concepts of Space: The large Desargues configuration and two figures illustrating Cullinane models of Galois geometry

For further details, see posts on Desargues and Galois.

Monday, February 10, 2014

Mystery Box III: Inside, Outside

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:28 PM

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

The Box

Inside the Box

Outside the Box

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

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

Saturday, August 17, 2013

Up-to-Date Geometry

Filed under: General,Geometry — Tags: , — m759 @ 7:24 PM

The following excerpt from a January 20, 2013, preprint shows that
a Galois-geometry version of the large Desargues 154203 configuration,
although based on the nineteenth-century work of Galois* and of Fano,** 
may at times have twenty-first-century applications.

IMAGE- James Atkinson, Jan. 2013 preprint on Yang-Baxter maps mentioning finite geometry

Some context —

Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 154203 configuration.
See my own Classical Geometry in Light of Galois Geometry.  Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.

* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
  Bulletin des Sciences Mathématiques de M. Férussac,
  Vol. 13, 1830, pp. 428-435.

** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
    Giornale di Matematiche, Vol. 30, 1892, pp. 106-132.

Sunday, July 28, 2013

Sermon

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

(Simplicity continued)

"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by  the
demonstration or the metaphor.

What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already-
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."

— Jan Zwicky, Wisdom & Metaphor , page 36 (left)

Zwicky illustrates this with Plato's diamond figure
​from the Meno  on the facing page— her page 36 (right).

A more sophisticated geometrical figure—

Galois-geometry key to
Desargues' theorem:

   D   E   F
 S'  P Q R
 S  P' Q' R'
 O  P1 Q1 R1

For an explanation, see 
Classical Geometry in Light of Galois Geometry.

Sunday, May 19, 2013

Sermon

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

Best vs. Bester

The previous post ended with a reference mentioning Rosenhain.

For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).

From the next day, April 2, 2013:

"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "

– Gian-Carlo Rota in Indiscrete Thoughts (1997)

Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
IMAGE- Rota's review of 'Sphere Packings, Lattices and Groups'-- in a word, 'best'

See, too, in the Conway-Sloane book, the Galois tesseract  
and, in this journal, Geometry for Jews and The Deceivers , by Bester.

Sunday, April 28, 2013

The Octad Generator

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM

… And the history of geometry  
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.

(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)

Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:

"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."

Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black  points and dashed  lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.

In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues '  theorem, but
rather of Brianchon 's theorem and of the Pascal  hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can  be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large  Desargues configuration. See Classical Geometry in Light of 
Galois Geometry
.)

For this large  Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large  Desargues configuration
to the Galois  geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator  and the large Mathieu group M24 —

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

See also Note on the MOG Correspondence from April 25, 2013.

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Wednesday, April 10, 2013

Caution: Slow Art

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."

— Geoff Dyer, review of Point Omega

Related material:

A phrase of critic Robert Hughes,
"slow art," in this journal.

A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:

The 3x3 square

The above 3×3 grid is embedded in a 
somewhat more sophisticated example
of conceptual art from April 1, 2013:

IMAGE- A Galois-geometry key to Desargues' theorem

Update of April 12, 2013

The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.

A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:

IMAGE- Desargues' theorem with three triangles (the large Desargues configuration) and Galois-geometry version

Wednesday, April 3, 2013

Museum Piece

Filed under: General,Geometry — Tags: — m759 @ 3:01 PM

Roberta Smith in 2011 on the American Folk Art Museum (see previous post):

"It could be argued that we need a museum of folk art
the way we need a museum of modern art,
to shine a very strong, undiluted light on
a very important achievement."

Some other aesthetic remarks:

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

— Robert Hughes, speech of June 2, 2004,
     quoted here June 15, 2007.

Perhaps, as well as museums of modern art and of folk art,
we need a Museum of Slow Art. 

One possible exhibit, from this journal Monday:

The diagram on the left is from 1922.  The 20 small squares at right
that each have 4 subsquares darkened were discussed, in a different
context, in 1905. They were re-illustrated, in a new context
(Galois geometry), in 1986. The "key" square, and the combined
illustration, is from April 1, 2013. For deeper background, see
Classical Geometry in Light of Galois Geometry.

Those who prefer faster art may consult Ten Years After.

Tuesday, April 2, 2013

Rota in a Nutshell

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "

— Gian-Carlo Rota in Indiscrete Thoughts (1997)

Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:

IMAGE- Rota's review of 'Sphere Packings, Lattices and Groups'-- in a word, 'best'

Related material:

Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.

See also Rota and Psychoshop.

Baker on Configurations

Filed under: General,Geometry — Tags: , — m759 @ 11:11 AM

The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as

Classical Geometry in Light of Galois Geometry.

Some background:

See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212-218)—

On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem

and Vol. II, Note II (pp. 219-236)—

On the Hexagrammum Mysticum  of Pascal.

Monday's elucidation of Baker's Desargues-theorem figure
treats the figure as a 15420configuration (15 points, 
4 lines on each, and 20 lines, 3 points on each).

Such a treatment is by no means new. See Baker's notes
referred to above, and 

"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.

What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).

See also Cremona's kernel, or nocciolo :

Baker on Cremona's approach to Pascal—

"forming, in Cremona's phrase, the nocciolo  of the whole."

IMAGE- Definition of 'nocciolo' as 'kernel'

A related nocciolo :

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Click on the nocciolo  for some
geometric background.

Tuesday, February 26, 2013

Publication

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and 
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'

'Didn’t publish!'

That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."

— George Steiner, 2002 talk at York University

Related material

See also Steiner on Galois.

Les Miserables  at the Academy Awards

Tuesday, February 5, 2013

Arsenal

The previous post discussed some fundamentals of logic.

The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not  the algebra
needed for Galois geometry . See below.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.

See also yesterday’s Universe of Discourse.

Monday, August 13, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 3:33 PM

(An episode of Mathematics and Narrative )

A report on the August 9th opening of Sondheim's Into the Woods

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning  "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

Compare and contrast…

1.  The following item from Walpurgisnacht 2012

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

2.  The six partitions of a tesseract's 16 vertices 
       into four parallel faces in Diamond Theory in 1937

Tuesday, May 29, 2012

The Shining of May 29

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

(Continued from May 29, 2002)

May 29, 1832—

http://www.log24.com/log/pix12A/120529-Galois-Signature-500w.jpg

Évariste Galois, Lettre de Galois à M. Auguste Chevalier

Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Martin Gardner on the above letter—

"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."

The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.

Commentary from Dec. 2011 on Gardner's word "published" —

(Click to enlarge.)

IMAGE- Peter M. Neumann, 'Galois and His Groups,' EMS Newsletter, Dec. 2011

Sunday, October 30, 2011

Sermon

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

Part I: Timothy Gowers on equivalence relations

Part II: Martin Gardner on normal subgroups

Part III: Evariste Galois on normal subgroups

"In all the history of science there is no completer example
 of the triumph of crass stupidity over untamable genius…."

— Eric Temple Bell, Men of Mathematics

See also an interesting definition and Weyl on Galois.

Update of 6:29 PM EDT Oct. 30, 2011—

For further details, see Herstein's phrase
"a tribute to the genius of Galois."

Thursday, September 8, 2011

Starring the Diamond

Filed under: General,Geometry — m759 @ 2:02 PM

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines.
"
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6

The Diamond Star

http://www.log24.com/log/pix11B/110905-StellaOctangulaView.jpg

The above is a version of a figure from Configurations and Squares.

Yesterday's post related the the Pappus configuration to this figure.

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.

Some excerpts from Coxeter—

http://www.log24.com/log/pix11B/110908-Coxeter83.jpg

The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—

http://www.log24.com/log/pix11B/110908-Coxeter83part2.jpg

(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues  configuration—

http://www.log24.com/log/pix11B/110908-Coxeter103.jpg

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts

"The value  of Desargues' theorem and the reason  why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities  that relate geometry and algebra in unexpected ways."

Wednesday, October 20, 2010

Celebration of Mind

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM
 

"Why the Celebration?"

"Martin Gardner passed away on May 22, 2010."

IMAGE-- Imaginary movie poster- 'The Galois Connection'- from stoneship.org

Imaginary movie poster from stoneship.org

Context— The Gardner Tribute.

Monday, September 27, 2010

The Social Network…

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

… In the Age of Citation

1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).

— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology,  Vol. 23, pp. 127–146 (1993)

From this paper's CiteSeer page

Citations

766  Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994
100 The act of creation – Koestler – 1964
 75 Visual Thinking – Arnheim – 1969

Visual Image of the Problem—

From a Google search today:

http://www.log24.com/log/pix10B/100927-GardnerGaloisSearch.jpg

Related material—

http://www.log24.com/log/pix10B/100927-GoogleBirthdayCake.jpg

"It is better to light one candle…"

"… the early favorite for best picture at the Oscars" — Roger Moore

Tuesday, July 6, 2010

What “As” Is

Filed under: General,Geometry — Tags: , , , , — m759 @ 8:00 PM

or:  Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)

“Dasein’s full existential structure is constituted by
the ‘as-structure’ or ‘well-joined structure’ of the rift-design*…”

— Gary Williams, post of January 22, 2010

Background—

Gian-Carlo Rota on Heidegger…

“… The universal as  is given various names in Heidegger’s writings….

The discovery of the universal as  is Heidegger’s contribution to philosophy….

The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”

— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts

… and projective points as separating rifts

Image-- The Three-Point Line: A Finite Projective Space

    Click image for details.

* rift-design— Definition by Deborah Levitt

Rift.  The stroke or rending by which a world worlds, opening both the ‘old’ world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”

— “Heidegger and the Theater of Truth,” in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998

Window, continued

“Simplicity, simplicity, simplicity!
I say, let your affairs be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail.”
— Henry David Thoreau, Walden

This quotation is the epigraph to
Section 1.1 of Alexandre V. Borovik’s
Mathematics Under the Microscope:
Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society,
Jan. 15, 2010, 317 pages).

From Peter J. Cameron’s review notes for
his new course in group theory

http://www.log24.com/log/pix10A/100705-CameronExample.jpg

From Log24 on June 24

Geometry Simplified

Image-- The Four-Point Plane: A Finite Affine Space
(an affine  space with subsquares as points
and sets  of subsquares as hyperplanes)

Image-- The Three-Point Line: A Finite Projective Space
(a projective  space with, as points, sets
of line segments that separate subsquares)

Exercise

Show that the above geometry is a model
for the algebra discussed by Cameron.

Monday, July 5, 2010

Window

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

“Examples are the stained-glass
windows of knowledge.” — Nabokov

Image-- Example of group actions on the set Omega of three partitions of a 4-set into two 2-sets

Related material:

Thomas Wolfe and the
Kernel of Eternity

Thursday, June 24, 2010

Midsummer Noon

Filed under: General,Geometry — Tags: , , — 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 —

http://www.log24.com/log/pix10A/100624-The4PointPlane.bmp
(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 —

http://www.log24.com/log/pix10A/100624-TrivialSpaces.bmp

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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