# Log24

## Saturday, October 24, 2020

### The Galois Tesseract

Filed under: General — Tags: — m759 @ 9:32 AM

Stanley E. Payne and J. A. Thas in 1983* (previous post) —

“… a 4×4 grid together with
the affine lines on it is AG(2,4).”

Payne and Thas of course use their own definition
of affine lines on a grid.

Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).

For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.

* And 26 years later,  in 2009.

## 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.

## 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 Andries Brouwer, preprint, 1982: "The Witt designs, Golay codes and Mathieu groups" (unpublished as of 2013) Pages 8-9: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24).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.

## Sunday, July 29, 2012

### The Galois Tesseract

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

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

— 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
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.

## 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—

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—

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

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.”

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

* For instance:

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

## Thursday, September 1, 2011

### The Galois Tesseract

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

## 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.

## 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.

## 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 —

"Open the pod bay doors, HAL."

## 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.

## 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:

## 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, 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.

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)).

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.

## Sunday, March 10, 2013

### Galois Space

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

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).

• 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:

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

The previous post suggests two sayings:

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

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

Illustrations—

## 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

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

## 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)—

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

Background—

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

## 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—

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.

## Saturday, March 7, 2020

### The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — Tags: — m759 @ 6:32 PM

From “Mathieu Moonshine and Symmetry Surfing” —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1)  Institute for Theoretical Physics, ETH Zurich
(2)  Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

“This presentation of the symmetry groups Gi  is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose.”

This “octad group” is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, “Symmetry invariance in a
diamond ring
,” by Steven H. Cullinane in Notices of the American Mathematical
Society
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the “octad group” in 1993 —

## Monday, January 27, 2020

### Jewel Box

Filed under: General — Tags: — m759 @ 9:02 PM

The phrase "jewel box" in a New York Times  obituary online this afternoon
suggests a review. See "And He Built a Crooked House" and Galois Tesseract.

## Monday, March 11, 2019

### Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 PM

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

## Monday, October 15, 2018

### History at Bellevue

Filed under: General,Geometry — Tags: , — m759 @ 9:38 PM

The previous post, "Tesserae for a Tesseract," contains the following
passage from a 1987 review of a book about Finnegans Wake

"Basically, Mr. Bishop sees the text from above
and as a whole — less as a sequential story than
as a box of pied type or tesserae for a mosaic,
materials for a pattern to be made."

A set of 16 of the Wechsler cubes below are tesserae that
may be used to make patterns in the Galois tesseract.

Another Bellevue story —

“History, Stephen said, is a nightmare
from which I am trying to awake.”

— James Joyce, Ulysses

## Thursday, June 21, 2018

### Models of Being

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

A Buddhist view —

“Just fancy a scale model of Being
made out of string and cardboard.”

— Nanavira Thera, 1 October 1957,
on a model of Kummer’s Quartic Surface
mentioned by Eddington

A Christian view —

A formal view —

From a Log24 search for High Concept:

## Monday, June 11, 2018

### Arty Fact

Filed under: General,Geometry — Tags: , , — m759 @ 10:35 PM

The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.

The Eye of ARTI —

— and a post of June 6, "Geometry for Goyim" —

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8

An arty fact I prefer, suggested by the triangular computer-eye forms above —

This is from the July 29, 2012, post The Galois Tesseract.

See as well . . .

## 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…."

## 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.

## 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.

## 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 —

## 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."

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 —

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

## Tuesday, May 2, 2017

### Image Albums

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

Pinterest boards uploaded to the new m759.net/piwigo

Update of May 2 —

Update of May 3 —

Update of May 8 —

Art Space board created at Pinterest

## 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)

## Tuesday, June 9, 2015

### Colorful Song

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

For geeks* —

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."

## 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).

## Tuesday, March 24, 2015

### Hirzebruch

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

(Continued from July 16, 2014.)

Some background from Wikipedia:

"Friedrich Ernst Peter Hirzebruch  ForMemRS[2]
(17 October 1927 – 27 May 2012)
was a
German mathematician, working in the fields of topology
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'

[3][4][5][6][7][8][9][10][11]"

A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch
, a book
reportedly published on September 19, 2012 —

 "Sadly, on May 27 this year, Friedrich Hirzebruch, on whose lectures this book is partially based, passed away. I would like to express my gratitude and my admiration by dedicating this book to his memory. Hannover, July 2012               Wolfgang Ebeling " (Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, Leibniz Universität Hannover, Germany)

## Monday, March 23, 2015

### Gallucci’s Möbius Configuration

Filed under: General,Geometry — Tags: — m759 @ 12:05 PM

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

(Click image to enlarge.)

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

Update of Thursday, March 26, 2015 —

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

## Monday, January 26, 2015

### Savior for Atheists…

Filed under: General,Geometry — m759 @ 5:26 PM

Continued from June 17, 2013
(
John Baez as a savior for atheists):

As an atheists-savior, I prefer Galois

The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois  geometry —

See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:

For the genesis of the figure,
see The Geometry of Logic.

## Friday, December 5, 2014

### Wittgenstein’s Picture

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

From Zettel  (repunctuated for clarity):

249. « Nichts leichter, als sich einen 4-dimensionalen Würfel
vorstellen! Er schaut so aus… »

"Nothing easier than to imagine a 4-dimensional cube!
It looks like this…

[Here the editor supplied a picture of a 4-dimensional cube
that was omitted by Wittgenstein in the original.]

« Aber das meine ich nicht, ich meine etwas wie…

"But I don't mean that, I mean something like…

…nur mit 4 Ausdehnungen! »

but with four dimensions!

« Aber das ist nicht, was ich dir gezeigt habe,
eben etwas wie…

"But isn't  what I showed you like

…nur mit 4 Ausdehnungen? »

…only with four dimensions?"

« Nein; das meine  ich nicht! »

"No, I don't mean  that!"

« Was aber meine ich? Was ist mein Bild?
Nun der 4-dimensionale Würfel, wie du ihn gezeichnet hast,
ist es nicht ! Ich habe jetzt als Bild nur die Worte  und
die Ablehnung alles dessen, was du mir zeigen kanst. »

"But what do I mean? What is my picture?
Well, it is not  the four-dimensional cube
as you drew it. I have now for a picture only
the words  and my rejection of anything
you can show me."

"Here's your damn Bild , Ludwig —"

Context: The Galois Tesseract.

## Friday, October 31, 2014

### Structure

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

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

## Wednesday, May 21, 2014

### The Tetrahedral Model of PG(3,2)

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

suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).

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

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

(Click to enlarge.)

## Tuesday, March 11, 2014

### Depth

Filed under: General,Geometry — Tags: , — m759 @ 11:16 AM

"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."

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

Part I:  An Inch Deep

Part II:  An Inch Wide

See a search for "square inch space" in this journal.

## Friday, January 17, 2014

### The 4×4 Relativity Problem

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

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

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

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

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

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

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

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

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

## Friday, December 20, 2013

### For Emil Artin

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

(On His Dies Natalis )

This is asserted in an excerpt from…

"The smallest non-rank 3 strongly regular graphs
​which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152-212—

(Click for clearer image)

Note that Theorem 46 of Klin et al.  describes the role
of the Galois tesseract  in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric  part of the above
exceptional geometric-combinatorial isomorphism.

## Wednesday, December 18, 2013

### A Hand for the Band

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

"How about another hand for the band?
They work real hard for it.
The Cherokee Cowboys, ladies and gentlemen."

— Ray Price, video, "Danny Boy Mid 80's Live"

Other deathly hallows suggested by today's NY Times

Click the above image for posts from December 14.

That image mentions a death on August 5, 2005, in
"entertainment Mecca" Branson, Missouri.

Another note from August 5, 2005, reposted here
on Monday

Happy birthday, Keith Richards.

## Monday, December 16, 2013

### Quartet

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

Happy Beethoven's Birthday.

Related material:  Abel 2005 and, more generally, Abel.

## Sunday, December 15, 2013

### Sermon

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

Odin's Jewel

Jim Holt, the author of remarks in yesterday's
Saturday evening post

"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."

Vanity Fair Daily , July 16, 2012

Backstory Odin + Jewel in this journal.

For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —

## Saturday, September 21, 2013

### Geometric Incarnation

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

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

Related material from finitegeometry.org —

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

## Monday, August 12, 2013

### Form

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

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

The Galois tesseract is the basis for a representation of the smallest
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

## Friday, July 5, 2013

### Mathematics and Narrative (continued)

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

Short Story — (Click image for some details.)

Parts of a longer story —

## Tuesday, June 4, 2013

### Cover Acts

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

The Daily Princetonian  today:

A different cover act, discussed here  Saturday:

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

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

## Tuesday, May 28, 2013

### Codes

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

The thirty-five 4×4 structures within the MOG:

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

## 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:

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

### Priority Claim

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

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis
in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."

[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc.
July 20, 1987).

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ,"
arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

 Update of the above paragraph on July 6, 2013— No. The vector space structure was described by (for instance) Peter J. Cameron in a 1976 Cambridge University Press book — Parallelisms of Complete Designs . See the proof of Theorem 3A.13 on pages 59 and 60. The vector space structure as it occurs in a 4×4 array of the sort that appears in the Curtis Miracle Octad Generator may first have been pointed out by me, Steven H. Cullinane, in an AMS abstract submitted in October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

## 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:

Related material:

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

## Thursday, March 7, 2013

### Proof Symbol

Filed under: General,Geometry — m759 @ 8:28 PM

Today's previous post recalled a post
from ten years before yesterday's  date.

The subject of that post was the
Galois tesseract.

Here is a post from ten years before
today's  date

The subject of that  post is the Halmos
tombstone:

"The symbol    is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."

Measure Theory  (1950)

For exact proportions, click on the tombstone.

For some classic mathematics related
to the proportions, see September 2003.

## Wednesday, February 13, 2013

### Form:

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

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

Related material:

 "… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance." – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult

1. today's previous post on the Penrose diamond
2. the remarks of Scott Aaronson on August 17, 2012
3. the remarks in this journal on that same date
4. the geometry of the 4×4 array in the context of M24.

## Saturday, January 5, 2013

### Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

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

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

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

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

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

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

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

For some related narrative, see tesseract  in this journal.

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

Update of August 9, 2013—

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

Update of August 13, 2013—

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

## Wednesday, April 11, 2012

### Steiner’s Systems

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

Background— George Steiner in this journal
and elsewhere—

"An intensity of outward attention —
interest, curiosity, healthy obsession —
was Steiner’s version of God’s grace."

Lee Siegel in The New York Times
March 12, 2009

Steiner in 1969  defined man as "a language animal."

Here is Steiner in 1974  on another definition—

Related material—

Also related — Kantor in 1981 on "exquisite finite geometries," and The Galois Tesseract.

## Tuesday, January 24, 2012

### The Screwing

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

"Debates about canonicity have been raging in my field
(literary studies) for as long as the field has been
around. Who's in? Who's out? How do we decide?"

— Stephen Ramsay, "The Hermeneutics of Screwing Around"

An example of canonicity in geometry—

"There are eight heptads of 7 mutually azygetic screws, each consisting of the screws having a fixed subscript (from 0 to 7) in common. The transformations of LF(4,2) correspond in a one-to-one manner with the even permutations on these heptads, and this establishes the isomorphism of LF(4,2) and A8. The 35 lines in S3 correspond uniquely to the separations of the eight heptads into two complementary sets of 4…."

— J.S. Frame, 1955 review of a 1954 paper by W.L. Edge,
"The Geometry of the Linear Fractional Group LF(4,2)"

Thanks for the Ramsay link are due to Stanley Fish
(last evening's online New York Times ).

For further details, see The Galois Tesseract.

## Monday, January 23, 2012

### How It Works

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

J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—

For earlier (and later) discussions of the geometry  (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.

## Saturday, December 31, 2011

Filed under: General,Geometry — Tags: — m759 @ 4:01 PM

"Design is how it works." — Steve Jobs

From a commercial test-prep firm in New York City—

 m759 @ 8:48 AM

From a New Year's Day, 2012, weblog post in New Zealand

From Arthur C. Clarke, an early version of his 2001  monolith

"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."

The numerical  (not crystal) pyramid above is related to a sort of
mathematical  block design known as a Steiner system.

For its relationship to the graphic  block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—

For graphic  block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.

For the barbed tail  of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.

## Tuesday, December 13, 2011

### Mathematics and Narrative, continued

Filed under: General,Geometry — m759 @ 11:01 PM

Mathematics —

(Some background for the Galois tesseract )

(Click to enlarge)

Narrative

An essay on science and philosophy in the January 2012
Notices of the American Mathematical Society .

Note particularly the narrative explanation of the double-slit experiment—

"The assertion that elementary particles have
some startling consequences. For instance, the
wave-particle duality paradox, in particular the baffling
results of the famous double slit experiment,
may now be reconsidered. In that experiment, first
conducted by Thomas Young at the beginning
of the nineteenth century, a point light source
illuminated a thin plate with two adjacent parallel
slits in it. The light passing through the slits
was projected on a screen behind the plate, and a
pattern of bright and dark bands on the screen was
observed. It was precisely the interference pattern
caused by the diffraction patterns of waves passing
through adjacent holes in an obstruction. However,
when the same experiment was carried out much
later, only this time with photons being shot at
the screen one at a time—the same interference
pattern resulted! But the Metaphysics of Quality
can offer an explanation: the photons each follow
Quality in their actions, and so either individually
or en masse (i.e., from a light source) will do the
same thing, that is, create the same interference
pattern on the screen."

This is from "a Ph.D. candidate in mathematics at the University of Calgary."
His essay is titled "A Perspective on Wigner’s 'Unreasonable Effectiveness
of Mathematics.'" It might better be titled "Ineffective Metaphysics."

## Thursday, November 3, 2011

### Ockham’s Bubbles–

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

Mathematics and Narrative, continued

"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"

— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th

"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."

— Dan Conover on September 1st, 2011

"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'

I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."

— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996

"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"

"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."

— Interviewer for the National Association of Scholars

From that conversation—

Wolfe : "People in academia should start insisting on objective scholarship, insisting  on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"

As for "mis-imagined quantum pins"…
This
journal on the date of the above interview— February 28, 2008

Illustration from a Perimeter Institute talk given on July 20, 2005

The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by  Conover, "The Last Epiphany," and four posts from September 1st, 2011—

Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.

"There is  such a thing as a tesseract." —A Wrinkle in Time

## Tuesday, September 13, 2011

### Day 256

Filed under: General,Geometry — m759 @ 2:56 PM

Today is day 256 of 2011, Programmers' Day.

Yesterday, Monday, R. W. Barraclough's website pictured the Octad of the Week—

" X never, ever, marks the spot."

## 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

## Sunday, November 15, 2020

### Map Methods

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

## 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.

### In Memoriam

Filed under: General — Tags: , — m759 @ 2:14 PM
 From the Vanderbilt University obituary of Vaughan F. R. Jones — “During the mid-1980s, while Jones was working on a problem in von Neumann algebra theory, which is related to the foundations of quantum mechanics, he discovered an unexpected link between that theory and knot theory, a mathematical field dating back to the 19th century. Specifically, he found a new mathematical expression—now known as the Jones polynomial—for distinguishing between different types of knots as well as links in three-dimensional space. Jones’ discovery had been missed by topologists during the previous 60 years, and his finding contributed to his selection as a Fields Medalist. ‘Now there is an area of mathematics called quantum topology, which basically followed from his original work,’ said Dietmar Bisch, professor of mathematics.” [Link added.]

Related to Jones’s work —

“Topological Quantum Information Theory” at
the website of Louis H. Kauffman —
http://homepages.math.uic.edu/~kauffman/Quanta.pdf.

### Welcome to AMS-LaTeX

Filed under: General — Tags: , — m759 @ 1:26 PM

Elsewhere on that same date —

## Monday, February 24, 2020

### Hidden Figure

Filed under: General — m759 @ 2:55 PM

“There is  such a thing as  &#x25A6;  .”

## Tuesday, January 28, 2020

### Very Stable Kool-Aid

Filed under: General — Tags: , — m759 @ 2:16 PM

Two of the thumbnail previews
from yesterday's 1 AM  post

Further down in the "6 Prescott St." post, the link 5 Divinity Avenue

 A Letter from Timothy Leary, Ph.D., July 17, 1961 Harvard University Department of Social Relations Center for Research in Personality Morton Prince House 5 Divinity Avenue Cambridge 38, Massachusetts July 17, 1961 Dr. Thomas S. Szasz c/o Upstate Medical School Irving Avenue Syracuse 10, New York Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . .

The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.

Related "jewel box" material for those who
prefer narrative to mathematics —

"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they

— Tim Doody in The Morning News  web 'zine on July 26, 2012**

Oracular advice related to yesterday evening's
"jewel box" post …

A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces
.  In its incarnation as a Galois  tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of  2,304 affine transformations.

Similar remarks apply to the I Ching  In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.

* The volume Wolfe mentions was, according to Fadiman, the I Ching.

** See also this  journal on that date — July 26, 2012.

## Monday, October 21, 2019

### Algebra and Space… Illustrated.

Filed under: General — Tags: , — m759 @ 4:26 PM

Related entertainment —

Detail:

George Steiner

"Perhaps an insane conceit."

Perhaps.

Perhaps Not.

See Dirac and Geometry .

## Wednesday, October 9, 2019

### The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).

Related narrative The "Quantum Tesseract Theorem."

## Friday, September 27, 2019

### The Black List

Filed under: General — Tags: , — m759 @ 11:46 AM

"… Max Black, the Cornell philosopher, and others have pointed out
algebra, and perhaps without the metaphor there would never have
been any algebra' …."

— Max Black, Models and Metaphors, Cornell U. Press, 1962,
page 242, as quoted in Dramas, Fields, and Metaphors, by
Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25

Metaphor —

Algebra —

 The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214): 1. , , , , , 2. , , , , , 3. , , , , , 4. , , , , , 5. , , , , , 6. , , , , . SEE ALSO:  Pauli Matrices REFERENCES: Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985. Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982. Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977. Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964. Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982. Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980. Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955. Referenced on Wolfram|Alpha:  Dirac Matrices CITE THIS AS: Weisstein, Eric W.  "Dirac Matrices." From MathWorld— A Wolfram Web Resource.  http://mathworld.wolfram.com/DiracMatrices.html

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

## Friday, August 16, 2019

### Nocciolo

Filed under: General — Tags: , , — m759 @ 10:45 AM

A revision of the above diagram showing

Related tables from August 10

See "Schoolgirl Space Revisited."

## 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.

## Saturday, August 10, 2019

### Schoolgirl Space* Revisited:

Filed under: General — Tags: , — m759 @ 10:51 PM

The Square "Inscape" Model of

Click image to enlarge.

* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.

## Tuesday, July 16, 2019

### Schoolgirl Space for Quantum Mystics

Filed under: General — Tags: , — m759 @ 2:16 PM

From a post on St. Andrew's Day, 2017

## Sunday, July 14, 2019

### Old Pathways in Science:

Filed under: General — Tags: , — m759 @ 12:37 PM

The Quantum Tesseract Theorem Revisited

"The secret  is that the super-mathematician expresses by the anticommutation
of  his operators the property which the geometer conceives as  perpendicularity
of displacements.  That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.

Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."

A related illustration, from posts tagged Dirac and Geometry —

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

## Thursday, February 28, 2019

### Wikipedia Scholarship

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

Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this  journal a Galois  tesseract.

## Saturday, December 22, 2018

### Cremona-Richmond

Filed under: General,Geometry — Tags: , — m759 @ 12:34 PM

The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005

A more recent appearance of the configuration —

## Wednesday, December 12, 2018

### An Inscape for Douthat

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:

Related material — the phrase
"Quantum Tesseract Theorem" and

A.  An image from the recent
film "A Wrinkle in Time" —

B.  A quote from the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

## Friday, December 7, 2018

### The Angel Particle

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 7:15 PM

(Continued from this morning)

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

## Tuesday, November 13, 2018

### Blackboard Jungle Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:19 PM

From the 1955 film "Blackboard Jungle" —

From a trailer for the recent film version of A Wrinkle in Time

Detail of the phrase "quantum tesseract theorem":

From the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —

Some background —

background on finite  geometry, see a web page
at Thas's institution, Ghent University.

## 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

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

## Wednesday, June 27, 2018

### Taken In

Filed under: General,Geometry — Tags: , , — m759 @ 9:36 AM

A passage that may or may not have influenced Madeleine L’Engle’s

But beware of being drawn into the personal life of the Happy Family .

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

And ends with

Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a ‘personality disorder.’  His anger was
paranoid and unremitting.”

## Thursday, June 21, 2018

### Dirac and Geometry (continued)

"Just fancy a scale model of Being
made out of string and cardboard."

Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington

"… a treatise on Kummer's quartic surface."

The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.

See Hudson + Kummer in this  journal.

## 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
.

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.

## 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 —

## Saturday, February 17, 2018

### The Binary Revolution

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

## Tuesday, January 9, 2018

### Koen Thas and Quantum Theory

Filed under: General — Tags: — m759 @ 9:23 AM

This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.

## Monday, January 8, 2018

### Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 11:15 PM

Raiders —

A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling

Time Magazine  December 25, 2017 – January 1, 2018

## Saturday, December 23, 2017

### The Right Stuff

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:12 PM

A figure related to the general connecting theorem  of Koen Thas —

Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."

### The Patterning

Filed under: General — Tags: — m759 @ 12:25 PM

See a Log24 search for "Patterning Windows."

Related material (Click for context) —

.

### IT Girl (for Sweet Home Alabama)

Filed under: General — Tags: — m759 @ 3:35 AM

## Friday, December 22, 2017

### IT

Filed under: General,Geometry — Tags: — m759 @ 4:08 PM

Related material from May 25, 2016 —

## Thursday, December 21, 2017

### Wrinkles

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

TIME magazine, issue of December 25th, 2017 —

" In 2003, Hand worked with Disney to produce a made-for-TV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "

Not the dream?  It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.

From a trailer for the latest version —

Detail:

From the 1962 book —

"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."

Song adapted from a 1960 musical —

"In short, there's simply not
A more congenial spot
For happy-ever-aftering
Than here in Camazotz!"

## Sunday, December 10, 2017

### Geometry

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
related remarks on the notion of  linear  (or line ) complex
in the finite projective space PG(3,2) —

## Tuesday, October 10, 2017

### Another 35-Year Wait

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

The title refers to today's earlier post "The 35-Year Wait."

A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

See also Inscape in this  journal and posts tagged Dirac and Geometry.

## 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 —

## Saturday, August 26, 2017

### Aesthetic Distance

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

Naive readers may suppose that this sort of thing is
related to what has been dubbed "geometric group theory."

It is not. See posts now tagged Aesthetic Distance.

## 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

Some context:  McLuhan + Analogy.

## Thursday, April 20, 2017

### Stone Logic

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

Related mathematics — Guitart in this journal —

## Sunday, April 16, 2017

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):

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 —

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.]

## Tuesday, November 22, 2016

### Jargon

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

See "sacerdotal jargon" in this journal.

For those who prefer scientific  jargon —

combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

For some science that is not just jargon, see

and, also from posts tagged Dirac and Geometry

The above line complex also illustrates an outer automorphism
of the symmetric group S6. See last Thursday's post "Rotman and
the Outer Automorphism
."

## 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 —

(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.

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."

## Friday, June 3, 2016

### Bruins and van Dam

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

A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:

The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:

## Wednesday, May 25, 2016

### Kummer and Dirac

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

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

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

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

[4]

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

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

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

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

See that structure in this  journal, for instance —

See as well yesterday morning's post.

## 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)

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

Related terminology describing the Göpel tetrads above

## 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.)”

## Friday, May 6, 2016

### Review

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

## 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.

## Monday, February 8, 2016

### A Game with Four Letters

Filed under: General,Geometry — Tags: , — m759 @ 2:56 PM

Related material — Posts tagged Dirac and Geometry.

For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).

## Wednesday, January 13, 2016

### Geometry for Jews

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

(Continued from previous episodes)

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 —

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