“Galois Field” « Search Results

Wednesday, January 23, 2013

DNA and a Galois Field

Filed under: General,Geometry — m759 @ 9:29 pm

From Ewan Birney's weblog today:

WEDNESDAY, 23 JANUARY 2013

Using DNA as a digital archive media

Today sees the publication in Nature of “Toward practical high-capacity low-maintenance storage of digital information in synthesised DNA,” a paper spearheaded by my colleague Nick Goldman and in which I played a major part, in particular in the germination of the idea.

Birney appeared in Log24 on Dec. 30, 2012, quoted as follows:

"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."

— Birney's weblog on July 3, 2012, "Galois and Sequencing."

Birney's widespread appearance in news articles today about the above Nature publication suggests a review of the "Galois-field"-"DNA" connection.

See, for instance, the following papers:

Gail Rosen and Jeff Moore. "Investigation of Coding Structure in DNA," IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [pdf]
Gail Rosen. "Finding Near-Periodic DNA Regions using a Finite-Field Framework," 2nd IEEE Genomic Signal Processing Workshop (GENSIPS), Baltimore, MD, May 2004. [pdf]
Gail Rosen. "Examining Coding Structure and Redundancy in DNA," IEEE Engineering in Medicine and Biology Magazine, Volume 25, Issue 1, January/February 2006. [pdf]

A Log24 post of Sept. 17, 2012, also mentions the phrases "Galois field" and "DNA" together.

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Saturday, November 6, 2010

Galois Field of Dreams, continued

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

Hollywood Reporter Exclusive

Martin Sheen Caught in
Spider-Man's Web

“… lux in tenebris lucet…"

Sally Field is in early talks
to play Aunt May.

Related material:

Birthdays in this journal,
Galois Field of Dreams,
and Class of 64.

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Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: Eightfold Cube, Springer — m759 @ 10:01 am

It is well known that the seven (2² + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (3² + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (4² + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G₂₀₁₆₀ in Mitchell 1910, LF(3,2²) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,2²),"
Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

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Sunday, June 23, 2024

Marks

Filed under: General — Tags: Abstraction and Structure, Art Space — m759 @ 2:12 pm

Some marks I find more interesting . . . Those of a Galois field.

See a June 5 post on cultural appropriation .

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Tuesday, December 13, 2022

In Memory of a Mississippi Coach

Filed under: General — Tags: Baez Monoids, Boole vs. Galois — m759 @ 1:10 pm

The "Boolean exclusive or" is the same as addition
in the two-element Galois field GF(2).

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Saturday, March 26, 2022

Box Geometry: Space, Group, Art (Work in Progress)

Filed under: General — Tags: Cubehenge, Six-Set — m759 @ 2:06 am

Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").

Here is a draft for a table of related material, arranged
as internet URL labels.

**Finite Geometry Notes — Summary Chart**
Name Tag	.Space	.Group	.Art
Box4	2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space)	S₄ = AGL(2,2) (Box4.group)	(Box4.art)
Box6	3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set.	S₆
Box8	2x2x2 cube or 4×2 (4-row, 2-column) array.	S₈ or A₈or AGL(3,2) of order 1344, or GL(3,2) of order 168
Box9	The 3×3 square.	AGL(2,3) or GL(2,3)
Box12	The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M₁₂.	Symmetries of the cube or elements of the group M₁₂
Box13	The 13 symmetry axes of the cube.	Symmetries of the cube.
Box15	The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field.	Collineations of PG(3,2)
Box16	The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field.	AGL(4,2), the affine group of 322,560 permutations of the parts of a 4×4 array (a Galois tesseract)
Box20	The configuration representing Desargues's theorem.
Box21	The 21 points and 21 lines of PG(2,4).
Box24	The 24 points of the Steiner system S(5, 8, 24).
Box25	A 5×5 array representing PG(2,5).
Box27	The 3-dimensional Galois affine space over the 3-element Galois field GF(3).
Box28	The 28 bitangents of a plane quartic curve.
Box32	Pair of 4×4 arrays representing orthogonal Latin squares.	Used to represent elements of AGL(4,2)
Box35	A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2)	PGL(3,2), order 20,160
Box36	Eurler's 36-officer problem.
Box45	The 45 Pascal points of the Pascal configuration.
Box48	The 48 elements of the group AGL(2,3).	AGL(2,3).
Box56	The 56 three-sets within an 8-set or 56 triangles in a model of Klein's quartic surface or the 56 spreads in PG(3,2).
Box60	The Klein configuration.
Box64	Solomon's cube.

— Steven H. Cullinane, March 26-27, 2022

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Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: Octad, Octad Generator — m759 @ 5:00 am

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

See the Cullinane diamond theorem .

If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.

See the Miracle Octad Generator of R. T. Curtis.

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Sunday, December 6, 2020

“Binary Coordinates”

Filed under: General — Tags: Boole vs. Galois — 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.

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Wednesday, November 11, 2020

Qube

Filed under: General — Tags: Fields, Qubes — m759 @ 8:30 pm

The new domain qube.link forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .

More generally, qubes.link forwards to this post,
which defines qubes .

Definition: A qube is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)

Examples: 8, 27, 64. See qubes.site.

Update on Nov. 18, 2020, at about 9:40 PM ET —

Problem:

For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and Affine Groups on Small
Binary Spaces.)

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Monday, November 2, 2020

Q Bits

Filed under: General — Tags: Q-Bits — m759 @ 6:34 pm

The new domain name q-bits.space does not refer to
the q in “quantum ,” but rather to the q that symbolizes
the order of a Galois field .

See the Wikipedia article “Finite field.”

The “space” suffix refers to a web page on geometry.

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Thursday, January 10, 2019

Toy Story Continues.

Filed under: General — Tags: Fields, Toy Story — m759 @ 11:13 am

Friday, September 14, 2018

Denkraum

Filed under: G-Notes,General,Geometry — Tags: Chessboard.space, Interstice — m759 @ 1:00 am

Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.

In this field, "1 + 1 = 0," as noted here Wednesday.

See also other posts now tagged Interstice.

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Saturday, February 17, 2018

The Binary Revolution

Filed under: General,Geometry — Tags: Boole vs. Galois, Geometric Theology, Shaw and Finite Geometry, Trinity Dice — m759 @ 5:00 pm

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:

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Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — Tags: Six-Set — m759 @ 11:29 pm

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular" might aptly be applied to . . .

The adjective "affine" might aptly be applied to . . .

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

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Tuesday, September 5, 2017

Annals of Critical Epistemology

Filed under: General,Geometry — Tags: BU Logic, Springer — m759 @ 5:36 pm

"But unlike many who left the Communist Party, I turned left
rather than right, and returned—or rather turned for the first time—
to a critical examination of Marx's work. I found—and still find—
that his analysis of capitalism, which for me is the heart of his work,
provides the best starting point, the best critical tools, with which—
suitably developed—to understand contemporary capitalism.
I remind you that this year is also the sesquicentennial of the
Communist Manifesto , a document that still haunts the capitalist world."

— From "Autobiographical Reflections," a talk given on June 5, 1998, by
John Stachel at the Max Planck Institute for the History of Science in Berlin
on the occasion of a workshop honoring his 70th birthday,
"Space-Time, Quantum Entanglement and Critical Epistemology."

From a passage by Stachel quoted in the previous post —

From the source for Stachel's remarks on Weyl and coordinatization —

Note that Stachel distorted Weyl's text by replacing Weyl's word
"symbols" with the word "quantities." —

This replacement makes no sense if the coordinates in question
are drawn from a Galois field — a field not of quantities , but rather
of algebraic symbols .

"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"

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Florence 2001

Filed under: General,Geometry — Tags: BU Logic — m759 @ 4:44 am

Or: Coordinatization for Physicists

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

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

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

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Thursday, August 31, 2017

A Conway-Norton-Ryba Theorem

Filed under: General,Geometry — Tags: Fields, Zero Sum — m759 @ 1:40 pm

In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —

A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.

See also Magic Square and Diamond Theorem in this journal.

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Wednesday, July 5, 2017

Imaginarium of a Different Kind

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

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May 22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry —

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).^[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

— User Rgdboer

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Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: Conceptualist Minimalism, Fields, The Coxeter Aleph — m759 @ 6:29 pm

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

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Thursday, September 15, 2016

The Smallest Perfect Number/Universe

Filed under: General,Geometry — Tags: Six-Set, Tetrahedron vs. Square, Whitehead 1906 — m759 @ 6:29 am

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

* For the definition of "perfect number," see any introductory
number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3-space over the 2-element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.

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Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: General,Geometry — Tags: Geometry, Kummerhenge, Octad, Octad Generator, Priority, Six-Set — m759 @ 10:00 pm

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

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

Hudson in 1905:

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

A recap by Cullinane in 2013:

Click images for further details.

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Monday, May 30, 2016

Perfect Universe

Filed under: General,Geometry — Tags: Six-Set — m759 @ 7:00 pm

(A sequel to the previous post, Perfect Number)

Since antiquity, six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its
proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set")
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same
adjective in a different sense, has called
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance,
that might please Yahoo CEO Marissa Mayer on her birthday —

The semantic turnaround here in the meaning of "perfect"
is accompanied by a model turnaround in the picture of PG(3,2) as
Polster's tetrahedral model is replaced by Cullinane's square model.

Further background from the previous post —

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: Dirac and Geometry, Facets, Kummerhenge, Octad, Octad Generator, Quantum Tesseract Theorem, Rosenhain and Göpel, The Omega Matrix, The Rosenhain Symmetry — m759 @ 8:23 am

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

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

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Monday, May 9, 2016

Search for the Lost Theorem

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

The three Solomons of the previous post (LeWitt,
Marcus, and Golomb) suggest the three figures
-1, 0, and 1 … symbols for the three elements
of the Galois field GF(3). This in turn suggests a
Search for The Lost Theorem. Some cross-cultural
context: The First of May, 2010.

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Friday, November 20, 2015

Anticommuting Dirac Matrices as Skew Lines

Filed under: General,Geometry — Tags: Dirac and Geometry, Quantum Tesseract Theorem — m759 @ 11:45 pm

(Continued from November 13)

The work of Ron Shaw in this area, ca. 1994-1995, does not
display explicitly the correspondence between anticommutativity
in the set of Dirac matrices and skewness in a line complex of
PG(3,2), the projective 3-space over the 2-element Galois field.

Here is an explicit picture —

Anticommuting Dirac matrices as spreads of projective lines

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Shaw, Ron, "Finite Geometry, Dirac Groups, and the Table of
Real Clifford Algebras," undated article at ResearchGate.net

Update of November 23:

See my post of Nov. 23 on publications by E. M. Bruins
in 1949 and 1959 on Dirac matrices and line geometry,
and on another author who gives some historical background
going back to Eddington.

Some more-recent related material from the Slovak school of
finite geometry and quantum theory —

The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.

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Sunday, October 18, 2015

Sunday School

Filed under: General,Geometry — m759 @ 8:30 am

A Unified Field —

Click the above image for further details.

See also a search in this journal for Jorie Graham.

Related dramatic dialogue for Emma Stone and
Joaquin Phoenix, actors in "Irrational Man" —

"Are you aware of what's going on at that table?"

Philosophical backstory by Hans Christian Andersen —

"He was quite frightened, and he tried to repeat the Lord's Prayer;
but all he could do, he was only able to remember the multiplication table."

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Thursday, July 9, 2015

Man and His Symbols

Filed under: General,Geometry — m759 @ 2:24 pm

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure
clearly is of fundamental importance.

The Galois field GF(3)

It is, however, perhaps too small to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array…

… hence as the traditional Chinese Holy Field.

Marketing the Holy Field

The above illustration of China's Holy Field occurred in the context of
Log24 posts on Child Buyers. For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

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Thursday, June 11, 2015

Omega

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 12:00 pm

Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts.

For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field Ω = GF(8).

Related fiction: The Eight , by Katherine Neville.

Related non-fiction: A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —

Mathematics

The Fano plane block design

Magic

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

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Tuesday, June 9, 2015

Colorful Song

Filed under: General,Geometry — Tags: Four Color, Galois's Space — 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."

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Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: Galois's Space, Geometry — m759 @ 12:31 am

The incidences of points and planes in the
Möbius 8₄ 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).

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Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: 5x5, Fields, Octad, Octad Generator — m759 @ 2:56 pm

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ₅).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ₅.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

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Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Geometry, The Rosenhain Symmetry — m759 @ 6:11 pm

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

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Tuesday, June 3, 2014

Robert Steinberg, 1922-2014

Filed under: General,Geometry — Tags: Steinberg Deathday — m759 @ 2:20 pm

Galois matrices, the subject of the previous post,
are of course not new. See, for instance, Steinberg in 1951:

IMAGE- Robert Steinberg, introduction to 'A Geometric Approach to the Representations of the Full Linear Group over a Galois Field'

The American Mathematical Society reports that Steinberg died
on May 25, 2014.

As the above 1951 paper indicates, Steinberg was well acquainted with
what Weyl called "the devil of abstract algebra." In this journal, however,
Steinberg himself appears rather as an angel of geometry.

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Wednesday, May 21, 2014

The Tetrahedral Model of PG(3,2)

Filed under: General,Geometry — Tags: on140521, Whitehead 1906 — m759 @ 10:15 pm

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

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

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

(Click to enlarge.)

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Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: Bauhaus Cube, Cube School, Cullinane Cube, Solomon's Cube — m759 @ 2:28 pm

(Continued)

Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

Click to enlarge.

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Thursday, March 13, 2014

Entartete Kunst

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

The title refers to a New York Times story about
an art exhibition that opened today.

This evening’s NY Lottery numbers: 016 and 2858.

Pictures from these links:

016 (Blackboard Jungle , 1955) —

IMAGE- Blackboard from 'Blackboard Jungle'

2858 (number of a Log24 post, 2007) —

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Monday, March 3, 2014

Blackboard Jungle Revisited

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

IMAGE- Blackboard from 'Blackboard Jungle'

Blackboard Jungle , 1955

"We are going to keep doing this
until we get it right." — June 15, 2007

"Her wall is filled with pictures,
she gets 'em one by one" — Chuck Berry

See too a more advanced geometry lesson
that also uses the diagram pictured above.

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Saturday, February 15, 2014

Rosenhain and Göpel

Filed under: General,Geometry — Tags: Depth, Rosenhain and Göpel — m759 @ 11:00 pm

(Continued)

See The Oslo Version in this journal and the New Year’s Day 2014 post.
The pictures of the 56 spreads in that post (shown below) are based on
the 20 Rosenhain and 15 Göpel tetrads that make up the 35 lines of
PG(3,2), the finite projective 3-space over the 2-element Galois field.

Click for a larger image.

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Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: Coordinatization, Finite Relativity, Geometry, Priority — 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—

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

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

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

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

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

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Saturday, September 21, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: Geometric Theology, Kummer-Sept-2013, Rosenhain and Göpel — m759 @ 1:00 am

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine." An example— the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M₂₄. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array. It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

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

(Continued from yesterday)

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

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

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

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

Narrative:

Aooo.

Happy birthday to Stephen King.

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Thursday, September 5, 2013

Moonshine II

Filed under: General,Geometry — Tags: Barth Moonshine, Depth, Geometric Theology, Geometry, Kummerhenge, Priority, Rosenhain and Göpel — m759 @ 10:31 am

(Continued from yesterday)

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

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

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

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Wednesday, September 4, 2013

Moonshine

Filed under: General,Geometry — Tags: Barth Moonshine, Geometric Theology, Kummerhenge, Octad, Octad Generator, Rosenhain and Göpel — m759 @ 4:00 pm

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M₂₄ via the book of Hudson and
the geometry of the 4×4 square.

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Tuesday, August 13, 2013

The Story of N

Filed under: General,Geometry — Tags: Story of N, The Coxeter Aleph — m759 @ 9:00 pm

(Continued from this morning)

The above stylized "N," based on
an 8-cycle in the 9-element Galois field
GF(9), may also be read as an Aleph.

Graphic designers may prefer a simpler,
bolder version:

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Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .

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Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: Geometry, Overarching Group, Priority — m759 @ 10:30 pm

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory monograph.

But such a survey might not find any such pre-1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4-space over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group M ₂₄.

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M ₂₄ ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material:

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

* A rather abstract 2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.

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Tuesday, May 28, 2013

Codes

Filed under: General,Geometry — Tags: Coding Theory, Diamond-Theory-tesseract, Geometry, Natural Diagram, Octad, Octad Generator, Priority — m759 @ 12:00 pm

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

IMAGE- A hyperspace model of the 4D vector space over 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).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

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

IMAGE- The 35 square patterns within the Curtis MOG

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

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

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):

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Sunday, March 31, 2013

For Baker

Filed under: General,Geometry — m759 @ 8:00 pm

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

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

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

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

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

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

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

Update of 9:15 PM March 31—

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

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Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator, Penrose diamond — 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—

IMAGE- The Penrose diamond and 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)…

IMAGE- Stroppel on the Klein quadric, 2008

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

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

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Sunday, January 20, 2013

In the Details

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

Part I: Synthesis

Part II: Iconic Symbols

IMAGE- Blackboard from 'Blackboard Jungle'

Blackboard Jungle , 1955

Part III: Euclid vs. Galois

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Sunday, December 30, 2012

Not Often

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

"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."

— Ewan Birney at his weblog on July 3, 2012.

Try a Google search. (And see such a search as of Dec. 30, 2012.)

See also "Context Part III" in a Log24 post of Sept. 17, 2012.

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Saturday, November 10, 2012

Descartes Field of Dreams

Filed under: General,Geometry — Tags: Field of Bad Dreams, Imago, Nov 10 2012 — m759 @ 2:01 pm

(A prequel to Galois Field of Dreams)

The opening of Descartes' Dream ,
by Philip J. Davis and Reuben Hersh—

"The modern world,
our world of triumphant rationality,
began on November 10, 1619,
with a revelation and a nightmare."

For a revelation, see Battlefield Geometry.

For a nightmare, see Joyce's Nightmare.

Some later work of Descartes—

From "What Descartes knew of mathematics in 1628,"
by David Rabouin, CNRS-Univ. Paris Diderot,
Historia Mathematica , Volume 37, Issue 3,
Contexts, emergence and issues of Cartesian geometry,
August 2010, pages 428–459 —

Fig. 5. How to represent the difference between white, blue, and red
according to Rule XII [from Descartes, 1701, p. 34].

A translation —

The 4×4 array of Descartes appears also in the Battlefield Geometry posts.
For its relevance to Galois's field of dreams, see (for instance) block designs.

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Tuesday, October 16, 2012

Cube Review

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

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

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

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

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

See

the 4x4x4 cube and An Invariance of Symmetry
the 4x4x4 cube and the nineteenth-century
geometers' "Solomon's seal"
the 4x4x4 cube and the Kummer surface
the 4x4x4 cube and the Klein quadric.

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

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Thursday, September 27, 2012

Kummer and the Cube

Filed under: General,Geometry — Tags: Cube Mine, Kummer-Sept-2012, Kummerhenge — m759 @ 7:11 pm

Denote the d-dimensional hypercube by γ_d .

"… after coloring the sixty-four vertices of γ₆
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."

— From "Kummer's 16₆," section 12 of Coxeter's 1950
"Self-dual Configurations and Regular Graphs"

Just as the 4×4 square represents the 4-dimensional
hypercube γ₄over the two-element Galois field GF(2),
so the 4x4x4 cube represents the 6-dimensional
hypercube γ₆ over GF(2).

For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).

See also two professors in The New York Times
discussing images of the sacred in an op-ed piece
dated Sept. 26 (Yom Kippur).

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Monday, September 17, 2012

Pattern Conception

Filed under: General,Geometry — Tags: Fitch, Four Color, Harvard Miller, on120917, Pattern Theory — m759 @ 10:00 am

( Continued from yesterday's post FLT )

Context Part I —

"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….

…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory (FLT)…."

— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B (2012) 367, 1925-1932

Context Part II —

Context Part III —

A four-color theorem describes the mathematics of
general structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.

Context Part IV —

A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—

"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."

Related material from a literate source—

"So we moved, and they, in a formal pattern"

Formal Patterns—

Not formal language theory but rather
finite projective geometry provides a graphic grammar
of abstract design—

See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.

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Wednesday, August 1, 2012

Elementary Finite Geometry

Filed under: General,Geometry — Tags: Four Gods, God + Lucy, Lexicon Omega, SymOmega — m759 @ 7:16 pm

I. General finite geometry (without coordinates):

A finite affine plane of order n has n^2 points.

A finite projective plane of order n has n^2 + n + 1

points because it is formed from an order-n finite affine

plane by adding a line at infinity that contains n + 1 points.

Examples—

Affine plane of order 3

Projective plane of order 3

II. Galois finite geometry (with coordinates over a Galois field):

A finite projective Galois plane of order n has n^2 + n + 1

points because it is formed from a finite affine Galois 3-space

of order n with n^3 points by discarding the point (0,0,0) and

identifying the points whose coordinates are multiples of the

(n-1) nonzero scalars.

Note: The resulting Galois plane of order n has

(n^3-1)/(n-1)= (n^2 + n + 1) points because

(n^2 + n + 1)(n – 1) =

(n^3 + n^2 + n – n^2 – n – 1) = (n^3 – 1) .

III. Related art:

Another version of a 1994 picture that accompanied a New Yorker
article, "Atheists with Attitude," in the issue dated May 21, 2007:

IMAGE- 'Four Gods,' by Jonathan Borofsky

The Four Gods of Borofsky correspond to the four axes of
symmetry of a square and to the four points on a line at infinity
in an order-3 projective plane as described in Part I above.

Those who prefer literature to mathematics may, if they like,
view the Borofsky work as depicting

"Blake's Four Zoas, which represent four aspects
of the Almighty God" —Wikipedia

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Thursday, June 7, 2012

The Field

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

"Time for you to see the field." —Bagger Vance

This post was suggested by a link from a post
in this journal seven years ago yesterday—

“Is the language of thought
any more than a dream?“

— Rimbaud

Yes.

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Saturday, March 3, 2012

Decomposition

Filed under: General,Geometry — Tags: Four Color — m759 @ 3:33 am

A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.

The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.

(See the article's revision history for mid-August 2007.)

Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.

Some related material involving Fass and 4×4 squares—

A 2003 paper he wrote with Jacob Feldman—

(Click to enlarge.)

"Design is how it works." — Steve Jobs

An assignment for Jobs in the afterlife—

Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."

The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.

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Monday, February 20, 2012

Coxeter and the Relativity Problem

Filed under: General,Geometry — Tags: The Coxeter Aleph — m759 @ 12:00 pm

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"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

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of the primitive root λ or 10 are

λ = 10, λ² = 21, λ³ = 22, λ⁴ = 02,
λ⁵ = 20, λ⁶ = 12, λ⁷ = 11, λ⁸ = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ⁴, λ⁵, λ⁷.)"

Coxeter's Figure 10 yields...

The Aleph

The details:

(Click to enlarge)

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

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Friday, September 9, 2011

Galois vs. Rubik

Filed under: General,Geometry — Tags: Cubehenge, Field Theology, Holy Field GF(3), Psycho History, Rubik Adoration, Rubik vs. Abel — m759 @ 2:56 pm

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from "Bleu"

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

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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

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—

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—

(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—

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."

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Saturday, September 3, 2011

The Galois Tesseract (continued)

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

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

Here is some supporting material—

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

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

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

The affine structure appears in the 1979 abstract mentioned above—

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

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

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

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

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

(Received July 20 1987)

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

* For instance:

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

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Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: Cubehenge, Eightfold Cube, Geometric Theology, Rubik Adoration, Sacred Space — m759 @ 6:29 pm

Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."

A simpler candidate for the "Cube" part of that phrase:

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."

— Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"

Borovik has a such a diagram—

The planes in Borovik's figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine's Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

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Sunday, June 26, 2011

Sunday Dinner

Filed under: General,Geometry — Tags: Fields, Security Story — m759 @ 2:22 pm

From "Sunday Dinner" in this journal—

"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"

—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010

Some table topics—

Today's midday New York Lottery numbers were 027 and 7002.

The former suggests a Galois cube, the latter a course syllabus—

CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science

An item from that syllabus:

Six

22 February 2011

DES

History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2.

Related material: A novel, PopCo , was required reading for the course.

Discuss a different novel by the same author—

The End of Mr. Y .

Discuss the author herself, Scarlett Thomas.

Background for the discussion—

Derrida in this journal versus Charles Williams in this journal.

Related topics from the above syllabus date—

Metaphor and Gestell and Quadrat.

Some context— Midsummer Eve's Dream.

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Wednesday, February 9, 2011

An Abstract Window

Filed under: General,Geometry — Tags: Symmetry Framed — m759 @ 12:00 pm

The sliding window in blue below

Click for the web page shown.

is an example of a more general concept.

Such a sliding window,* if one-dimensional of length n , can be applied to a sequence of 0's and 1's to yield a sequence of n-dimensional vectors. For example— an "m-sequence" (where the "m" stands for "maximum length") of length 63 can be scanned by a length-6 sliding window to yield all possible 6-dimensional binary vectors except (0,0,0,0,0,0).

For details, see A Galois Field—

The image is from Bert Jagers at his page on the Galois field GF(64) that he links to as "A Field of Honor."

For a discussion of the m-sequence shown in circular form above, see Jagers's "Pseudo-Random Sequences from GF(64)." Here is a noncircular version of the length-63 m-sequence described by Jagers (with length scale below)—

100000100001100010100111101000111001001011011101100110101011111

123456789012345678901234567890123456789012345678901234567890123

This m-sequence may be viewed as a condensed version of 63 of the 64 I Ching hexagrams. (See related material in this journal.)

For a more literary approach to the window concept, see The Seventh Symbol (scroll down after clicking).

* Moving windows also appear (in a different way) In image processing, as convolution kernels .

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Saturday, July 3, 2010

Beyond the Limits

Filed under: General,Geometry — Tags: JAF, Khora, Mythspace — m759 @ 7:29 pm

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…."

– Don DeLillo, Point Omega

Capitalized, the letter omega figures in the theology of two Jesuits, Teilhard de Chardin and Gerard Manley Hopkins. For the former, see a review of DeLillo. For the latter, see James Finn Cotter's Inscape and "Hopkins and Augustine."

The lower-case omega is found in the standard symbolic representation of the Galois field GF(4)—

$GF(4) = {0, 1, ω, ω 2}$

A representation of GF(4) that goes beyond the standard representation—

Here the four diagonally-divided two-color squares represent the four elements of GF(4).

The graphic properties of these design elements are closely related to the algebraic properties of GF(4).

This is demonstrated by a decomposition theorem used in the proof of the diamond theorem.

To what extent these theorems are part of "a saga of created reality" may be debated.

I prefer the Platonist's "discovered, not created" side of the debate.

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Devising Entities

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

or, Darkness and Brightness at Noon

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional."

— Don DeLillo, Point Omega

$GF(4) = {0, 1, ω, ω 2}$

$— Symbolic representation of a Galois field$

"One two three four,
who are we for?"

— Cheerleaders' chant

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Thursday, June 24, 2010

Midsummer Noon

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

Geometry Simplified

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

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

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

Homogeneous coordinates for the
points of this line —

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

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

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

(an affine space)

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

For further details, see Galois Geometry.

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

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

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Sunday, January 24, 2010

Today’s Sermon

Filed under: General,Geometry — Tags: Octad, Venn Lotus — m759 @ 11:00 am

More Than Matter

Wheel in Webster’s Revised Unabridged Dictionary, 1913

(f) Poetry

The burden or refrain of a song.

⇒ “This meaning has a low degree of authority, but is supposed from the context in the few cases where the word is found.” Nares.

You must sing a-down a-down, An you call him a-down-a. O, how the wheel becomes it! Shak.

“In one or other of G. F. H. Shadbold’s two published notebooks, Beyond Narcissus and Reticences of Thersites, a short entry appears as to the likelihood of Ophelia’s enigmatic cry: ‘Oh, how the wheel becomes it!’ referring to the chorus or burden ‘a-down, a-down’ in the ballad quoted by her a moment before, the aptness she sees in the refrain.”

— First words of Anthony Powell’s novel “O, How the Wheel Becomes It!” (See Library Thing.)

Anthony Powell's 'O, How the Wheel Becomes It!' along with Laertes' comment 'This nothing's more than matter.'

Related material:

Photo uploaded on January 14, 2009
with caption “This nothing’s more than matter”

and the following nothings from this journal
on the same date– Jan. 14, 2009—

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Tuesday, September 8, 2009

Tuesday September 8, 2009

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Super-8 — m759 @ 12:25 pm

Froebel's
Magic Box

Box containing Froebel's Third Gift-- The Eightfold Cube

Continued from Dec. 7, 2008,
and from yesterday.

Non-Euclidean
Blocks

Passages from a classic story:

… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….

Tesseract

"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."

"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.

"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child– especially a baby– might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

"Hardening of the thought-arteries," Jane interjected.

Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–"

"Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–"

"Poor kid," Jane said.

Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–"

"Blocks? What kind?"

Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."

— "Mimsy Were the Borogoves," Lewis Padgett, 1943

Padgett (pseudonym of a husband-and-wife writing team) says that alphabet blocks are the intuitive "basis of Euclid." Au contraire; they are the basis of Gutenberg.

For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…

Friedrich Froebel
(1782-1852), who
invented kindergarten.

His "third gift" —

Froebel's Third Gift-- The Eightfold Cube

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring

Go figure.

* i.e., other than Euclidean

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Tuesday, August 18, 2009

Tuesday August 18, 2009

Filed under: General,Geometry — Tags: Modal Diamond Box, Sein Feld, Shadow Work — m759 @ 12:00 pm

Prima Materia

(Background: Art Humor: Sein Feld (March 11, 2009) and Ides of March Sermon, 2009)

From Cardinal Manning's review of Kirkman's Philosophy Without Assumptions—

"And here I must confess… that between something and nothing I can find no intermediate except potentia, which does not mean force but possibility."

— Contemporary Review, Vol. 28 (June-November, 1876), page 1017

Furthermore….

Cardinal Manning, Contemporary Review, Vol. 28, pages 1026-1027:

The following will be, I believe, a correct statement of the Scholastic teaching:–

1. By strict process of reason we demonstrate a First Existence, a First Cause, a First Mover; and that this Existence, Cause, and Mover is Intelligence and Power.

2. This Power is eternal, and from all eternity has been in its fullest amplitude; nothing in it is latent, dormant, or in germ: but its whole existence is in actu, that is, in actual perfection, and in complete expansion or actuality. In other words God is Actus Purus, in whose being nothing is potential, in potentia, but in Him all things potentially exist.

3. In the power of God, therefore, exists the original matter (prima materia) of all things; but that prima materia is pura potentia, a nihilo distincta, a mere potentiality or possibility; nevertheless, it is not a nothing, but a possible existence. When it is said that the prima materia of all things exists in the power of God, it does not mean that it is of the existence of God, which would involve Pantheism, but that its actual existence is possible.

4. Of things possible by the power of God, some come into actual existence, and their existence is determined by the impression of a form upon this materia prima. The form is the first act which determines the existence and the species of each, and this act is wrought by the will and power of God. By this union of form with the materia prima, the materia secunda or the materia signata is constituted.

5. This form is called forma substantialis because it determines the being of each existence, and is the root of all its properties and the cause of all its operations.

6. And yet the materia prima has no actual existence before the form is impressed. They come into existence simultaneously;

[p. 1027 begins]

as the voice and articulation, to use St. Augustine's illustration, are simultaneous in speech.

7. In all existing things there are, therefore, two principles; the one active, which is the form– the other passive, which is the matter; but when united, they have a unity which determines the existence of the species. The form is that by which each is what it is.

8. It is the form that gives to each its unity of cohesion, its law, and its specific nature.*

When, therefore, we are asked whether matter exists or no, we answer, It is as certain that matter exists as that form exists; but all the phenomena which fall under sense prove the existence of the unity, cohesion, species, that is, of the form of each, and this is a proof that what was once in mere possibility is now in actual existence. It is, and that is both form and matter.

When we are further asked what is matter, we answer readily, It is not God, nor the substance of God; nor the presence of God arrayed in phenomena; nor the uncreated will of God veiled in a world of illusions, deluding us with shadows into the belief of substance: much less is it catter [pejorative term in the book under review], and still less is it nothing. It is a reality, the physical kind or nature of which is as unknown in its quiddity or quality as its existence is certainly known to the reason of man.

* "… its specific nature"
(Click to enlarge) —

The Catholic physics expounded by Cardinal Manning above is the physics of Aristotle.

For a more modern treatment of these topics, see Werner Heisenberg's Physics and Philosophy. For instance:

"The probability wave of Bohr, Kramers, Slater, however, meant… a tendency for something. It was a quantitative version of the old concept of 'potentia' in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality."

Compare to Cardinal Manning's statement above:

"… between something and nothing I can find no intermediate except potentia…"

To the mathematician, the cardinal's statement suggests the set of real numbers between 1 and 0, inclusive, by which probabilities are measured. Mappings of purely physical events to this set of numbers are perhaps better described by applied mathematicians and physicists than by philosophers, theologians, or storytellers. (Cf. Voltaire's mockery of possible-worlds philosophy and, more recently, The Onion's mockery of the fictional storyteller Fournier's quantum flux. See also Mathematics and Narrative.)

Regarding events that are not purely physical– those that have meaning for mankind, and perhaps for God– events affecting conception, birth, life, and death– the remarks of applied mathematicians and physicists are often ignorant and obnoxious, and very often do more harm than good. For such meaningful events, the philosophers, theologians, and storytellers are better guides. See, for instance, the works of Jung and those of his school. Meaningful events sometimes (perhaps, to God, always) exhibit striking correspondences. For the study of such correspondences, the compact topological space [0, 1] discussed above is perhaps less helpful than the finite Galois field GF(64)– in its guise as the I Ching. Those who insist on dragging God into the picture may consult St. Augustine's Day, 2006, and Hitler's Still Point.

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Monday, July 27, 2009

Monday July 27, 2009

Filed under: General,Geometry — Tags: I Ching Wheel, Waymark Prize — m759 @ 2:29 pm

Field Dance

The New York Times
on June 17, 2007:

Design Meets Dance,
and Rules Are Broken

Yesterday's evening entry was
on the fictional sins of a fictional
mathematician and also (via a link
to St. Augustine's Day, 2006), on
the geometry of the I Ching* —

The eternal
combined with
the temporal:

Circular arrangement of I Ching hexagrams based on Singer 63-cycle in the Galois field GF(64)

The fictional mathematician's
name, noted here (with the Augustine-
I Ching link as a gloss) in yesterday's
evening entry, was Summerfield.

From the above Times article–
"Summerspace," a work by
choreographer Merce Cunningham
and artist Robert Rauschenberg
that offers a competing
vision of summer:

Summerspace — Set by Rauschenberg, choreography by Cunningham

Cunningham died last night.

John Cage, Merce Cunningham, Robert Rauschenberg in the 1960's

From left, composer John Cage,
choreographer Merce Cunningham,
and artist Robert Rauschenberg
in the 1960's

"When shall we three meet again?"

* Update of ca. 5:30 PM 7/27– today's online New York Times (with added links)– "The I Ching is the 'Book of Changes,' and Mr. Cunningham's choreography became an expression of the nature of change itself. He presented successive images without narrative sequence or psychological causation, and the audience was allowed to watch dance as one might watch successive events in a landscape or on a street corner."

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Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: General,Geometry — Tags: Cubehenge, Facets, Heinlein Space, Knight's Move, Octad, Octad Generator, Stevens Rock — m759 @ 1:00 pm

Hollywood Nihilism
Meets
Pantheistic Solipsism

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

Tina Fey and Steve Martin at the 2009 Oscars

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

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

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

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

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

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

Heinlein:

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

Stevens:

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

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

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

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

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

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

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

                    (Collected Poems, 528)

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

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

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

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

More positively…

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

The 4x4x4 cube

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

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

Cara:

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

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Wednesday, January 14, 2009

Wednesday January 14, 2009

Filed under: General,Geometry — Tags: Octad, Venn Lotus — m759 @ 2:45 am

Eight is a Gate

From the most highly
rated negative review:

“I never did figure out
what ‘The Eight’ was.”

Various approaches
to this concept
(click images for details):

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Tuesday, October 3, 2006

Tuesday October 3, 2006

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

Serious

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

— Charles Matthews at Wikipedia, Oct. 2, 2006

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

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

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

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

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

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

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

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Monday, August 28, 2006

Monday August 28, 2006

Filed under: General,Geometry — Tags: I Ching Wheel — m759 @ 1:00 am

Today's Sinner:

Augustine of Hippo, who is said to
have died on this date in 430 A.D.

"He is, after all, not merely taking over a Neoplatonic ontology, but he is attempting to combine it with a scriptural tradition of a rather different sort, one wherein the divine attributes most prized in the Greek tradition (e.g. necessity, immutability, and atemporal eternity) must somehow be combined with the personal attributes (e.g. will, justice, and historical purpose) of the God of Abraham, Isaac, and Jacob."

— Stanford Encyclopedia of Philosophy on Augustine

Here is a rather different attempt
to combine the eternal with the temporal:

The Eternal

Symbol of necessity,
immutability, and
atemporal eternity:

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

For details, see
finite geometry of
the square and cube.

The Temporal

Symbol of the
God of Abraham,
Isaac, and Jacob:

The image “http://www.log24.com/log/pix06A/060828-Cloud.jpg” cannot be displayed, because it contains errors.

For details, see
Under God
(Aug. 11, 2006)

The eternal
combined with
the temporal:

Singer 63-cycle in the Galois field GF(64) used to order the I Ching hexagrams

Related material:

Hitler's Still Point and
the previous entry.

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Friday, June 23, 2006

Friday June 23, 2006

Filed under: General,Geometry — Tags: Boole vs. Galois, Fields — m759 @ 2:56 pm

Binary Geometry

There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2ⁿ elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).

Examples:

Charles Sanders Peirce, “The Simplest Mathematics.”
Donald E. Knuth’s discussion of binary hypercubes in “Boolean Basics,” a draft of section 7.1.1 in The Art of Computer Programming, Volume 4: Combinatorial Algorithms
My own discussion of a binary hypercube in Geometry of the 4x4x4 Cube
A more sophisticated example: the geometry of elliptic curves over a binary Galois field. For an excellent introduction, see the Certicom online elliptic curve tutorial. This has an applet illustrating elliptic curves in a space of 256 points (256=16×16, with the x and y variables of a curve each having 16 possible values).
In summary, apart from the fact that the native language of computers has characteristic 2, “binary” mathematics, i.e. mathematics in characteristic 2, is of special interest both in the study of finite geometry (Finite Geometry of the Square and Cube) and in algebraic geometry (see, for instance, the work of Brian Conrad).

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Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: Knight Move, Springer — m759 @ 4:04 pm

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to the “Diamond Theory” of truth in The Non-Euclidean Revolution

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”

— Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

In 1490, The White Knight
(Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com—

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

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Thursday, August 4, 2005

Thursday August 4, 2005

Filed under: General,Geometry — Tags: Alperin, Lo Shu — m759 @ 1:00 pm

Visible Mathematics, continued

Today's mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Related material: Visualizing GL(2,p)

"The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121

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