Sunday, September 8, 2019
The ChildResistant 4×4
Thursday, June 27, 2019
Group Actions on the 4x4x4 Cube
For affine group actions, see Ex Fano Appollinis (June 24)
and Solomon's Cube.
For one approach to Mathieu group actions on a 24cube subset
of the 4x4x4 cube, see . . .
For a different sort of Mathieu cube, see Aitchison.
Thursday, February 7, 2019
Geometry of the 4×4 Square: The Kummer Configuration
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finitegeometry properties of the 4×4 square as
a finite affine 4space — properties that are of use in studying the Mathieu
group M_{24 }with the aid of the MOG.
Tuesday, September 13, 2016
Parametrizing the 4×4 Array
The previous post discussed the parametrization of
the 4×4 array as a vector 4space over the 2element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2subsets of a 6set, 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 2element 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.
Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteendot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16point affine geometry over the twoelement Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79TA37.)
See also a 2011 publication of the Mathematical Association of America —
Saturday, November 16, 2019
Logic in the Spielfeld
"A great many other properties of Eoperators
have been found, which I have not space
to examine in detail."
— Sir Arthur Eddington, New Pathways in Science ,
Cambridge University Press, 1935, page 271.
The following 4×4 space, from a post of Aug. 30, 2015,
may help:
The next time she visits an observatory, Emma Stone
may like to do a little dance to …
Tuesday, October 8, 2019
Kummer at Noon
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
Saturday, September 14, 2019
Landscape Art
From "Six Significant Landscapes," by Wallace Stevens (1916) —
VI
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To rightangled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the halfmoon —
Rationalists would wear sombreros.
But see "cones, waving lines, ellipses" in Kummer's Quartic Surface
(by R. W. H. T. Hudson, Cambridge University Press, 1905) and their
intimate connection with the geometry of the 4×4 square.
Sunday, September 8, 2019
A Harvard*Resistant Book Search
(The above title was suggested by today's "ChildResistant 4×4.")
* Specifically, resistant to the above revelation attributed to Barbara Johnson
by Harvard University Press —
"Deconstruction calls attention to gaps and reveals
that their claims upon us are fraudulent."
A related search at Amazon.com:
Barbara Johnson explains Lacan:
" The 'gap' in the real is the leap
from the empirical to
the signifying articulation
of the object of desire;
it cannot be perceived empirically.
It is 'nothing.' "
— Persons and Things , paperback (2010),
Harvard University Press, page 213
Saturday, September 7, 2019
Fashion Calendar
"New York Fashion Week is scheduled Friday, September 6 through
Wednesday, September 11. View the preliminary schedule here. …
UPDATED AS OF: Thu 08/29/19"
— https://cfda.com/fashioncalendar/officialnyfwschedule
View also a fashion photo from this journal on 08/29/19 —
I do not know where the above tank top can be purchased.
Thursday, August 29, 2019
As Well
For some backstory, see
http://m759.net/wordpress/?s="I+Ching"+48+well .
See as well "elegantly packaged" in this journal.
"Well" in written Chinese is the hashtag symbol,
i.e., the framework of a 3×3 array.
My own favorite 3×3 array is the ABC subsquare
at lower right in the figure below —
For Carol Danvers* (Battle Angel )**
Saturday, July 6, 2019
Mythos and Logos
Mythos
Logos
The six square patterns which, applied as above to the faces of a cube,
form "diamond" and "whirl" patterns, appear also in the logo of a coal
mining company —
Related material —
Monday, June 3, 2019
Jar Story
". . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”
— T. S. Eliot, Four Quartets
From Writing Chinese Characters:
“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can… be… subdivided, usually to
9 or 16 squares. . . ."
These “Chinese jars” (as opposed to their contents)
are as follows:
.
See as well Eliot's 1922 remarks on "extinction of personality"
and the phrase "egoextinction" in Weyl's Philosophy of Mathematics —
Saturday, May 4, 2019
The Chinese Jars of ShingTung Yau
The title refers to CalabiYau spaces.
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A less "cosmic" but still noteworthy code — The Golay code.
This resides in a 12dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related CalabiYau "Chinese jar" first described in detail in 1905 —
A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —
ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3pix.pdf
Monday, March 11, 2019
AntMan Meets Doctor Strange
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
Thursday, February 28, 2019
Wikipedia Scholarship
Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this journal a Galois tesseract.
Tuesday, February 26, 2019
Citation
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K_{6} and to the 15 2subsets of a 6set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2subsets of a 6set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6set
in the work of other authors in 1994 and 2002 . . .

GonzalezDorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P^{3}.
American Mathematical Society, Providence, RI, 1994. 
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 30313057.
Wednesday, December 12, 2018
An Inscape for Douthat
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Thursday, November 22, 2018
Rosenhain and Göpel Meet Kummer in Projective 3Space
For further details, see finitegeometry.org/sc/35/hudson.html.
Geometric Incarnation
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
Note also the four 4×4 arrays surrounding the central diamond
in the chi of the chirho page of the Book of Kells —
From a Log24 post
of March 17, 2012
"Interlocking, interlacing, interweaving"
— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science
Thursday, November 8, 2018
Geometry Lesson
From "The Trials of Device" (April 24, 2017) —
See also Wittgenstein in a search for "Ein Kampf " in this journal.
Monday, July 16, 2018
Greatly Exaggerated Report
"The novel has a parallel narrative that eventually
converges with the main story."
— Wikipedia on a book by Foer's novelist brother
Public Squares
An image from the online New York Times
on the date, July 6,
of the above Atlantic article —
An image from "Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
Saturday, July 14, 2018
Expanding the Spiel
The walkerart.org passage above is from Feb. 17, 2011.
See also this journal on Feb. 17, 2011 —
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
Thursday, July 12, 2018
Kummerhenge Illustrated
“… the utterly real thing in writing is the only thing that counts…."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"Omega is as real as we need it to be."
— Burt Lancaster in "The Osterman Weekend"
Thursday, July 5, 2018
Paz:
Some context for what Heidegger called
das SpiegelSpiel des Gevierts
From Helen Lane's translation of El Mono Gramático ,
a book by Nobel winner Octavio Paz first published
in Barcelona by Seix Barral in 1974 —
Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oftrepeated and everdifferent metaphor of identity.
— Paz, Octavio. The Monkey Grammarian 
A related 1960 meditation from Claude LéviStrauss taken from a
Log24 post of St. Andrew's Day 2017, "The Matrix for Quantum Mystics":
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —
Monday, July 2, 2018
In Memoriam
This post is in memory of dancerchoreographer Gillian Lynne,
who reportedly died at 92 on Sunday, July 1, 2018.
For a scene from her younger days, click on Errol Flynn above.
The cube contemplated by Flynn is from Log24 on Sunday.
"This is how we enter heaven, enter dancing."
— Paraphrase of Lorrie Moore (See Oct. 18, 2003.)
Saturday, June 23, 2018
Plan 9 from Inner Space
From Nanavira Thera, "Early Letters," in Seeking the Path —
"nine possibilities arising quite naturally" —
Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15 2subsets of a 6set —
Monday, May 28, 2018
Skewers
A piece cowritten by Ivanov, the author noted in the previous post, was cited
in my "Geometry of the 4×4 Square."
Also cited there — A paper by Pasini and Van Maldeghem that mentions
the Klein quadric.
Those sources suggested a search —
The link is to some geometry recently described by Tabachnikov
that seems rather elegant:
For another, more direct, connection to the geometry of the 4×4 square,
see Richard Evan Schwartz in this journal.
This same Schwartz appears also in the above Tabachnikov paper:
Saturday, April 7, 2018
Sides
The FBI holding cube in "The Blacklist" —
" 'The Front' is not the whole story . . . ."
— Vincent Canby, New York Times film review, 1976,
as quoted in Wikipedia.
See also Solomon's Cube in this journal.
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was allmeaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
See also a related remark by LéviStrauss in 1955:
"…three different readings become possible:
left to right, top to bottom, front to back."
Saturday, February 24, 2018
The Ugly Duck
"What of the night
That lights and dims the stars?
Do you know, Hans Christian,
Now that you see the night?"
— The concluding lines of
"Sonatina to Hans Christian,"
by Wallace Stevens
(in Harmonium (second edition, 1931))
From "Mathmagic Land" (May 22, 2015)
Donald in Mathmagic Land
From "The Trials of Device" (April 24, 2017)
Saturday, February 17, 2018
The Binary Revolution
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement 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 yearend 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:
Friday, February 16, 2018
Two Kinds of Symmetry
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 highenergy 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 2subsets of a 6set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2subsets with the 16
element affine space over the twoelement Galois field GF(2), see my note
of May 26, 1986, "The 2subsets of a 6set 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 (19801981), Talk no. 577, pp. 258276.
Thursday, December 21, 2017
Friday, December 1, 2017
The Architect and the Matrix
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss, 1976
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
From a Log24 post of June 2627, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
Friday, November 24, 2017
The Matrix Meets the Grid
The Matrix —
The Grid —
^{ Picturing the Witt Construction —}
"Read something that means something." — New Yorker ad
Tuesday, October 24, 2017
Visual Insight
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
Saturday, October 14, 2017
Tuesday, September 5, 2017
Florence 2001
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 sevensyllable words.
Sunday, September 3, 2017
Broomsday Revisited
Ivars Peterson in 2000 on a sort of conceptual art —
" Brill has tried out a variety of gridscrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "
— "Scrambled Grids," August 28, 2000
Or not.
If aesthetic sensibilities lead to a 23cycle on a 4×6 grid, the results
may not be pretty —
From "Geometry of the 4×4 Square."
See a Log24 post, Noncontinuous Groups, on Broomsday 2009.
Saturday, September 2, 2017
A Touchstone
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Interuniversal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
Thursday, August 31, 2017
A ConwayNortonRyba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on orderfour 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.
Wednesday, July 26, 2017
Friday, July 14, 2017
March 26, 2006 (continued)
The above image, posted here on March 26, 2006, was
suggested by this morning's post "Black Art" and by another
item from that date in 2006 —
Thursday, July 6, 2017
A Pleasing Situation
The 4x4x4 cube is the natural setting
for the finite version of the Klein quadric
and the eight "heptads" discussed by
Conwell in 1910.
As R. Shaw remarked in 1995,
"The situation is indeed quite pleasing."
Wednesday, June 21, 2017
Concept and Realization
Remark on conceptual art quoted in the previous post —
"…he’s giving the concept but not the realization."
A concept — See a note from this date in 1983:
A realization —
Not the best possible realization, but enough for proof of concept .
Tuesday, May 23, 2017
Saturday, May 20, 2017
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Monday, May 15, 2017
Appropriation at MoMA
For example, Plato's diamond as an object to be transformed —
Versions of the transformed object —
See also The 4×4 Relativity Problem in this journal.
Wednesday, April 26, 2017
A Tale Unfolded
A sketch, adapted tonight from Girl Scouts of Palo Alto —
From the April 14 noon post High Concept —
From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
Monday, April 24, 2017
The Trials of Device
"A blank underlies the trials of device"
— Wallace Stevens, "An Ordinary Evening in New Haven" (1950)
A possible meaning for the phrase "the trials of device" —
See also Log24 posts mentioning a particular device, the pentagram .
For instance —
Friday, April 21, 2017
Music Box
Friday, April 14, 2017
Hudson and Finite Geometry
The above fourelement sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4space (or, equivalently, as lines in the corresponding finite projective
3space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 twoelement subsets of a sixelement set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetricdifference (Galois binary)
sums of the 15 twoelement subsets, and identifying any resulting four
element (or, summing three disjoint twoelement subsets, sixelement)
subsets with their complements. This process was described in my note
"The 2subsets of a 6set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
Monday, April 3, 2017
Sunday, March 5, 2017
The Omega Matrix
Monday, February 20, 2017
Mathematics and Narrative
Mathematics —
Hudson's parametrization of the
4×4 square, published in 1905:
A later parametrization, from this date in 1986:
A note from later in 1986 shows the equivalence of these
two parametrizations:
Narrative —
Posts tagged MemoryHistoryGeometry.
The mathematically challenged may prefer the narrative of the
Creation Matrix from the religion of the Transformers:
"According to religious legend, the core of the Matrix
was created from Solomus, the god of wisdom,
trapped in the form of a crystal by Mortilus, the god
of death. Following the defeat of Mortilus, Solomus
managed to transform his crystal prison into the Matrix—
a conduit for the energies of Primus, who had himself
transformed into the lifegiving computer Vector Sigma."
Saturday, February 18, 2017
Verbum
The Log24 version (Nov. 9, 2005, and later posts) —
VERBUM

See also related material in the previous post, Transformers.
Sunday, January 8, 2017
A Theory of Everything
The title refers to the Chinese book the I Ching ,
the Classic of Changes .
The 64 hexagrams of the I Ching may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.
A related post —
Wednesday, January 4, 2017
A Drama of Many Forms
According to art historian Rosalind Krauss in 1979,
the grid's earliest employers
"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."
The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the nextlargest square grid, the 4×4 array:
See instances of the above image.
Monday, December 19, 2016
Tetrahedral CayleySalmon Model
The figure below is one approach to the exercise
posted here on December 10, 2016.
Some background from earlier posts —
Click the image below to enlarge it.
Sunday, December 11, 2016
Complexity to Simplicity via Hudson and Rosenhain*
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
Friday, November 25, 2016
Priority
Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
twocolor square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
Tuesday, October 18, 2016
Parametrization
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vectorspace nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 twosubsets of a sixset (Hudson, 1905) or by a
blank plus the 5 elements and the 10 twosubsets of a fiveset
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“This is the relativity problem: to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
Note, however, that Weyl's definition of "coordinatization"
is not limited to vectorspace coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vectorspace
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the pointspace
being coordinatized.)
Sunday, October 2, 2016
Westworld
On a new HBO series that opens at 9 PM ET tonight —
Watching Westworld , you can sense a grand mythology unfolding before your eyes. The show’s biggest strength is its worldbuilding, an aspect of screenwriting that many television series have botched before. Often shows will rush viewers into plot, forgetting to instill a sense of place and of history, that you’re watching something that doesn’t just exist in a vacuum but rather is part of some larger ecosystem. Not since Lost can I remember a TV show so committed to immersing its audience into the physical space it inhabits. (Indeed, Westworld can also be viewed as a meta commentary on the art of screenwriting itself: brainstorming narratives, building characters, all for the amusement of other people.) Westworld is especially impressive because it builds two worlds at once: the Western theme park and the futuristic workplace. The Western half of Westworld might be the more purely entertaining of the two, with its shootouts and heists and chases through sublime desert vistas. Behind the scenes, the theme park’s workers show how the robot sausage is made. And as a dystopian office drama, the show does something truly original. — Adam Epstein at QUARTZ, October 1, 2016 
"… committed to immersing its audience
into the physical space it inhabits…."
See also, in this journal, the Mimsy Cube —
"Mimsy Were the Borogoves," "… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play." 
Saturday, September 24, 2016
Core Structure
For the director of "Interstellar" and "Inception" —
At the core of the 4x4x4 cube is …
Cover modified.
Monday, September 19, 2016
Squaring the Pentagon
The "points" and "lines" of finite geometry are abstract
entities satisfying only whatever incidence requirements
yield noncontradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, nonEuclidean)
space.
Models of finite geometries may, however, embed the
points and lines within non finite geometries in order
to aid visualization.
For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not* lines of the Euclidean plane.
Example — See "4×4" in this journal.
Some impose on configurations from finite geometry
the rather artificial requirement that both points and lines
must be representable as those of a Euclidean plane.
Example: A CremonaRichmond pentagon —
A square version of these 15 "points" —
A 1905 square version of these 15 "points"
with digits instead of letters —
See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).
Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2subsets of a 6set." (See one such search.)
* But in some models are subsets of the grid lines
that separate squares within an array.
Friday, September 16, 2016
A CountingPattern
Monday, September 12, 2016
The Kummer Lattice
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M_{24 }"
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 twosubsets of a sixset. See the post of Sept. 13.
Wednesday, August 24, 2016
Core Statements
"That in which space itself is contained" — Wallace Stevens
An image by Steven H. Cullinane from April 1, 2013:
The large Desargues configuration of Euclidean 3space can be
mapped canonically to the 4×4 square of Galois geometry —
On an Auckland University of Technology thesis by Kate Cullinane —
The thesis reportedly won an Art Directors Club award on April 5, 2013.
Thursday, July 28, 2016
The Giglmayr Foldings
Giglmayr's transformations (a), (c), and (e) convert
his starting pattern
1 2 5 6
3 4 7 8
9 10 13 14
11 12 15 16
to three length16 sequences. Putting these resulting
sequences back into the 4×4 array in normal reading
order, we have
1 2 3 4 1 2 4 3 1 4 2 3
5 6 7 8 5 6 8 7 7 6 8 5
9 10 11 12 13 14 16 15 15 14 16 13
13 14 15 16 9 10 12 11 9 12 10 11
(a) (c) (e)
Four length16 basis vectors for a Galois 4space consisting
of the origin and 15 weight8 vectors over GF(2):
0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1
0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1
1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1
1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 .
(See "Finite Relativity" at finitegeometry.org/sc.)
The actions of Giglmayr's transformations on the above
four basis vectors indicate the transformations are part of
the affine group (of order 322,560) on the affine space
corresponding to the above vector space.
For a description of such transformations as "foldings,"
see a search for Zarin + Folded in this journal.
Tuesday, June 7, 2016
Wednesday, May 25, 2016
Framework
"Studies of spin½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 ConwaySloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
Kummer and Dirac
From "Projective Geometry and PTSymmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. ^{1 }"
" ^{1} These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the PluckerKlein correspondence between lines of
a threedimensional projective space and points of a quadric
in a fivedimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 16_{6} configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 16_{6} configuration . . . .
See as well yesterday morning's post.
Tuesday, May 24, 2016
Rosenhain and Göpel Revisited
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted 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 4space over
the twoelement 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
Tuesday, May 3, 2016
Symmetry
A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —
The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was allmeaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above web page* —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
Update of Sept. 5, 2016 — See also a related remark
by LéviStrauss in 1955: "…three different readings
become possible: left to right, top to bottom, front
to back."
* For the underlying mathematics, see a June 21, 1983, research note.
Monday, May 2, 2016
Subjective Quality
The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.
Siobhan Roberts recently wrote a book about the first of these
authors, Conway. I just discovered that last fall she also had an
article about the second author, Sloane, published:
"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.
Meanwhile, in this journal …
Log24 on that same date, Oct. 22, 2015 —
Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective quality, as opposed to the objective quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here last October.
Sunday, May 1, 2016
Sunday Appetizer from 1984
Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —
“What would an organic Christian Sabbath look like today?”
The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich 
That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."
The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the
1988 first edition of Sphere Packings, Lattices and Groups ,
by J. H. Conway and N. J. A. Sloane —
A passage from the 1984 document —
Monday, April 25, 2016
Seven Seals
An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —
"More poetically …"
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twentyone. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
The seven seals from the previous post, with some context —
These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .
Friday, April 8, 2016
Space Cross
For George Orwell
Illustration from a book on mathematics —
This illustrates the Galois space AG(4,2).
For some related spaces, see a note from 1984.
"There is such a thing as a space cross."
— Saying adapted from a youngadult novel
Saturday, November 14, 2015
Space Program
A quote that appeared here on April 14, 2013 —
"I know what 'nothing' means." — Joan Didion
Dirac on the 4×4 matrices of an underlying nothingness —
"Corresponding to the four rows and columns,
the wave function ψ must contain a variable
that takes on four values, in order that the matrices
shall be capable of being multiplied into it."
— P. A. M. Dirac, Principles of Quantum Mechanics,
Fourth Edition, Oxford University Press, 1958,
page 257
Monday, October 5, 2015
Forms that Rhyme:
Friday, August 7, 2015
Parts
Spielerei —
"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."
— Daniel Kehlmann, F: A Novel (2014),
translated from the German by
Carol Brown Janeway
Nicht Spielerei —
A figure from this journal at 2 AM ET
on Monday, August 3, 2015
Also on August 3 —
FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."
MANHATTAN — "Carol Brown Janeway, a Scottishborn
publishing executive, editor and awardwinning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."
Related material — Heisenberg on beauty, Munich, 1970
Monday, August 3, 2015
Text and Context*
"The ORCID organization offers an open and
independent registry intended to be the de facto
standard for contributor identification in research
and academic publishing. On 16 October 2012,
ORCID launched its registry services… and
started issuing user identifiers." — Wikipedia
This journal on the above date —
A more recent identifier —
Related material —
See also the recent posts Ein Kampf and Symplectic.
* Continued.
Wednesday, June 17, 2015
Slow Art, Continued
The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.
Example from mathematics:

Göpel tetrads as subsets of a 4×4 square in the classic
1905 book Kummer's Quartic Surface by R. W. H. T. Hudson.
These subsets were constructed as helpful schematic diagrams,
without any reference to the concept of finite geometry they
were later to embody.

Göpel tetrads (not named as such), again as subsets of
a 4×4 square, that form the 15 isotropic projective lines of the
finite projective 3space PG(3,2) in a note on finite geometry
from 1986 — 
Göpel tetrads as these figures of finite geometry in a 1990
foreword to the reissued 1905 book of Hudson:
Click the Barth passage to see it with its surrounding text.
Related material:
Monday, June 15, 2015
Omega Matrix
See that phrase in this journal.
See also last night's post.
The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the fourdimensional
affine space over the twoelement Galois
field, the appropriate Ω is the 4×4 grid above.
Saturday, April 4, 2015
Harrowing of Hell (continued)
Holy Saturday is, according to tradition, the day of
the harrowing of Hell.
Notes:
The above passage on "Die Figuren der vier Modi
im Magischen Quadrat " should be read in the context of
a Log24 post from last year's Devil's Night (the night of
October 3031). The post, "Structure," indicates that, using
the transformations of the diamond theorem, the notorious
"magic" square of Albrecht Dürer may be transformed
into normal reading order. That order is only one of
322,560 natural reading orders for any 4×4 array of
symbols. The above four "modi" describe another.
Wednesday, April 1, 2015
Manifest O
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. 
Thursday, March 26, 2015
The Möbius Hypercube
The incidences of points and planes in the
Möbius 8_{4 } 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 faceplanes 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 pointplane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," 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 (x_{4}, x_{3}, x_{2}, x_{1}) over the twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 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 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Monday, March 23, 2015
Gallucci’s Möbius Configuration
From H. S. M. Coxeter's 1950 paper
"SelfDual Configurations and Regular Graphs,"
a 4×4 array and a more perspicuous rearrangement—
(Click image to enlarge.)
The above rearrangement brings Coxeter's remarks into accord
with the webpage The Galois Tesseract.
Update of Thursday, March 26, 2015 —
For an explanation of Coxeter's Fig. 24, see Thursday's later
post titled "The Möbius Hypercube."
Saturday, March 14, 2015
Unicode Diamonds
The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.
◢◣
◥◤
Some variations:
◤◥
◣◢
◤◥
◢◣
◤◣
◢◥
◤◣
◥◢
Such combined Unicode characters —
◢
black lower right triangle,
◣
black lower left triangle,
᭘
black upper left triangle,
᭙
black upper right triangle
— might be used for a textonly version of the Diamond 16 Puzzle
that is more easily programmed than the current version.
The tricky part would be coding the letterspacing and
lineheight to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.
Update of 11:20 PM ET March 15, 2015 —
Seekers of simplicity should note that there is
a simple program in the Processing.js language, not using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamondtheorem array when the
display window is clicked.
Saturday, February 14, 2015
Valentine Dance
Wednesday, January 14, 2015
Serial Box
Enotes.com on Herman Wouk's 1985 novel Inside, Outside :
"The 'outside' of the title is the goyish world
into which David’s profession has drawn him;
the 'inside' is the warm life of his Russian
Jewish family on which he, as narrator, reflects
in the course of the novel."
For a different sort of 'inside' life, see this morning's post
Gesamtkunstwerk , and Nathan Shields's Feb. 8, 2011,
tribute to a serial composer "In Memoriam, Milton Babbitt."
Some other context for Shields's musical remarks —
Doctor Faustus and Dürer Square.
For a more interesting contrast of inside with outside
that has nothing to do with ethnicity, see the Feb. 10,
2014, post Mystery Box III: Inside, Outside, about
the following box:
.
Saturday, October 25, 2014
Foundation Square
In the above illustration of the 345 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
twodimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ_{5}).
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 HoffmanSingleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and 
the Web pages "HoffmanSingleton Graph" and "HigmanSims Graph"
of A. E. Brouwer.
Hafner's abstract:
We describe the HoffmanSingleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ_{5}.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5related 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 fourdimensional vector space over GF(2).)
Tuesday, October 21, 2014
Tools
(Night at the Museum continues.)
"Strategies for making or acquiring tools
While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"
— Jamie CohenCole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science vol. 40, no. 4 (2007): 567597.
Obituary of a cofounder, in 1960, of the Center for Cognitive Studies at Harvard:
"Disciplinary Exchange" —
In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —
The Kaleidoscope Puzzle, The Diamond 16 Puzzle,
The 2x2x2 Cube, and The 4x4x4 Cube
"Intellectual Economies" —
In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —
This illustrates not geometry, but rather creative capitalism.
Related material from the date of the above Harvard death: Art Wars.
Monday, October 13, 2014
Raiders of the Lost Theorem
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.
One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:
In context (click to enlarge):
Update of later the same day —
Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —
“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the wouldbe constructor.”
* Defined by Sallows later in the book:
“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”
** See some geometric matrices by Cullinane in a March 2001 webpage.
Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:
“51 pp. on the symmetries & algebra of
matrices with geometricfigure entries.”
— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society
The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.
Sunday, September 21, 2014
Sermon
The previous post discussed the anatomy of the sum 9 + 6.
A different approach: “A” and “The 6 spreads in A” below —
Wednesday, September 17, 2014
Raiders of the Lost Articulation
Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
Tuesday, September 9, 2014
Smoke and Mirrors
This post is continued from a March 12, 2013, post titled
"Smoke and Mirrors" on art in Tromsø, Norway, and from
a June 22, 2014, post on the nineteenthcentury
mathematicians Rosenhain and Göpel.
The latter day was the day of death for
mathematician Loren D. Olson, Harvard '64.
For some background on that June 22 post, see the tag
Rosenhain and Göpel in this journal.
Some background on Olson, who taught at the
University of Tromsø, from the American Mathematical
Society yesterday:
Olson died not long after attending the 50th reunion of the
Harvard Class of 1964.
For another connection between that class (also my own)
and Tromsø, see posts tagged "Elegantly Packaged."
This phrase was taken from today's (print)
New York Times review of a new play titled "Smoke."
The phrase refers here to the following "package" for
some mathematical objects that were named after
Rosenhain and Göpel — a 4×4 array —
For the way these objects were packaged within the array
in 1905 by British mathematician R. W. H. T. Hudson, see
a page at finitegometry.org/sc. For the connection to the art
in Tromsø mentioned above, see the diamond theorem.
Sunday, August 31, 2014
Sunday School
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
Wednesday, August 13, 2014
Stranger than Dreams*
Illustration from a discussion of a symplectic structure
in a 4×4 array quoted here on January 17, 2014 —
See symplectic structure in this journal.
* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.
Monday, August 4, 2014
A Wrinkle in Space
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
Sunday, July 20, 2014
Sunday School
Paradigms of Geometry:
Continuous and Discrete
The discovery of the incommensurability of a square's
side with its diagonal contrasted a wellknown discrete
length (the side) with a new continuous length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous configuration at
left is embodied in the discrete unit cells of the square at right.
See Desargues via Galois (August 6, 2013).
Wednesday, May 21, 2014
The Tetrahedral Model of PG(3,2)
The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3space PG(3,2) over the
twoelement 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.)
Thursday, March 27, 2014
Diamond Space
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the sixdimensional affine space
over the twoelement Galois field.
The reason for the name:
Click to enlarge.
Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the TurynCurtis construction
from the University of Cambridge —
— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M_{24},” in slides for lectures 18 from lectures
at Cambridge in 20102011 on “Sporadic and Related Groups.”
See also the Parker lectures of 20122013 on the same topic.
A third construction of Curtis’s 35 4×6 1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4partitions of an 8set with the 35 lines of the projective 3space
over the 2element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22March 23 —
Adding together as (0,1)matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S_{3} on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this “byhand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction, not by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.
* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
Saturday, March 8, 2014
Conwell Heptads in Eastern Europe
“Charting the Real FourQubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [mathph] 26 Jun 2012 —
P. 4— “It was found that Q ^{+}(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad [18].
The set of 28 points lying off Q ^{+}(5,2) comprises
eight such heptads, any two having exactly one
point in common.”
P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q ^{+}(5,2).”
[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76
A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R^{8} discussed there.
For more about Conwell heptads, see The Klein Correspondence,
Penrose SpaceTime, and a Finite Model.
For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —
Walter Gropius
Sunday, March 2, 2014
Sermon
Raiders of the Lost (Continued)
"Socrates: They say that the soul of man is immortal…."
From August 16, 2012—
In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.
An 1892 figure by Jowett illustrating Plato's Meno—
A more correct version, from hermespress.com —
Socrates: He only guesses that because the square is double, the line is double.Meno: True.
Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of thisthat is to say of eight feet; and I want to know whether you still say that a double square comes from double line? [Boy] Yes. Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)? [Boy] Certainly.
Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet? [Boy] Yes. Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet? [Boy] Yes. Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN) [Boy] True. Socrates: And is not that four times four? [Boy] Certainly. Socrates: And four times is not double? [Boy] No, indeed. Socrates: But how much? [Boy] Four times as much. Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much. [Boy] True. Socrates: Four times four are sixteen— are they not? [Boy] Yes. 
As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:
A 4×4 square version, by Coxeter in 1950, of a tesseract—
This square version we may call the Galois tesseract.
Monday, February 10, 2014
Mystery Box III: Inside, Outside
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Thursday, February 6, 2014
The Representation of Minus One
For the late mathematics educator Zoltan Dienes.
"There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities."
— Article by "Melanie" at Zoltan Dienes's website
Dienes reportedly died at 97 on Jan. 11, 2014.
From this journal on that date —
A star figure and the Galois quaternion.
The square root of the former is the latter.
Update of 5:01 PM ET Feb. 6, 2014 —
An illustration by Dienes related to the diamond theorem —
See also the above 15 images in …
… and versions of the 4×4 coordinatization in The 4×4 Relativity Problem
(Jan. 17, 2014).
Saturday, February 1, 2014
The Delft Version
My webpage "The Order4 Latin Squares" has a rival—
"Latin squares of order 4: Enumeration of the
24 different 4×4 Latin squares. Symmetry and
other features."
The author — Yp de Haan, a professor emeritus of
materials science at Delft University of Technology —
The main difference between de Haan's approach and my own
is my use of the fourcolor decomposition theorem, a result that
I discovered in 1976. This would, had de Haan known it, have
added depth to his "symmetry and other features" remarks.
Saturday, January 18, 2014
The Triangle Relativity Problem
A sequel to last night's post The 4×4 Relativity Problem —
In other words, how should the triangle corresponding to
the above square be coordinatized ?
See also a post of July 8, 2012 — "Not Quite Obvious."
Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly item published
in 1984.
Thursday, January 16, 2014
Confession of a Sucker
Today's 11 AM (ET) post was suggested by a New York Times
article, online yesterday, about art gallery owner Lisa Cooley.
A check of Cooley's website yields the image below,
related to Beckett's Molloy .
For the relevant passage from Molloy , click the following:
"I took advantage of being at the seaside
to lay in a store of suckingstones."
For posts on Molloy in this journal, click Beckett + Molloy .
Cynthia Daignault, 2011:
The one I shall now describe, if I can…
Oil on linen, in 2 parts: 40 x 30 inches, 96 x 75 inches
Related art theory —
Friday, November 1, 2013
Cameron’s Group Theory Notes
In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—
 Definition 1.3.1, Group actions, and example on partitions of a 4set, p. 19.
 Exercise 1.1, The group of Fanoplane symmetries, p. 35.
 Exercise 2.17, The group of the empty set and the 15 twosubsets of a sixset, p. 66.
 Section 3.1.2, The holomorph of a group, p. 70.
 Exercise 3.7, The groups A_{8} and AGL(4,2), p. 78.
Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.
But note also Log24 posts tagged Not Theology.
Monday, October 28, 2013
Stella’s Goal
The Whitney Museum of American Art has stated
that artist Frank Stella in 1959
"wanted to create work that was methodical,
intellectual and passionless."
Source: Whitney Museum, transcript of audio guide.
Related material:
A figure from this journal on July 13, 2003…
… and some properties of that figure.
Sunday, September 22, 2013
Incarnation, Part 2
"… a list of group theoretic invariants
and their geometric incarnation…"
— David Lehavi on the Kummer 16_{6} configuration in 2007
Related material —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
"This is not theology; this is mathematics."
— Steven H. Cullinane on four quartets
To wit:
Saturday, September 21, 2013
Geometric Incarnation
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Thursday, September 5, 2013
Wednesday, September 4, 2013
Saturday, August 17, 2013
UptoDate Geometry
The following excerpt from a January 20, 2013, preprint shows that
a Galoisgeometry version of the large Desargues 15_{4}20_{3} configuration,
although based on the nineteenthcentury work of Galois* and of Fano,**
may at times have twentyfirstcentury applications.
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 15_{4}20_{3} configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106132.