Saturday, October 24, 2020
Stanley E. Payne and J. A. Thas in 1983* (previous post) —
“… a 4×4 grid together with
the affine lines on it is AG(2,4).”
Payne and Thas of course use their own definition
of affine lines on a grid.
Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).
For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.
* And 26 years later, in 2009.
Comments Off on The Galois Tesseract
Monday, October 15, 2018
The source —
Comments Off on Tesserae for a Tesseract
Monday, March 12, 2018
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Comments Off on “Quantum Tesseract Theorem?”
Saturday, May 20, 2017
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.
Comments Off on van Lint and Wilson Meet the Galois Tesseract*
Thursday, May 11, 2017
Dialogue from the film "Interstellar" —
Cooper: Did it work?
TARS: I think it might have.
Cooper: How do you know?
TARS: Because the bulk beings
are closing the tesseract.
Related material — "Bulk apperception"
in this journal, and …
Comments Off on Reopening the Tesseract
Wednesday, December 28, 2016
Click to enlarge.
Related material: Folk Etymology (Dec. 10, 2016).
Comments Off on Rosetta Tesseracts
Tuesday, March 24, 2015
Yesterday's post suggests a review of the following —
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)
Pages 89:
Substructures of S(5, 8, 24)
An octad is a block of S(5, 8, 24).
Theorem 5.1
Let B_{0} be a fixed octad. The 30 octads disjoint from B_{0}
form a selfcomplementary 3(16,8,3) design, namely
the design of the points and affine hyperplanes in AG(4, 2),
the 4dimensional affine space over F_{2}.
Proof….
… (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say DembowskiWagner or Veblen & Young.
An explicit construction of the vector space is also easy….)

Related material: Posts tagged Priority.
Comments Off on Brouwer on the Galois Tesseract
Tuesday, December 10, 2013
See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)
See also the tag Diamond Theory tesseract in this journal.
Comments Off on Wittgenstein’s Tesseract
Saturday, July 6, 2013
From Andries Brouwer —
* Related material: Yesterday's evening post and The People's Cube.
(By the way, any 4×4 array is a tesseract .)
Comments Off on The People’s Tesseract*
Thursday, August 16, 2012
(Continued from August 13. See also Coxeter Graveyard.)
Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."
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—
Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.
A similar 1950 figure by Coxeter illustrating a tesseract—
For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.
Comments Off on Raiders of the Lost Tesseract
Monday, August 13, 2012
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
Comments Off on Raiders of the Lost Tesseract
Sunday, July 29, 2012
(Continued)
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vectorspace structure:
0 
c 
d 
c + d 
a 
a + c 
a + d 
a + c + d 
b 
b + c 
b + d 
b + c + d 
a + b 
a + b + c 
a + b + d 
a + b +
c + d 
(This vectorspace a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "SelfDual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 ConwaySloane diagram.
Comments Off on The Galois Tesseract
Monday, June 4, 2012
Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3set.*
This suggests a review of a post from last January—
* See a comment on yesterday's post relating it to earlier,
very similar, remarks by Margaret Masterman.
I was unaware yesterday that those remarks exist.
Comments Off on Cube to Tesseract
Tuesday, January 31, 2012
“… a finite set with n elements
is sometimes called an nset ….”
Tesseract formed from a 4set—
The same 16 subsets or points can
be arranged in a 4×4 array that has,
when the array’s opposite edges are
joined together, the same adjacencies
as those of the above tesseract.
“There is such a thing as a 4set.”
— Saying adapted from a novel

Update of August 12, 2012:
Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—
Comments Off on Tesseract
Saturday, September 3, 2011
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, 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 4space over the 2element 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 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The “35 structures” of the abstract were listed, with an application to
Latinsquare 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 M_{24}, 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 misnamed 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: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
Comments Off on The Galois Tesseract
Wednesday, November 18, 2020
Harold Edwards, a founding coeditor of The Mathematical Intelligencer ,
reportedly died at 84 on Tuesday, Nov. 10, 2020.
Images from this journal on that date —
“Surprise Party” revisited —
A Philippine meditation by Alex Garland quoted here on May 6, 2010 —
Comments Off on La Chanson Fatale
Friday, September 11, 2020
From the Vanderbilt University obituary of Vaughan F. R. Jones —
“During the mid1980s, while Jones was working on a problem in von Neumann algebra theory, which is related to the foundations of quantum mechanics, he discovered an unexpected link between that theory and knot theory, a mathematical field dating back to the 19th century.
Specifically, he found a new mathematical expression—now known as the Jones polynomial—for distinguishing between different types of knots as well as links in threedimensional space. Jones’ discovery had been missed by topologists during the previous 60 years, and his finding contributed to his selection as a Fields Medalist.
‘Now there is an area of mathematics called
quantum topology, which basically followed
from his original work,’
said Dietmar Bisch, professor of mathematics.” [Link added.] 
Related to Jones’s work —
“Topological Quantum Information Theory” at
the website of Louis H. Kauffman —
http://homepages.math.uic.edu/~kauffman/Quanta.pdf.
Kauffman —
Comments Off on In Memoriam
Elsewhere on that same date —
Comments Off on Welcome to AMSLaTeX
Monday, July 13, 2020
The above novel uses extensively the term “inscape.”
The term’s originator, a 19thcentury Jesuit poet,
is credited . . . sort of. For other uses of the term,
search for Inscape in this journal. From that search —
A quote from a 1962 novel —
“There’s something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz.”
Addendum for the Church of Synchronology —
The Joe Hill novel above was published (in hardcover)
on Walpurgisnacht —April 30, 2013. See also this journal
on that date.
Comments Off on Unpoetic License
Tuesday, April 21, 2020
The Tesseract Timeline:
Where The Cube Has Been In The …
www.cinemablend.com › news › thetesseracttimeline…
Mar 13, 2019 – With HYDRA. In 1942, Johann Schmidt, a.k.a.
the Red Skull, arrived in Tønsberg to procure the Tesseract
from an ancient church. While he …
Related material from posts tagged Aqua
(suggested by a name in the previous post) —
Lene from Tønsberg —
Comments Off on Living Water
Saturday, March 7, 2020
From “Mathieu Moonshine and Symmetry Surfing” —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
“This presentation of the symmetry groups G_{i} is
particularly welladapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the socalled octad group
G = (Z_{2})^{4}^{ }⋊ A_{8}_{ }.
It can be described as a maximal subgroup of M_{24}
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O_{9 }= {3,5,6,9,15,19,23,24} ∈ 𝒢_{24}. The octad
subgroup is of order 322560, and its index in M_{24}
is 759, which is precisely the number of
different reference octads one can choose.”
This “octad group” is in fact the symmetry group of the affine 4space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79TA37, “Symmetry invariance in a
diamond ring,” by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the “octad group” in 1993 —
Comments Off on The “Octad Group” as Symmetries of the 4×4 Square
Monday, February 24, 2020
“There is such a thing as ▦ .”
— Saying adapted from a 1962 youngadult novel.
Comments Off on Hidden Figure
Tuesday, January 28, 2020
Two of the thumbnail previews
from yesterday's 1 AM post …
"Hum a few bars"
"For 6 Prescott Street"
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University
Department of Social Relations
Center for Research in Personality
Morton Prince House
5 Divinity Avenue
Cambridge 38, Massachusetts
July 17, 1961
Dr. Thomas S. Szasz
c/o Upstate Medical School
Irving Avenue
Syracuse 10, New York
Dear Dr. Szasz:
Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun.
The Myth of Mental Illness is the most important book in the history of psychiatry.
I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century.
It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty.
. . . .

The small Morton Prince House in the above letter might, according to
the abovequoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric KoolAid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three IChing coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster
inChief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4dimensional hypercube H (a tesseract ) has 24 square
2dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4space over the finite (i.e., Galois) twoelement
field GF(2)), the 24 faces transform into 140 4point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbitstabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
Comments Off on Very Stable KoolAid
Monday, January 27, 2020
The phrase "jewel box" in a New York Times obituary online this afternoon
suggests a review. See "And He Built a Crooked House" and Galois Tesseract.
Comments Off on Jewel Box
Monday, October 21, 2019
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Comments Off on Algebra and Space… Illustrated.
Tuesday, October 15, 2019
(The title refers to Log24 posts now tagged Fire Temple.)
In memory of a New Yorker cartoonist who
reportedly died at 97 on October 3, 2019 …
"Read something that means something."
— New Yorker advertising slogan
From posts tagged Tetrahedron vs. Square —
This journal on October 3 —
"There is such a thing as a 4set."
— Saying adapted from a 1962 youngadult novel.
Illustration (central detail a from the above tetrahedral figure) —
Comments Off on Inside the Fire Temple
Wednesday, October 9, 2019
Note that in the pictures below of the 15 twosubsets of a sixset,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
Comments Off on The Joy of Six
Friday, September 27, 2019
"… Max Black, the Cornell philosopher, and others have pointed out
how 'perhaps every science must start with metaphor and end with
algebra, and perhaps without the metaphor there would never have
been any algebra' …."
— Max Black, Models and Metaphors, Cornell U. Press, 1962,
page 242, as quoted in Dramas, Fields, and Metaphors, by
Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
Metaphor —
Algebra —
The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):
1. , , , , ,
2. , , , , ,
3. , , , , ,
4. , , , , ,
5. , , , , ,
6. , , , , .
SEE ALSO: Pauli Matrices
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211217, 1985.
Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 8084, 1982.
Bethe, H. A. and Salpeter, E. Quantum Mechanics of One and TwoElectron Atoms. New York: Plenum, pp. 4748, 1977.
Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGrawHill, 1964.
Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: AddisonWesley, p. 580, 1980.
Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187211, 1955.
Referenced on WolframAlpha: Dirac Matrices
CITE THIS AS:
Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource.
http://mathworld.wolfram.com/DiracMatrices.html

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
— Wallace Stevens, "The Motive for Metaphor"
Comments Off on The Black List
Sunday, September 22, 2019
(Continued from November 27, 2010.)
The previous post suggests a review of a Tilman Piesk
illustration, with the general form of a 4simplex, from
the Wikipedia article titled Simplex . As the article
notes, the lines shown connecting points are those of a
tesseract.
Comments Off on Simplex Sigillum Veri
Wednesday, September 18, 2019
"There is such a thing as a 4set." — Saying adapted
from a 1962 youngadult novel.
Midrash — An image posted here on August 6 —
Comments Off on The Perpetual Identity Crisis
Friday, August 16, 2019
(Continued)
A revision of the above diagram showing
the Galoisadditiontable structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
Comments Off on Nocciolo
Saturday, August 10, 2019
The Square "Inscape" Model of
the Generalized Quadrangle W(2)
Click image to enlarge.
* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.
Comments Off on Schoolgirl Space* Revisited:
Tuesday, August 6, 2019
"There is such a thing as a desktop."
— Saying adapted from a 1962 youngadult novel.
Comments Off on Mathematics and Narrative: The Crosswicks Curse Continues.
Tuesday, July 16, 2019
Comments Off on Schoolgirl Space for Quantum Mystics
Sunday, July 14, 2019
The Quantum Tesseract Theorem Revisited
From page 274 —
"The secret is that the supermathematician expresses by the anticommutation
of his operators the property which the geometer conceives as perpendicularity
of displacements. That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.
Thus far we have touched only the fringe of the structure of our set of sixteen Eoperators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."
A related illustration, from posts tagged Dirac and Geometry —
Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.
Comments Off on Old Pathways in Science:
Tuesday, July 9, 2019
(Continued)
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
Comments Off on Perception of Space
Monday, July 8, 2019
Comments Off on Exploring Schoolgirl Space
Thursday, July 4, 2019
And now, General, time presses; and America is in a hurry.
Have you realized that though you may occupy towns and win battles,
you cannot conquer a nation? — The Devil's Disciple
A figure related to Dürer's "magic" square posted during Devil's Night —
Comments Off on From Devil’s Night 2014
Tuesday, April 9, 2019
The Crosswicks Curse Continues . . .
"There is such a thing as geometry."
— Saying adapted from a 1962 youngadult novel.
Comments Off on Zero Dark Nine:
Monday, March 11, 2019
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
"Think outside the tesseract."
Comments Off on AntMan Meets Doctor Strange
See also "Overarching + Tesseract" in this journal. From the results
of that search, some context for the "inscape" of the previous post —
Comments Off on Overarching Metanarratives
Wednesday, March 6, 2019
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. 535541
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

For some context, see Relativity Problem in this journal.
In the case of PG(3,2), there is a choice of geometric models
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.
The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.
For comparison of the two models, see a figure posted here on
May 21, 2014 —
Labeling the Tetrahedral Model (Click to enlarge) —
"Citation needed" —
The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)," a "reliable source."
Comments Off on The Relativity Problem and Burkard Polster
Thursday, February 28, 2019
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.
Comments Off on Wikipedia Scholarship
Sunday, February 17, 2019
"And the Führer digs for trinkets in the desert."
"See also Acht "
— Cambridge GermanEnglish Dictionary, article on "Elf "
Comments Off on See Also …
Saturday, December 22, 2018
The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005 —
A more recent appearance of the configuration —
Comments Off on CremonaRichmond
Wednesday, December 12, 2018
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."
Comments Off on An Inscape for Douthat
Friday, December 7, 2018
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
Comments Off on The Angel Particle
Tuesday, November 13, 2018
From the 1955 film "Blackboard Jungle" —
From a trailer for the recent film version of A Wrinkle in Time —
Detail of the phrase "quantum tesseract theorem":
From the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —
Some background —
See also posts tagged Dirac and Geometry. For more
background on finite geometry, see a web page
at Thas's institution, Ghent University.
Comments Off on Blackboard Jungle Continues.
Monday, October 15, 2018
The previous post, "Tesserae for a Tesseract," contains the following
passage from a 1987 review of a book about Finnegans Wake —
"Basically, Mr. Bishop sees the text from above
and as a whole — less as a sequential story than
as a box of pied type or tesserae for a mosaic,
materials for a pattern to be made."
A set of 16 of the Wechsler cubes below are tesserae that
may be used to make patterns in the Galois tesseract.
Another Bellevue story —
“History, Stephen said, is a nightmare
from which I am trying to awake.”
— James Joyce, Ulysses
Comments Off on History at Bellevue
Tuesday, September 4, 2018
(Suggested by the previous post)
* For the church, see Kermode + Garber.
Comments Off on MBTI at the Church of St. Frank*
Wednesday, June 27, 2018
A passage that may or may not have influenced Madeleine L’Engle’s
writings about the tesseract :
From Mere Christianity , by C. S. Lewis (1952) —
“Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity”
. . . .
I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.
You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.
Do you see the point? A world of one dimension would be a straight line. In a twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.
Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.
In God’s dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something superpersonal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.
You may ask, “If we cannot imagine a threepersonal Being, what is the good of talking about Him?” Well, there isn’t any good talking about Him. The thing that matters is being actually drawn into that threepersonal life, and that may begin any time —tonight, if you like.
. . . . 
But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
“The colorful story of this undertaking begins with a bang.”
And ends with …
Martin Gardner on Galois—
“Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a ‘personality disorder.’ His anger was
paranoid and unremitting.”
Comments Off on Taken In
Thursday, June 21, 2018
A Buddhist view —
“Just fancy a scale model of Being
made out of string and cardboard.”
— Nanavira Thera, 1 October 1957,
on a model of Kummer’s Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
Comments Off on Models of Being
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "supermathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
Comments Off on Dirac and Geometry (continued)
Monday, June 11, 2018
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "UhOh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Comments Off on Arty Fact
Sunday, June 10, 2018
This journal on April 16, 2018 —
Happy birthday to Pope Emeritus Benedict XVI.
Related material from another weblog in a post also dated April 16, 2018 —
"As I write this, it’s April 5, midway through the eightday
festival of Passover. During this holiday, we Jews air our
grievances against the ancient Pharaoh who enslaved
and oppressed us, and celebrate the feats of strength
with which the Almighty delivered us from bondage —
wait a minute, I think I’m mixing up Passover with Festivus."
. . . .
"Next month: Time and Tesseracts."
From that next post, dated May 16, 2018 —
"The tesseract entered popular culture through
Madeleine L’Engle’s 'A Wrinkle in Time' . . . ."
The post's author, James Propp, notes that
" L’Engle caused some of her readers confusion
when one of the characters … the prodigy
Charles Wallace Murray [sic ] , declared 'Well, the fifth
dimension’s a tesseract.' "
Propp is not unfamiliar with prodigies:
"When I was a kid living in the Long Island suburbs,
I sometimes got called a math genius. I didn’t think
the label was apt, but I didn’t mind it; being put in
the genius box came with some pretty good perks."
— "The Genius Box," a post dated March 16, 2018
To me, Propp seems less like Charles Wallace
and more like the Prime Coordinator —
For further details, see the following synchronicity checks:
Propp March 16 Log24 March 16
Propp April 16 Log24 April 16
Propp May 16 Log24 May 16 .
Comments Off on Pieces of April
Tuesday, May 1, 2018
Remarks on space from 1998 by scifi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by scifi author M. A. Foster
(see the tag "Foster's Space"):
Quoted here on September 26, 2012 —
"All she had to do was kick off and flow."
— The Gameplayers of Zan
"I'se so silly to be flowing but I no canna stay."
— Finnegans Wake
Another work by Sawyer —
Comments Off on Wake
Sunday, April 29, 2018
Wednesday, April 25, 2018
Comments Off on A Deathly Triangle
Thursday, March 29, 2018
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from SixSet Geometry in this journal . . .
Comments Off on “Before Creation Itself . . .”
Sunday, March 11, 2018
. . . With intolerable disrespect for the word …
In particular, the word "theorem."
See also "Quantum Tesseract Theorem" in this journal.
Comments Off on Blackboard Jungle Continues . . .
Thursday, March 8, 2018
Comments Off on Thanking the Academy…
Thursday, January 25, 2018
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Comments Off on Beware of Analogical Extension
Wednesday, January 24, 2018
(Continued)
From a Log24 post of March 4, 2008 —
SINGER, ISAAC:
"Are Children the Ultimate Literary Critics?"
— Top of the News 29 (Nov. 1972): 3236.
"Sets forth his own aims in writing for children and laments
'slice of life' and chaos in children's literature. Maintains that
children like good plots, logic, and clarity, and that they
have a concern for 'socalled eternal questions.'"
— An Annotated Listing of Criticism
by Linnea Hendrickson
"She returned the smile, then looked across the room to
her youngest brother, Charles Wallace, and to their father,
who were deep in concentration, bent over the model
they were building of a tesseract: the square squared,
and squared again: a construction of the dimension of time."
— A Swiftly Tilting Planet,
by Madeleine L'Engle
For "the dimension of time," see A Fold in Time, Time Fold,
and Diamond Theory in 1937.
A Swiftly Tilting Planet is a fantasy for children
set partly in Vespugia, a fictional country bordered by
Chile and Argentina.

Ibid. —
The pen's point:
John Trever, Albuquerque Journal, 2/29/08
Note the figure on the cover of National Review above —
A related figure from Pentagram Design —
See, more generally, Isaac Singer in this journal.
Comments Off on The Pentagram Papers
Tuesday, January 9, 2018
This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.
Comments Off on Koen Thas and Quantum Theory
Monday, January 8, 2018
The Quantum Tesseract Theorem —
Raiders —
A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling
Time Magazine December 25, 2017 – January 1, 2018
Comments Off on Raiders of the Lost Theorem
Thursday, December 28, 2017
The above prose suggests a musical alternative to the Dec. 21
Camazotz song in the posts tagged Quantum Tesseract Theorem . . .
Comments Off on Rocky Start
Saturday, December 23, 2017
Comments Off on Search Result
A figure related to the general connecting theorem of Koen Thas —
See also posts tagged Dirac and Geometry in this journal.
Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."
Comments Off on The Right Stuff
See a Log24 search for "Patterning Windows."
Related material (Click for context) —
.
Comments Off on The Patterning
Comments Off on IT Girl (for Sweet Home Alabama)
Friday, December 22, 2017
From a Log24 post of October 10, 2017 —
Related material from May 25, 2016 —
Comments Off on IT
Thursday, December 21, 2017
TIME magazine, issue of December 25th, 2017 —
" In 2003, Hand worked with Disney to produce a madeforTV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "
Not the dream? It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.
From a trailer for the latest version —
Detail:
From the 1962 book —
"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."
Song adapted from a 1960 musical —
"In short, there's simply not
A more congenial spot
For happyeveraftering
Than here in Camazotz!"
Comments Off on Wrinkles
Sunday, December 10, 2017
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 5979 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
The above symplectic figure appears in remarks on
the diamondtheorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —
Comments Off on Geometry
Thursday, October 19, 2017
The elementary shapes at the top of the figure below mirror
the lookingglass property of the classical Lo Shu square.
The nine shapes at top left* and their lookingglass reflection
illustrate the lookingglass reflection relating two orthogonal
Latin squares over the three digits of modulothree arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 19982008.
** A wellknown construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
Comments Off on Design Grammar***
Tuesday, October 10, 2017
The title refers to today's earlier post "The 35Year Wait."
A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude LéviStrauss.
The LéviStrauss formula —
My "inscape" formula, from a note of Sept. 22, 1982 —
S = f ( f ( X ) ) .
Some mathematics from last year related to the 1982 formula —
See also Inscape in this journal and posts tagged Dirac and Geometry.
Comments Off on Another 35Year Wait
Saturday, September 23, 2017
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Comments Off on The Turn of the Frame
Sunday, August 27, 2017
The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Comments Off on Black Well
Saturday, August 26, 2017
Naive readers may suppose that this sort of thing is
related to what has been dubbed "geometric group theory."
It is not. See posts now tagged Aesthetic Distance.
Comments Off on Aesthetic Distance
Sunday, July 23, 2017
Comments Off on The Partitioned Self
Tuesday, July 11, 2017
At scifi.stackexchange.com —
Why was the Cosmic Cube named the Tesseract
in the Marvel movie series? Is there any specific reason
for the name change? According to me, Cosmic Cube
seems a nice and cooler name.
— Asked March 14, 2013, by Dhwaneet Bhatt
At least it wasn't called 'The AllSpark.'
It's not out of the realm of possibility.
— Solemnity, March 14, 2013
Comments Off on Dialogue from Plato’s Cave
Saturday, June 3, 2017
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Comments Off on Expanding the Spielraum (Continued*)
Tuesday, May 23, 2017
Comments Off on Pursued by a Biplane
Saturday, May 20, 2017
From a review of the 2016 film "Arrival" —
"A seemingly offhand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squidlike aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
SapirWhorf taken to the ludicrous extreme."
— Jordan Brower in the Los Angeles Review of Books
Further on in the review —
"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picturelanguage.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."
For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —
van Lint and Wilson Meet the Galois Tesseract.
The death process of van Lint occurred on Sept. 28, 2004.
See this journal on that date.
Comments Off on The Ludicrous Extreme
Tuesday, May 2, 2017
Comments Off on Image Albums
Saturday, March 25, 2017
The phrase "twin pillars" in a New York Times Fashion & Style
article today suggests a look at another pair of pillars —
This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as "parodies of ideas (which are ghosts)." (See the previous post.)
For a relationship between a 3dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.
Comments Off on Twin Pillars of Symmetry
Sunday, March 19, 2017
"And the Führer digs for trinkets in the desert."
See also the previous post.
Comments Off on Norwegian Sermon
Saturday, December 10, 2016
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the smallconfiguration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 3839 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
Comments Off on Folk Etymology
Friday, December 9, 2016
See Ballet Blanc in this journal.
For a darker perspective, click on the image below.
See also Cartier in The Hexagon of Opposition.
Happy birthday to Kirk Douglas.
Comments Off on Snow Dance
Tuesday, November 22, 2016
See "sacerdotal jargon" in this journal.
For those who prefer scientific jargon —
"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
For some science that is not just jargon, see …
and, also from posts tagged Dirac and Geometry …
The above line complex also illustrates an outer automorphism
of the symmetric group S_{6}. See last Thursday's post "Rotman and
the Outer Automorphism."
Comments Off on Jargon
Wednesday, October 5, 2016
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Comments Off on Sources
Tuesday, August 16, 2016
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
Boulez and Boole and…
Cross and Boolean lattice.
The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —
Update of 10 am on August 16, 2016 —
See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on Midnight Narrative
Friday, July 15, 2016
Robert Nye, author of the novel Falstaff , reportedly died
at 77 on July 2, 2016.
Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.
In memoriam —
From the date of Nye's death:
From Nye's book:
From the date of Heinz's death:
* See also a search for the title in this journal.
Comments Off on Autistic Enchantment*
Friday, June 3, 2016
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
Comments Off on Bruins and van Dam
Wednesday, May 25, 2016
"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.
Comments Off on Framework
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 that structure in this journal, for instance —
See as well yesterday morning's post.
Comments Off on Kummer and Dirac
Tuesday, May 24, 2016
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
Comments Off on Rosenhain and Göpel Revisited
Wednesday, May 11, 2016
The tesseract in last night's post Game Theory
suggests a search in Log24 for "Jewel of Odin."
See also Trinkets.
Comments Off on Jewel of Odin
Tuesday, May 10, 2016
Comments Off on Game Theory
Monday, April 18, 2016
… is a novel by James Morrow reviewed in The New York Times
on March 23, 2008:
"Morrow’s inventiveness is beguiling, as are his delight
in Western philosophy and his concern for the sorry state
of the world. Yet there’s also something comicbookish
about his novel…."
— Siddhartha Deb
"Something comicbookish"
in memory of Albert Einstein,
who reportedly died on this date
in 1955 —
Comments Off on The Philosopher’s Apprentice…
Saturday, April 16, 2016
Today is Kelli O'Hara's last Saturday matinee in "The King and I."
A show that some may prefer —
Related to the plot of Dante's film —
"…it would be quite a long walk
for him if he had to walk straight across."
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
"he would be there, without that long trip.
That is how we travel."
– A Wrinkle in Time , Chapter 5, "The Tesseract"

Comments Off on Matinee (continued)
Monday, February 22, 2016
and versions of "Both Sides Now"
See a New York Times version of "Both Sides Now."
I prefer a version by Umberto Eco.
Related material for storytellers and the Church of Synchronology —
This journal on the date of the above shooting script, 03/19/15.
Comments Off on Schoolgirl Problems…
Monday, February 8, 2016
Related material — Posts tagged Dirac and Geometry.
For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).
Comments Off on A Game with Four Letters
Friday, January 29, 2016
(Continued from Dec. 9, 2013)
"…it would be quite a long walk
for him if he had to walk straight across."
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
"he would be there, without that long trip.
That is how we travel."
– A Wrinkle in Time ,
Chapter 5, "The Tesseract"

From a media weblog yesterday, a quote from the video below —
"At 12:03 PM Eastern Standard Time, January 12th, 2016…."
This weblog on the previous day (January 11th, 2016) —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a youngadult novel
* For some backstory, see a Caltech page.
Comments Off on Excellent Adventure*
Thursday, January 14, 2016
See Triumph of the Will and Box of Nothing.
"And the Führer digs for trinkets in the desert."
Comments Off on Raiders of the Lost Box
Monday, January 11, 2016
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 4227 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239249, Mar. 1978
“Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
“There is such a thing as harmonic analysis of switching functions.”
— Saying adapted from a youngadult novel
Comments Off on Space Oddity
Friday, January 8, 2016
"And the Führer digs for trinkets in the desert."
Comments Off on Triumph of the Will
Monday, November 23, 2015
Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —
His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.
For a biography of Bruins translated by Google, click here.
For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 3952 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992 . (Springer, 2012,
originally published by Kluwer in 1993.)
For morerecent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
Comments Off on Dirac and Line Geometry
Friday, November 20, 2015
(Continued from November 13)
The work of Ron Shaw in this area, ca. 19941995, 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 3space over the 2element Galois field.
Here is an explicit picture —
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
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 morerecent 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.
Comments Off on Anticommuting Dirac Matrices as Skew Lines
Friday, November 13, 2015
Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation ). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Related material:
The 6set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —
Background reading:
Ron Shaw on finite geometry, Clifford algebras, and Dirac groups
(undated compilation of publications from roughly 19941995)—
Comments Off on A Connection between the 16 Dirac Matrices and the Large Mathieu Group
Wednesday, October 21, 2015
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Comments Off on Algebra and Space
Monday, October 12, 2015
“By groping toward the light
we are made to realize
how deep the darkness
is around us.”
— Arthur Koestler,
The Call Girls: A TragiComedy,
Random House, 1973,
page 118

"The Tesseract is where it belongs: out of our reach."
— Samuel L. Jackson as Nick Fury,
quoted here on Epiphany 2013
Earlier … (See Jan. 27, 2012) …
"And the Führer digs for trinkets in the desert."
Comments Off on Ex Tenebris
Monday, September 28, 2015
Click to enlarge:
For the hypercube as a vector space over the twoelement field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 19781986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 57.
Comments Off on Hypercube Structure
Friday, June 19, 2015
There is such a thing as geometry.*
* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.
Comments Off on Footnote
Tuesday, June 9, 2015
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the fourelement 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."
Comments Off on Colorful Song
Saturday, May 23, 2015
On the artist Hilma af Klint (18621944):
"She belonged to a group called 'The Five'…."
Related material — Real Life (Log24, May 20, 2015).
From that post:
Comments Off on Group
Wednesday, May 20, 2015
From Amazon.com —
From the Milwaukee Journal Sentinel Tuesday afternoon —
A 46yearold Jesuit priest who was a Marquette University
assistant professor of theology collapsed on campus
Tuesday morning and died, President Michael Lovell
announced to the campus community in an email….
"Rev. Lúcás (Yiu Sing Luke) Chan, S.J., died after
collapsing this morning in Marquette Hall. Just last Sunday,
Father Chan offered the invocation at the Klingler College
of Arts and Sciences graduation ceremony…."
Synchronicity check…
From this journal on the above publication date of
Chan's book — Sept. 20, 2012 —
From a Log24 post on the preceding day, Sept. 19, 2012 —
“The Game in the Ship cannot be approached as a job,
a vocation, a career, or a recreation. To the contrary,
it is Life and Death itself at work there. In the Inner Game,
we call the Game Dhum Welur , the Mind of God."
— The Gameplayers of Zan
Comments Off on Real Life
Thursday, March 26, 2015
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).
Comments Off on The Möbius Hypercube
Tuesday, March 24, 2015
(Continued from July 16, 2014.)
Some background from Wikipedia:
"Friedrich Ernst Peter Hirzebruch ForMemRS^{[2]}
(17 October 1927 – 27 May 2012)
was a German mathematician, working in the fields of topology,
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'
^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]"}
A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch , a book
reportedly published on September 19, 2012 —
"Sadly, on May 27 this year, Friedrich Hirzebruch,
on whose lectures this book is partially based,
passed away. I would like to express my gratitude
and my admiration by dedicating this book
to his memory.
Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry,
Leibniz Universität Hannover, Germany)

Also sadly …
Comments Off on Hirzebruch
Monday, March 23, 2015
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."
Comments Off on Gallucci’s Möbius Configuration
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