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.
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.
Yesterday's post suggests a review of the following —
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups" 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}
the design of the points and affine hyperplanes in AG(4, 2), Proof…. … (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2) An explicit construction of the vector space is also easy….) 
Related material: Posts tagged Priority.
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.
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.
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
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."
This is a sequel to yesterday's post Cube Space Continued.
The above sketch indicates, in a vague, handwaving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
For the first two pages, click here.
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
Illustrations—
(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—
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Euclidean square and triangle—
Galois square and triangle—
Background—
This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.
Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.
That approach will appeal to few mathematicians, so here is another.
Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace is a book by Leonard Mlodinow published in 2002.
More recently, Mlodinow is the coauthor, with Stephen Hawking, of The Grand Design (published on September 7, 2010).
A review of Mlodinow's book on geometry—
"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society, May 2002
The Langlands remark is an apt introduction to Mlodinow's more recent work.
It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.
For the latter, see a Google search done this morning—
Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.
Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…
For n=4, there is only one factorisation, which we can write concisely as 1234, 1324, 1423. Its automorphism group is the symmetric group S_{4}, and acts as S_{3} on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.
This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.
See also, in this journal, Window and Window, continued (July 5 and 6, 2010).
Gardner scoffs at the importance of Galois's last letter —
"Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers."
— Last Recreations, page 156
For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
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
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.
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 . . .
"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 .
"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.
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.
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.
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.
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
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)
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."
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).
(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, Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, 
Also sadly …
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."
Continued from June 17, 2013
(John Baez as a savior for atheists):
As an atheistssavior, I prefer Galois…
The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois geometry —
See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:
For the genesis of the figure,
see The Geometry of Logic.
From Zettel (repunctuated for clarity):
249. « Nichts leichter, als sich einen 4dimensionalen Würfel
vorstellen! Er schaut so aus… »
"Nothing easier than to imagine a 4dimensional cube!
It looks like this…
[Here the editor supplied a picture of a 4dimensional cube
that was omitted by Wittgenstein in the original.]
« Aber das meine ich nicht, ich meine etwas wie…
"But I don't mean that, I mean something like…
…nur mit 4 Ausdehnungen! »
but with four dimensions!
« Aber das ist nicht, was ich dir gezeigt habe,
eben etwas wie…
"But isn't what I showed you like…
…nur mit 4 Ausdehnungen? »
…only with four dimensions?"
« Nein; das meine ich nicht! »
"No, I don't mean that!"
« Was aber meine ich? Was ist mein Bild?
Nun der 4dimensionale Würfel, wie du ihn gezeichnet hast,
ist es nicht ! Ich habe jetzt als Bild nur die Worte und
die Ablehnung alles dessen, was du mir zeigen kanst. »
"But what do I mean? What is my picture?
Well, it is not the fourdimensional cube
as you drew it. I have now for a picture only
the words and my rejection of anything
you can show me."
"Here's your damn Bild , Ludwig —"
Context: The Galois Tesseract.
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vectorspace substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.
(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)
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.)
"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."
— G. H. Hardy, A Mathematician's Apology
Part I: An Inch Deep
Part II: An Inch Wide
See a search for "square inch space" in this journal.
See also recent posts with the tag depth.
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 —
(On His Dies Natalis )…
This is asserted in an excerpt from…
"The smallest nonrank 3 strongly regular graphs
which satisfy the 4vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152212—
(Click for clearer image)
Note that Theorem 46 of Klin et al. describes the role
of the Galois tesseract in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric part of the above
exceptional geometriccombinatorial isomorphism.
"How about another hand for the band?
They work real hard for it.
The Cherokee Cowboys, ladies and gentlemen."
— Ray Price, video, "Danny Boy Mid 80's Live"
Other deathly hallows suggested by today's NY Times—
Click the above image for posts from December 14.
That image mentions a death on August 5, 2005, in
"entertainment Mecca" Branson, Missouri.
Another note from August 5, 2005, reposted here
on Monday—
Happy birthday, Keith Richards.
Happy Beethoven's Birthday.
Related material: Abel 2005 and, more generally, Abel.
See also Visible Mathematics.
Odin's Jewel
Jim Holt, the author of remarks in yesterday's
Saturday evening post—
"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."
— Vanity Fair Daily , July 16, 2012
Backstory: Odin + Jewel in this journal.
See also Odin on the Kyoto school —
For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —
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 .
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
Short Story — (Click image for some details.)
Parts of a longer story —
The Daily Princetonian today:
A different cover act, discussed here Saturday:
See also, in this journal, the Galois tesseract and the Crosswicks Curse.
"There is such a thing as a tesseract." — Crosswicks saying
The hypercube model of the 4space over the 2element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galoistesseract model of the 4space over GF(2).
The thirtyfive 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book coauthored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galoistesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliamsSloane book was first published):
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– GianCarlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the ConwaySloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hepth > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array 
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— GianCarlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
Today's previous post recalled a post
from ten years before yesterday's date.
The subject of that post was the
Galois tesseract.
Here is a post from ten years before
today's date.
The subject of that post is the Halmos
tombstone:
"The symbol is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."
— Measure Theory (1950)
For exact proportions, click on the tombstone.
For some classic mathematics related
to the proportions, see September 2003.
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the fivedimensional projective space
over a field. If the field is the twoelement Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M_{24}. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose SpaceTime, and a Finite Model.
Related material:
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) 
Those who prefer story to structure may consult
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4space over GF(2)—
The same field, again disguised as an affine 4space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vectorspace structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“SelfDual Configurations and Regular Graphs.”
Background— George Steiner in this journal
and elsewhere—
"An intensity of outward attention —
interest, curiosity, healthy obsession —
was Steiner’s version of God’s grace."
— Lee Siegel in The New York Times ,
March 12, 2009
(See also Aesthetics of Matter in this journal on that date.)
Steiner in 1969 defined man as "a language animal."
Here is Steiner in 1974 on another definition—
Related material—
Also related — Kantor in 1981 on "exquisite finite geometries," and The Galois Tesseract.
"Debates about canonicity have been raging in my field
(literary studies) for as long as the field has been
around. Who's in? Who's out? How do we decide?"
— Stephen Ramsay, "The Hermeneutics of Screwing Around"
An example of canonicity in geometry—
"There are eight heptads of 7 mutually azygetic screws, each consisting of the screws having a fixed subscript (from 0 to 7) in common. The transformations of LF(4,2) correspond in a onetoone manner with the even permutations on these heptads, and this establishes the isomorphism of LF(4,2) and A_{8}. The 35 lines in S_{3} correspond uniquely to the separations of the eight heptads into two complementary sets of 4…."
— J.S. Frame, 1955 review of a 1954 paper by W.L. Edge,
"The Geometry of the Linear Fractional Group LF(4,2)"
Thanks for the Ramsay link are due to Stanley Fish
(last evening's online New York Times ).
For further details, see The Galois Tesseract.
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M_{24}. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2element field), see The Galois Tesseract.
"Design is how it works." — Steve Jobs
From a commercial testprep firm in New York City—
From the date of the above uploading—

From a New Year's Day, 2012, weblog post in New Zealand—
From Arthur C. Clarke, an early version of his 2001 monolith—
"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."
The numerical (not crystal) pyramid above is related to a sort of
mathematical block design known as a Steiner system.
For its relationship to the graphic block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M_{24}," which contains the following
version of the above numerical pyramid—
For graphic block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.
For the barbed tail of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.
Mathematics —
(Some background for the Galois tesseract )
Narrative —
An essay on science and philosophy in the January 2012
Notices of the American Mathematical Society .
Note particularly the narrative explanation of the doubleslit experiment—
"The assertion that elementary particles have
free will and follow Quality very closely leads to
some startling consequences. For instance, the
waveparticle duality paradox, in particular the baffling
results of the famous double slit experiment,
may now be reconsidered. In that experiment, first
conducted by Thomas Young at the beginning
of the nineteenth century, a point light source
illuminated a thin plate with two adjacent parallel
slits in it. The light passing through the slits
was projected on a screen behind the plate, and a
pattern of bright and dark bands on the screen was
observed. It was precisely the interference pattern
caused by the diffraction patterns of waves passing
through adjacent holes in an obstruction. However,
when the same experiment was carried out much
later, only this time with photons being shot at
the screen one at a time—the same interference
pattern resulted! But the Metaphysics of Quality
can offer an explanation: the photons each follow
Quality in their actions, and so either individually
or en masse (i.e., from a light source) will do the
same thing, that is, create the same interference
pattern on the screen."
This is from "a Ph.D. candidate in mathematics at the University of Calgary."
His essay is titled "A Perspective on Wigner’s 'Unreasonable Effectiveness
of Mathematics.'" It might better be titled "Ineffective Metaphysics."
Mathematics and Narrative, continued
"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"
— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th
"… our laughable abstractions, our wryly ironic pomo angels dancing on the heads of so many misimagined quantum pins."
— Dan Conover on September 1st, 2011
"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'
I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."
— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996
"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"
— Stanford Encyclopedia of Philosophy
"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."
— Interviewer for the National Association of Scholars
From that conversation—
Wolfe : "People in academia should start insisting on objective scholarship, insisting on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"
As for "misimagined quantum pins"…
This journal on the date of the above interview— February 28, 2008—
Illustration from a Perimeter Institute talk given on July 20, 2005
The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by Conover, "The Last Epiphany," and four posts from September 1st, 2011—
Boundary, How It Works, For Thor's Day, and The Galois Tesseract.
Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.
"There is such a thing as a tesseract." —A Wrinkle in Time
Today is day 256 of 2011, Programmers' Day.
Yesterday, Monday, R. W. Barraclough's website pictured the Octad of the Week—
" X never, ever, marks the spot."
See also The Galois Tesseract.
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.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by GianCarlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
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: 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 …
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusementpark ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.
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
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 ."
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!"
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
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.
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
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:
In memory of New Yorker artist Anatol Kovarsky,
who reportedly died at 97 on June 1.
Note the Santa, a figure associated with Macy's at Herald Square.
See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.
A note related both to Galois space and to
the "Herald Square"tagged posts —
"There is such a thing as a length16 sequence."
— Saying adapted from a youngadult novel.
Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.
A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."
From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE "The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two and threedimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed." "Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (JulyAugust 1967): 54. (LeWitt’s contribution was originally untitled.)" 
See also the Cullinane models of some small Galois spaces —
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
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."
— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575
Or gold letters.
From a search for Seagram in this journal —
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building (Oxford University Press, 1979).
A quote from the publisher:
"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."
Three paragraphs from the book (pp. xiiixiv):
19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.
20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.
21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.
Compare to, and contrast with, these illustrations of "Boolean space":
(See also similar illustrations from Berkeley and Purdue.)
Detail of the above image —
Note the "unfolding," as Christopher Alexander would have it.
These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois spaces. See the diamond theorem —
(A review)
There is such a thing as geometry.*
* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.
Wednesday, March 13, 2013

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:
I showed my masterpiece to the
grownups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
* For the title, see Plato Thanks the Academy (Jan. 3).
Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.
For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:
For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.
(Five by Five continued)
As the 3×3 grid underlies the order3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged GaloisPlane Models.
Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).
My response —
Wikipedia's definition of a tetrahedron as a
"trianglebased pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
Norway dance (as interpreted by an American)
I prefer a different, Norwegian, interpretation of "the dance of four."
Related material: 
See also some of Burkard Polster's trianglebased pyramids
and a 1983 trianglebased pyramid in a paper that Polster cites —
(Click image below to enlarge.)
Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :
From On Art and Magic (May 5, 2011) —

(Updated at about 7 PM ET on Dec. 3.)
The seven symmetry axes of the regular tetrahedron
are of two types: vertextoface and edgetoedge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertextoface axes and one edgetoedge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edgetoedge axes.
(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 1617.)
There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetricdifference sum of the
other two members.
(This is the eightfold cube discussed at finitegeometry.org.)
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 1617 of A Geometrical Picture Book , by
Burkard Polster (Springer, 1998).
"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"
—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya
For a similar but more difficult problem involving the
31point projective plane, see yesterday's post
"EuclideanGalois Interplay."
The above new [see update above] Fanoplane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "EuclideanGalois Interplay"
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.
Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former. [9] I am aware only of a series of inhouse publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie IX.
— Stephen Eberhart, Dept. of Mathematics, 
Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. nonEuclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful. Steve 
See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
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.]
* For related remarks, see posts of May 2628, 2012.
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.
Or: The Naked Blackboard Jungle
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , 
Related material: Machete Math and…
Starring the late Eleanor Parker as Swiftly Mrs. Who.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013. The talk's subtitle was
"Oswald Veblen and the Sixhundredacre Woods."
Meanwhile…
Thursday, March 21, 2013

Related material for those who prefer narrative
to mathematics:
Log24 on June 6, 2006:
The Omen :

Related material for those who prefer mathematics
to narrative:
What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry. As the British
geometer H. F. Baker pointed out, 6 is a rather important number
in that discipline. For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.
See also last night's Veblen and Young in Light of Galois.
"There is such a thing as a tesseract." — Madeleine L'Engle
From the prologue to the new Joyce Carol Oates
novel Accursed—
"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.
1905!—the very year of the Curse."
Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract of Madeleine L'Engle.
The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —
"There is such a thing as a tesseract."
A tesseract is a 4dimensional hypercube that
(as pointed out by Coxeter in 1950) may also
be viewed as a 4×4 array (with opposite edges
identified).
Meanwhile, back in 1905…
For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15point projective
Galois space PG(3,2).
See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.
"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'
'Didn’t publish!'
That cold tenure joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."
— George Steiner, 2002 talk at York University
See also Steiner on Galois.
Les Miserables at the Academy Awards
The subtitle of Jack Kerouac's novel Doctor Sax
is Faust Part Three.
Related material—
Types of Ambiguity— Galois Meets Doctor Faustus
(this journal, December 14, 2010).
From an obituary for Helen Nicoll, author
of a popular series of British children's books—
"They feature Meg, a witch whose spells
always seem to go wrong, her cat Mog,
and their friend Owl."
For some (very loosely) related concepts that
have been referred to in this journal, see…
See, too, "Kids grow up" (Feb. 13, 2012).
(Continued from May 29, 2002)
May 29, 1832—
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
– The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.
Commentary from Dec. 2011 on Gardner's word "published" —
Part I: Timothy Gowers on equivalence relations
Part II: Martin Gardner on normal subgroups
Part III: Evariste Galois on normal subgroups
"In all the history of science there is no completer example
of the triumph of crass stupidity over untamable genius…."
— Eric Temple Bell, Men of Mathematics
See also an interesting definition and Weyl on Galois.
Update of 6:29 PM EDT Oct. 30, 2011—
For further details, see Herstein's phrase
"a tribute to the genius of Galois."
"Martin Gardner passed away on May 22, 2010."
Imaginary movie poster from stoneship.org
Context— The Gardner Tribute.
… In the Age of Citation
1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a squareactor by actorbinary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).
— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology, Vol. 23, pp. 127–146 (1993)
From this paper's CiteSeer page—
Citations
766  Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994 
100  The act of creation – Koestler – 1964 
75  Visual Thinking – Arnheim – 1969 
Visual Image of the Problem—
From a Google search today:
Related material—
"It is better to light one candle…"
"… the early favorite for best picture at the Oscars" — Roger Moore
"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
A 1973 review of Koestler's book—
"Koestler's 'call girls,' summoned here and there
by this university and that foundation
to perform their expert tricks, are the butts
of some chilling satire."
Examples of Light—
Felix Christian Klein (1849 June 22, 1925) and Évariste Galois (18111832)
Klein on Galois—
"… in France just about 1830 a new star of undreamtof brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."
— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.
"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."
— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.
Examples of Darkness—
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."
Gardner was reviewing a recent book about Galois by one Amir Alexander.
Alexander himself has written some reviews relevant to the Koestler book above.
See Alexander on—
The 2005 Mykonos conference on Mathematics and Narrative
A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and filmmakers to work together on mathematicallyinspired literary works."
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the fourdiamond figure
with the Pythagorean theorem.
A more relevant association is the
fourdiamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
NonEuclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
FourPart Tesseract Divisions—
The above figure shows how fourpart partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to fourpart partitions
of the 16 points in a finite Galois space
Euclidean spaces versus Galois spaces in a larger context—
Infinite versus Finite The central aim of Western religion — "Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
 Martha Cooley in The Archivist (1998)
The central aim of Western philosophy — Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." 
Another picture related to philosophy and religion—
Jung's FourDiamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 195760) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 195761) 
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.” If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multidimensionally^{*} whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multidimensional symbols beyond our grasp. 
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. 
"I wonder if there's just been a critical mass
of creepy stories about Harvard
in the last couple of years…
A kind of piling on of
nastiness and creepiness…"
— Margaret Soltan, October 23, 2006
Harvard University Press
on Facebook—
http://ping.fm/YrgOh  
May 26 at 6:28 pm via Ping.f 
The book that the late Gardner was reviewing
was published in April by Harvard University Press.
If Gardner's remark were true,
Galois would fit right in at Harvard. Example—
The Harvard math department's pieeating contest—
Wikipedia—
"On June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown."
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
— The Last Recreations, by Martin Gardner, published by Springer in 2007, page 156.
"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue."
– Roger Kimball of The New Criterion, May 23, 2010.
The Gardner piece is now online. It contains…
Gardner's tribute to Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called a 'personality disorder.' His anger was paranoid and unremitting." 
NonEuclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads…. Tesseract "Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Hardening of the thoughtarteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–" "Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–" "Blocks? What kind?" Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid." — "Mimsy Were the Borogoves," Lewis Padgett, 1943 
For the intuitive basis of one type of nonEuclidean* geometry– finite geometry over the twoelement Galois field– see the work of…
Friedrich Froebel
(17821852), who
invented kindergarten.
His "third gift" —
"And take upon's
the mystery of things
as if we were God's spies"
— King Lear
From Log24 on Aug. 19, 2003
and on Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::
"Perhaps we are meant to
see the story as a cubist
retelling of the crucifixion."
From Log24 on
Michaelmas 2007:
Google searches suggested by
Sunday's PA lottery numbers
(midday 170, evening 144)
and by the above
figure of Kate Beckinsale
pointing to an instance of
the number 144 —
Related material:
Beckinsale in another film
(See At the Crossroads,
Log24, Dec. 8, 2006):
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
From the conclusion of
Joan Didion's 1970 novel
Play It As It Lays —
"I know what 'nothing' means,
and keep on playing."
From Play It As It Lays,
the paperback edition of 1990
(Farrar, Straus and Giroux) —
Page 170:
"By the end of a week she was thinking constantly
170
even one microsecond she would have what she had 
"The page numbers
are generally reliable."
The New York Times online,
Friday, Sept. 7, 2007:
Madeleine L’Engle,
Children’s Writer,
Is Dead
Her death, of natural causes, was announced today by her publisher, Farrar, Straus and Giroux."
Related material:
Log24 entries of
August 31—
"That is how we travel."
— A Wrinkle in Time,
Chapter 5,
"The Tesseract"
— and of
September 2
(with update of
September 5)–
"There is such a thing
as a tesseract."
— A Wrinkle in Time
On Spekkens’ toy system and finite geometry
Background–
On finite geometry:
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quantph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4space over the twoelement field.
Related material:
See also arXiv:0707.0074v1 [quantph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)
Powered by WordPress