The phrase "jewel box" in a New York Times obituary online this afternoon
suggests a review. See "And He Built a Crooked House" and Galois Tesseract.
Monday, January 27, 2020
Jewel Box
Wednesday, October 24, 2018
Mystery Box
Tuesday, June 5, 2018
Mystery Box*
From The Force Awakens —
See also other posts now tagged Mystery Box.
* A phrase of filmmaker J.J. Abrams, director
of The Force Awakens (2015). See Abrams
and a different mystery box in The New York
Times on June 2, 2011.
Wednesday, May 2, 2018
Galois’s Space
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Saturday, May 20, 2017
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Tuesday, May 31, 2016
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
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.
Tuesday, March 24, 2015
Brouwer on the Galois Tesseract
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.
Monday, January 19, 2015
Wednesday, January 14, 2015
Serial Box
Enotes.com on Herman Wouk's 1985 novel Inside, Outside :
"The 'outside' of the title is the goyish world
into which David’s profession has drawn him;
the 'inside' is the warm life of his Russian
Jewish family on which he, as narrator, reflects
in the course of the novel."
For a different sort of 'inside' life, see this morning's post
Gesamtkunstwerk , and Nathan Shields's Feb. 8, 2011,
tribute to a serial composer "In Memoriam, Milton Babbitt."
Some other context for Shields's musical remarks —
Doctor Faustus and Dürer Square.
For a more interesting contrast of inside with outside
that has nothing to do with ethnicity, see the Feb. 10,
2014, post Mystery Box III: Inside, Outside, about
the following box:
.
Tuesday, November 25, 2014
EuclideanGalois Interplay
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.
Monday, February 10, 2014
Mystery Box III: Inside, Outside
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Wednesday, February 5, 2014
Mystery Box II
Continued from previous post and from Sept. 8, 2009.
Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the nonEuclidean geometry of Galois space.
In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.
Tuesday, February 4, 2014
Mystery Box
In honor of the tenth anniversary of Facebook
Viewed in the Chrome browser, a Facebook post from
January 29, 2014, displays an artist's Mystery Box*…
In the Internet Explorer browser, the mystery is solved:
Further details —
Related material — Lyche + Geometry in this journal.
See also the cat and triangle pictured by David Justice yesterday—
.
* A phrase of filmmaker J.J. Abrams. Click the link
for further details. See also a mystery box
in The New York Times on June 2, 2011.
Tuesday, August 6, 2013
Desargues via Galois
The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.
Update of Aug. 7, 2013: See also an expanded PDF version.
Sunday, March 10, 2013
Galois Space
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:
 Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 720 (1986).For the first two pages, click here.
 Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
Monday, March 4, 2013
Occupy Galois Space
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.
Thursday, February 21, 2013
Galois Space
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—
Sunday, July 29, 2012
The Galois Tesseract
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.
Thursday, July 12, 2012
Galois Space
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.)
Tuesday, July 10, 2012
Euclid vs. Galois
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.
Saturday, September 3, 2011
The Galois Tesseract (continued)
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.
Thursday, September 1, 2011
Friday, September 17, 2010
The Galois Window
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.
Friday, January 29, 2016
Thursday, May 14, 2020
For Mask Aficionados
Saturday, September 17, 2016 
For those who prefer comedy —
Other toys: Archimedes at Hiroshima and related posts.
Tony Award
“Tony Stark: That’s how I wished it happened.
Binarily Augmented RetroFraming, or BARF.
God, I gotta work on that acronym.
An extremely costly method of hijacking the
hippocampus to . . . clear traumatic memories. Huh.”
Another acronym — AIEEE !
Tuesday, January 28, 2020
Very Stable KoolAid
Two of the thumbnail previews
from yesterday's 1 AM post …
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 July 17, 1961
Dr. Thomas S. Szasz 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.
Saturday, January 18, 2020
Interplay
"This interplay of necessity and contingency
produces our anxious— and highly pleasurable—
speculation about the future path of the story."
— Michel Chaouli in "How Interactive Can Fiction Be?"
(Critical Inquiry 31, Spring 2005, page 613.)
See also . . .
Continuing previous Modal Diamond Box posts:
Sunday, December 16, 2018
Sunday School News
See as well Friday night's post "Lone Star Wars."
Wednesday, October 24, 2018
The Cracked Potter
Monday, October 15, 2018
History at Bellevue
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
Sunday, September 9, 2018
Plan 9 Continues.
"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
Sunday, September 2, 2018
Monday, June 11, 2018
Arty Fact
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 . . .
Sunday, June 10, 2018
Pieces of April
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:
Sunday, May 20, 2018
Not So Cryptic
From the date of the New York Times James Bond video
referenced in the previous post, "A Cryptic Message" —
Sunday, April 29, 2018
Amusement
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.
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
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."
Sunday, March 4, 2018
The Square Inch Space: A Brief History
Thursday, January 25, 2018
Beware of Analogical Extension
"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 .
Friday, December 8, 2017
Logos (Continued)
"Denn die Welt braucht ewig die Wahrheit,
also braucht sie ewig Heraklit:
obschon er ihrer nicht bedarf.
Was geht ihn sein Ruhm an?
Der Ruhm bei »immer fortfließenden Sterblichen!«,
wie er höhnisch ausruft.
Sein Ruhm geht die Menschen etwas an, nicht ihn,
die Unsterblichkeit der Menschheit braucht ihn,
nicht er die Unsterblichkeit des Menschen Heraklit.
Das, was er schaute, die Lehre vom Gesetz im Werden
und vom Spiel in der Notwendigkeit , muß von jetzt
ab ewig geschaut werden: er hat von diesem größten
Schauspiel den Vorhang aufgezogen."
Logos for Philosophers
(Suggested by Modal Logic) —
Saturday, September 23, 2017
The Turn of the Frame
"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.
Friday, September 15, 2017
Space Art
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.
Sunday, August 27, 2017
Black Well
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.
Sunday, August 13, 2017
Compare and Contrast
From The Atlantic , September 2017 issue, online —
"How America Lost Its Mind," by former Harvard Lampoon
writer Kurt Andersen —
The Atlantic 's embedded Google ad for "Quantum Space Elements"
is, by the way, completely unrelated to similarsounding work on
models of space in finite geometry (cf. tsimtsum ) . . .
Saturday, June 3, 2017
Expanding the Spielraum (Continued*)
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Tuesday, May 23, 2017
Saturday, May 20, 2017
The Ludicrous Extreme
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.
Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
Wednesday, October 5, 2016
Sources
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)
Wednesday, August 24, 2016
Core Statements
"That in which space itself is contained" — Wallace Stevens
An image by Steven H. Cullinane from April 1, 2013:
The large Desargues configuration of Euclidean 3space can be
mapped canonically to the 4×4 square of Galois geometry —
On an Auckland University of Technology thesis by Kate Cullinane —
The thesis reportedly won an Art Directors Club award on April 5, 2013.
Monday, August 1, 2016
Cube
From this journal —
See (for instance) Sacred Order, July 18, 2006 —
From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —
The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelvefoot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralowfrequency vibration of a massive engine. I drift toward the box, mesmerized.
— Crouch, Blake. Dark Matter: A Novel 
See also Log24 on the publication date of Dark Matter .
Saturday, June 18, 2016
Midnight in Herald Square
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.
Sunday, May 8, 2016
The Three Solomons
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 —
Friday, May 6, 2016
Monday, January 11, 2016
Space Oddity
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
Wednesday, December 23, 2015
Splitting Apart
Sunday, December 13, 2015
The Monster as Big as the Ritz
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
Monday, November 2, 2015
Colorful Story
"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
Wednesday, October 21, 2015
Algebra and Space
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Wednesday, August 26, 2015
“The Quality Without a Name”
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 —
Friday, August 14, 2015
Discrete Space
(A review)
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Tuesday, June 9, 2015
Colorful Song
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."
Thursday, March 26, 2015
The Möbius Hypercube
The incidences of points and planes in the
Möbius 8_{4 } configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and faceplanes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the pointplane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x_{4}, x_{3}, x_{2}, x_{1}) over the twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Monday, January 19, 2015
Product 19:
Revisionism
From Wikipedia as of today:
"In fiction, revisionism is the retelling of a story
or type of story with substantial alterations in
character or environment, to 'revise' the view
shown in the original work. Unlike most usages
of the term revisionism, this is not generally
considered pejorative.
The film Dances with Wolves is a revisionist
Western because it portrays the Native Americans
sympathetically instead of as the savages of
traditional Westerns, which have been criticized
as racist. Similarly, the novel Wicked by
Gregory Maguire is a revisionist account of
The Wonderful Wizard of Oz , which portrays the
Wicked Witch of the West fighting for what she
believes is right, and the Wizard as a ruthless
dictator of Oz."
See also another Wikipedia article's Revision History.
Sunday, January 18, 2015
Double Cross
Cross of Gold:
"I would tell them about Rhiannon,
and about my treasured gold cross…."
— Stevie Nicks
Dagger Cross:
See Dagger Definitions, by James Joyce:
"Hold to the now, the here, through which
all future plunges to the past."
A Jew's View:
Thursday, January 15, 2015
Princeton Music continues…
A post yesterday linked to a discussion
of the Faustian music of Milton Babbitt,
a serial composer who reportedly died
in Princeton on Saturday, Jan. 29, 2011.
Related material from this journal in
January 2005:
See also "me into you, you into me"
("Taking Lucifer Seriously," Jan. 24, 2004)
and the Saturday night "cold open" in this
journal on the date of Babbitt's death.
Wednesday, January 14, 2015
Kulturkampf for Princeton*
Einstein and Thomas Mann, Princeton, 1938
A sequel to Princeton Requiem,
Gesamtkunstwerk , and Serial Box —
Fearful Symmetry, Princeton Style:
* See as well other instances of Kulturkampf in this journal.
Monday, January 5, 2015
Gitterkrieg*
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).
Monday, December 29, 2014
Dodecahedron Model of PG(2,5)
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.
Thursday, December 18, 2014
Platonic Analogy
(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.
Wednesday, December 3, 2014
Pyramid Dance
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.)
Sunday, November 30, 2014
Two Physical Models of the Fano Plane
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.)
Wednesday, November 26, 2014
A Tetrahedral FanoPlane Model
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.
Class Act
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.
Monday, September 22, 2014
Sunday, September 21, 2014
Uncommon Noncore
This post was suggested by Greg Gutfeld’s Sept. 4 remarks on Common Core math.
Problem: What is 9 + 6 ?
Here are two approaches suggested by illustrations of Desargues’s theorem.
Solution 1:
9 + 6 = 10 + 5,
as in Common Core (or, more simply, as in common sense), and
10 + 5 = 5 + 10 = 15 as in Veblen and Young:
Solution 2:
In the figure below,
9 + 6 = no. of V’s + no. of A’s + no. of C’s =
no. of nonempty squares = 16 – 1 = 15.
(Illustration from Feb. 10, 2014.)
The silly educationists’ “partner, anchor, decompose” jargon
discussed by Gutfeld was their attempt to explain “9 + 6 = 10 + 5.”
As he said of the jargon, “That’s not math, that’s the plot from ‘Silence of the Lambs.'”
Or from Richard, Frank, and Marcus in last night’s “Intruders”
(BBC America, 10 PM).
Sunday, September 14, 2014
Sensibility
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:
Sunday, August 31, 2014
Sunday School
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
Friday, February 21, 2014
Night’s Hymn of the Rock
One way of interpreting the symbol _{}
at the end of yesterday's post is via
the phrase "necessary possibility."
See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .
The Tablet post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—
On that day (July 23) another weblog had
a post titled
Wallace Stevens: Night's Hymn of the Rock.
Some related narrative —
I prefer the following narrative —
Part I: Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"
Part II: Mystery Box III: Inside, Outside (2014)
Thursday, February 20, 2014
Relativity Blues
A review of this date in 2005 —
Modal Theology
"We symbolize logical necessity
with the box
and logical possibility
with the diamond
— Keith Allen Korcz
And what do we
symbolize by _{} ?
Tuesday, July 9, 2013
Vril Chick
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 .
Tuesday, February 26, 2013
Publication
"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
Friday, February 1, 2013
Get Quotes
For Tony Kushner fans:
For logic fans:
In the boxdiamond notation, the axiom Searle quotes is
"The euclidean property guarantees the truth of this." — Wikipedia
Linking to Euclid
Clicking on "euclidean" above yields another Wikipedia article…
"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."
Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.
Monday, August 13, 2012
Raiders of the Lost Tesseract
(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—
Tuesday, May 29, 2012
The Shining of May 29
(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" —
Sunday, October 30, 2011
Sermon
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."
Sunday, July 10, 2011
Wittgenstein’s Diamond
Philosophical Investigations (1953)—
97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest thing there is
(Tractatus LogicoPhilosophicus No. 5.5563).
— Translation by G.E.M. Anscombe
All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.
(Our problems are not abstract but perhaps
the most concrete that there are.)
97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach sein. Sie ist vor aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh. No. 5.5563.)
Related language in Łukasiewicz (1937)—
* Updates of 9:29 PM ET July 10, 2011—
A mnemonic from a course titled "Galois Connections and Modal Logics"—
"Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written (square) for necessarily
and (diamond) for possibly. Then, for example, P can be read as
'it is possibly the case that P .'"
See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition— "The diamond ⋄ symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬⋄¬. The dual symbol □ was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich ‘possible.’" Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159. Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen. Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century. 
Sunday, June 26, 2011
Sunday Dinner
From "Sunday Dinner" in this journal—
"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"
—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010
Some table topics—
Today's midday New York Lottery numbers were 027 and 7002.
The former suggests a Galois cube, the latter a course syllabus—
CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science
An item from that syllabus:
Six  22 February 2011  DES  History of DES; Encryption process; Decryption; Expander function; Sboxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Presentday lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2. 
Related material: A novel, PopCo , was required reading for the course.
Discuss a different novel by the same author—
Discuss the author herself, Scarlett Thomas.
Background for the discussion—
Derrida in this journal versus Charles Williams in this journal.
Related topics from the above syllabus date—
Metaphor and Gestell and Quadrat.
Some context— Midsummer Eve's Dream.
Wednesday, October 20, 2010
Celebration of Mind
"Martin Gardner passed away on May 22, 2010."
Imaginary movie poster from stoneship.org
Context— The Gardner Tribute.
Monday, September 27, 2010
The Social Network…
… 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
Tuesday, June 22, 2010
Mathematics and Narrative, continued
"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."
Saturday, June 19, 2010
Imago Creationis
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. 
Friday, June 4, 2010
A Better Story
Continued from May 8
(Feast of Saint Robert Heinlein)
“Wells and trees were dedicated to saints. But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden. Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy.”
— Charles Williams, Witchcraft, Faber and Faber, London, 1941
Why "Saint" Robert? See his accurate depiction of evil– the Eater of Souls in Glory Road.
For more on Williams's "other capacities," see Heinlein's story "Lost Legacy."
A related story– Fritz Leiber's "The Mind Spider." An excerpt:
The conference—it was much more a hyperintimate
gabfest—proceeded.
"My static box bugged out for a few ticks this morning,"
Evelyn remarked in the course of talking over the
trivia of the past twentyfour hours.
The static boxes were an invention of Grandfather
Horn. They generated a tiny cloud of meaningless brain
waves. Without such individual thoughtscreens, there was
too much danger of complete loss of individual personality
—once Grandfather Horn had "become" his infant daughter
as well as himself for several hours and the unfledged
mind had come close to being permanently lost in its own
subconscious. The static boxes provided a mental wall be
– hind which a mind could safely grow and function, similar
to the wall by which ordinary minds are apparently
always enclosed.
In spite of the boxes, the Horns shared thoughts and
emotions to an amazing degree. Their mental togetherness
was as real and as mysterious—and as incredible—as
thought itself . . . and thought is the original angelcloud
dancing on the head of a pin. Their present conference
was as warm and intimate and tart as any actual family
gathering in one actual room around one actual table.
Five minds, joined together in the vast mental darkness
that shrouds all minds. Five minds hugged together for
comfort and safety in the infinite mental loneliness that
pervades the cosmos.
Evelyn continued, "Your boxes were all working, of
course, so I couldn't get your thoughts—just the blurs of
your boxes like little old dark grey stars. But this time
if gave me a funny uncomfortable feeling, like a spider
Crawling down my—Grayl! Don't feel so wildly! What
Is it?”
Then… just as Grayl started to think her answer…
something crept from the vast mental darkness and infinite
cosmic loneliness surrounding the five minds of the
Horns.
Grayl was the first to notice. Her panicky thought had
ttie curling tookeen edge of hysteria. "There are six of
us now! There should only be five, but there are six.
Count! Count, I tell you! Six!"
To Mort it seemed that a gigantic spider was racing
across the web of their thoughts….
See also this journal on May 30– "720 in the Book"– and on May 31– "Memorial for Galois."
("Obnoxious nerds"— a phrase Martin Gardner recently applied to Galois— will note that 720
Wednesday, June 2, 2010
The Harvard Style
"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—
Rite of Passage
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.
Tuesday, June 1, 2010
The Gardner Tribute
"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." 
Tuesday, September 8, 2009
Tuesday September 8, 2009
Magic Box
Continued from Dec. 7, 2008,
and from yesterday.
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 
Padgett (pseudonym of a husbandandwife writing team) says that alphabet blocks are the intuitive "basis of Euclid." Au contraire; they are the basis of Gutenberg.
For the intuitive basis of one type of nonEuclidean* geometry– finite geometry over the twoelement Galois field– see the work of…
Friedrich Froebel
(17821852), who
invented kindergarten.
His "third gift" —
© 2005 The Institute for Figuring
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
Saturday, March 7, 2009
Saturday March 7, 2009
One or Two Ideas
From James Joyce's A Portrait of the Artist as a Young Man:
he hearth and began to stroke his chin. –When may we expect to have something from you on the esthetic question? he asked. –From me! said Stephen in astonishment. I stumble on an idea once a fortnight if I am lucky. –These questions are very profound, Mr Dedalus, said the dean. It is like looking down from the cliffs of Moher into the depths. Many go down into the depths and never come up. Only the trained diver can go down into those depths and explore them and come to the surface again. –If you mean speculation, sir, said Stephen, I also am sure that there is no such thing as free thinking inasmuch as all thinking must be bound by its own laws. –Ha! –For my purpose I can work on at present by the light of one or two ideas of Aristotle and Aquinas. –I see. I quite see your point. 
Besides being Mondrian's birthday, today is also the dies natalis (in the birthintoheaven sense) of St. Thomas Aquinas and, for those who believe worthy preChristians also enter heaven, possibly of Aristotle.
Pope Benedict XVI explained the dies natalis concept on Dec. 26, 2006:
"For believers the day of death, and even more the day of martyrdom, is not the end of all; rather, it is the 'transit' towards immortal life. It is the day of definitive birth, in Latin, dies natalis."
were in St. Peter's Square.
Pictorial version
of Hexagram 20,
Contemplation (View)
symbolizing art itself.
(See Nov.30 – Dec.1, 2008.)
In honor of
Aristotle and Aquinas,
here is a new web site,
illuminatidiamond.com,
with versions of the diamond shape
made famous by Mondrian —
— a shape symbolizing
possibility within modal logic
as well as the potentiality of
Aristotle's prima materia.
Monday, December 29, 2008
Monday December 29, 2008
Robert Stone,
"'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.' Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion." For further details, click on the diamond. 
Today's online Times on
the Saturday, Dec. 27,
death of an artist:
Mr. Wasserman wrote more than 75 scripts for television, the stage and the movies, including screenplays for 'The Vikings' (1958), a seafaring epic with Tony Curtis and Kirk Douglas, and 'A Walk With Love and Death' (1969), a John Huston film set in 14thcentury Europe….
He feuded with… John Huston, who gave the lead female role in 'Walk' to his teenage daughter, Anjelica, against Mr. Wasserman's wishes. And he never attended ceremonies to receive the awards he won."
Accepting for Mr. Wasserman:
Mr. Graham's widow,
Anjelica Huston —
"Well…"
Tuesday, January 9, 2007
Tuesday January 9, 2007
(continued from
January 9, 2003)
George Balanchine

"What on earth is
a concrete universal?"
— Robert M. Pirsig
Review:
From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:
"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."
There is, however,
"the field of reason"–
the 3×3 grid:
Click on grid
for details.
As Rosalind Krauss
has noted, some artists
regard the grid as
"a staircase to
the Universal."
Other artists regard
Epiphany itself as an
approach to
the Universal:
— Richard Kearney, 2005,
in The New Arcadia Review
Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.
— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 20052006
For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:
"We symbolize
logical necessity with the box and logical possibility with the diamond
"The possibilia that exist,
— Michael Sudduth, 
Saturday, January 6, 2007
Saturday January 6, 2007
for the Birthday
of E. L. Doctorow,
Author of
City of God
(Doctorow wrote about
New York. A city more
closely associated with
God is Jerusalem.)
This morning’s entry reboards the Galois train of thought.
Here are some relevant links:
Galois Connections (a French weblog entry providing an brief overview of Galois theory and an introduction to the use of Galois lattices in “formal concept analysis“)
Ontology (an introduction to formal concept analysis linked to on 3/31/06)
One motive for resuming consideration of Galois lattices today is to honor the late A. Richard Newton, a pioneer in engineering design who died at 55– also on Tuesday, Jan. 2, the date of Kollek’s death. Today’s New York Times obituary for Newton says that “most recently, Professor Newton championed the study of synthetic biology.”
A check of syntheticbiology.org leads to a web page on– again– ontology.
For the relationship between ontology (in the semanticweb sense) and Galois lattices, see (for instance)
“Knowledge Organisation and Information Retrieval Using Galois Lattices” (ps) and its references.
An epiphany within all this that Doctorow might appreciate is the following from Wikipedia, found by following a link to “upper ontology” in the syntheticbiology.org ontology page:
 There is no selfevident way of dividing the world up into concepts.
 There is no neutral ground that can serve as a means of translating between specialized (lower) ontologies.
 Human language itself is already an arbitrary approximation of just one among many possible conceptual maps. To draw any necessary correlation between English words and any number of intellectual concepts we might like to represent in our ontologies is just asking for trouble.
Related material:
The intellectual concepts
mentioned by Richard Powers
at the end of tomorrow’s
New York Times Book Review.
(See the links on these concepts
in yesterday’s “Goldberg Variation.”)
See also Old School Tie.
Friday, December 29, 2006
Friday December 29, 2006
of Christ Church
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon
Click on picture for details.
Today is the feast
of St. Thomas Becket.
In his honor, a meditation
on tools and causation:
— Review by H. Allen Orr in
The New York Review of Books,
Vol. 54, No. 1, January 11, 2007
"An odd extension"–
Wolpert's title is, of course,
from Lewis Carroll.
Related material:
"It's a poor sort of memory
that only works backwards."
— Through the LookingGlass
An event at the Kennedy Center
broadcast on
December 26, 2006
(St. Steven's Day):
from the conclusion to
(Log24, Aug. 22, 2005):
"At times, bullshit can
only be countered
with superior bullshit."
— Norman Mailer
"The concept of possible worlds dates back to at least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
"Il faut cultiver notre jardin."
— Voltaire
"We symbolize
logical necessity
with the box
and logical possibility
with the diamond
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity,
Christ Church College, Oxford
(the home of Lewis Carroll)
For further details,
click on the
Christ Church diamond.
Tuesday, October 31, 2006
Tuesday October 31, 2006
From 7/07, an art review from The New York Times:
Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College
"The show has an endgame, endtime mood….
I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
— Roberta Smith
would consider the
following "found" art an
example of originality.
It nevertheless does
"announce a faith."
"First for yourself"
Today's midday
Pennsylvania number:
707
See Log24 on 7/07
and the above review.
"Second, perhaps,
for the rest of us"
Today's evening
Pennsylvania number:
384
This number is an
example of what the
reviewer calls "compression"–
"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."
"Experiences"
See (for instance)
Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."
"Ideas"
"Feelings"
See A Wrinkle in Time.
"Whole"
The automorphisms
of the tesseract
form a group
of order 384.
"Committed"
See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.
"Visually comprehensible"
See "Diamond Theory in 1937,"
an excerpt from which
is shown below.
The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:
Tuesday, October 10, 2006
Tuesday October 10, 2006
Two Seconds
From Oct. 13 last year
(Yom Kippur):
A Poem for Pinter
Oct. 13, 2005 The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature: "Earlier this year, he announced his decision to retire from playwriting in favour of poetry," Michael Muskal in today's Los Angeles Times: "Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits." Robert Stone, A Flag for Sunrise (See Jan. 25): "'That old Jew gave me this here.' Egan looked at the diamond…. 'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal….'"
"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth. The words 'necessary' and 'possible' are called modal connectives, or modalities. A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true. In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"
Commentary:
"Waka" also means Japanese poem or Maori canoe. (For instance, this Japanese poem and this Maori canoe.)
For a meditation on "bang splat," see Sept. 2529. For the meaning of "tick tick," see Emily Dickinson on "degreeless noon." "Hash," of course, signifies "checkmate." (See previous three entries.) 
For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.
That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.
For more on the former, see
Death in Two Seconds.
For more on the latter, see
The Halmos Tombstone.
Wednesday, August 30, 2006
Wednesday August 30, 2006
A Multicultural Farewell
to a winner of the
Nobel Prize for Literature,
the Egyptian author of
The Seventh Heaven:
Supernatural Stories —
"Jackson has identified
the seventh symbol."
— Stargate
Other versions of
the seventh symbol —
"… 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
Friday, May 26, 2006
Friday May 26, 2006
A Living Church
continued from March 27
— G. K. Chesterton
Shakespearean Fool 
Related material:
Yesterday's entries
and their link to
The Line
as well as
and the remarks
of Oxford professor
Marcus du Sautoy,
who claims that
"the right side of the brain
is responsible for mathematics."
Let us hope that Professor du Sautoy
is more reliable on zeta functions,
his real field of expertise,
than on neurology.
The picture below may help
to clear up his confusion
between left and right.
His confusion about
pseudoscience may not
be so easily remedied.
flickr.com/photos/jaycross/3975200/
(Any resemblance to the film
"Hannibal" is purely coincidental.)
Wednesday, March 29, 2006
Wednesday March 29, 2006
Note: Carmichael's reference is to
A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.
— A Wrinkle in Time
Monday, January 23, 2006
Monday January 23, 2006
The Case
An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:
From The Hustler, by Walter Tevis:
The only light in the room was from the lamp over the couch where she was reading.
He looked at her face. She was very drunk. Her eyes were swollen, pink at the corners. "What's the book?" he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
She blinked up at him, smiled sleepily, and said nothing.
"What's the book?" His voice had an edge now.
"Oh," she said. "It's Kierkegaard. Soren Kierkegaard." She pushed her legs out straight on the couch, stretching her feet. Her skirt fell back a few inches from her knees. He looked away.
"What's that?" he said.
"Well, I don't exactly know, myself." Her voice was soft and thick.
He turned his face away from her again, not knowing what he was angry with. "What does that mean, you don't know, yourself?"
She blinked at him. "It means, Eddie, that I don't exactly know what the book is about. Somebody told me to read it, once, and that's what I'm doing. Reading it."
He looked at her, tried to grin at her– the old, meaningless, automatic grin, the grin that made everybody like him– but he could not. "That's great," he said, and it came out with more irritation than he had intended.
She closed the book, tucked it beside her on the couch. "I guess this isn't your night, Eddie. Why don't we have a drink?"
"No." He did not like that, did not want her being nice to him, forgiving. Nor did he want a drink.
Her smile, her drunk, amused smile, did not change. "Then let's talk about something else," she said. "What about that case you have? What's in it?" Her voice was not prying, only friendly. "Pencils?"
"That's it," he said. "Pencils."
She raised her eyebrows slightly. Her voice seemed thick. "What's in it, Eddie?"
"Figure it out yourself." He tossed the case on the couch.
Related material:
Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,
the Baseball Almanac,
and this morning's entry, "Natural Hustler."
Thursday, January 19, 2006
Thursday January 19, 2006
Sunday, November 20, 2005
Sunday November 20, 2005
of Power
Johnny Cash:
“And behold,
a white horse.”
Adapted from
illustration below:
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”
— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to the “Diamond Theory” of truth in The NonEuclidean Revolution
“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”
— Richard J. Trudeau in
The NonEuclidean Revolution
“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”
— Jim Holt in The New Yorker.
Exercise of Power:
Show that a white horse–
a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.
This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).
Related material
On this date:
In 1490, The White Knight
(Tirant lo Blanc )–
a major influence on Cervantes–
was published, and in 1910
the Mexican Revolution began.
Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York
“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of Godsupplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an allencompassing reality.'”
Thursday, October 13, 2005
Thursday October 13, 2005
A Poem for Pinter
The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:
"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"
Michael Muskal in today's Los Angeles Times:
"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."
Robert Stone, A Flag for Sunrise (See Jan. 25):
"'That old Jew gave me this here.' Egan looked at the diamond…. 'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal….'"
"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth. The words 'necessary' and 'possible' are called modal connectives , or modalities . A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true. In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"
A Poem for Pinter

Commentary:
"Waka" also means Japanese poem or Maori canoe.
(For instance, this Japanese poem and this Maori canoe.)
For a meditation on "bang splat," see Sept. 2529.
For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."
"Hash," of course, signifies "checkmate." (See previous three entries.)
Monday, August 22, 2005
Monday August 22, 2005
Apostolos Doxiadis on last month's conference on "mathematics and narrative"–
Doxiadis is describing how talks by two noted mathematicians were related to
"… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "
Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. [In one part] I’ll tell you a few things about proofs. [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the link… [connecting math and narrative]… There is this very wellknown Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’ But the similarity was not clearly explained: the hole, you see, the ‘miracle.’ Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer
Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
Background:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
PossibleWorlds Theory, by MarieLaure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)
'…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 4344)
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
— Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)
"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."
"We symbolize
logical necessity with the box and logical possibility with the diamond
"The possibilia that exist,
— Michael Sudduth, 
Tuesday, March 22, 2005
Tuesday March 22, 2005
Make a Différance
From Frida Saal's
Lacan_{ }Derrida:
From a Contemporary Literary Theory website:
"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."
Make a Difference Day, Oct. 23, 1999:
22. Without using the Pythagorean Theorem prove that the hypotenuse of an isosceles right triangle will have the length if the equal legs have the length 1. Suggestion: Consider the similar triangles in Fig. 39. 23. The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas. We cannot do so now because we have not yet considered the idea of area. Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = .
— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941) 
Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics* — they may in other respects be quite superior to men:
The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.
† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005. See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.
Thursday, March 3, 2005
Thursday March 3, 2005
Matrix group actions,
March 26, 1985
"We symbolize logical necessity
with the box
and logical possibility
with the diamond
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by _{} ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Sunday, February 20, 2005
Sunday February 20, 2005
Relativity Blues
Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry. Here is some related material.
In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the manyworlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).
(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the abovementioned note and in a generalization, Solomon's Cube.)
On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:
"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."
One possible response to such statements was recently provided in some detail by a Princeton philosophy professor. See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.
A more thoughtful response would take into account the following:
1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson
2. The physics underlying Einstein's remarks on free will, God, and dice
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels Secret Passages and Broken Symmetries
4. The physics underlying the recent socalled "free will theorem" of John Conway and Simon Kochen of Princeton University
5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.
From a review of Gilead by Jane Vandenburgh:
"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"
For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries. If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003, and those of November 5, 2002.
From Many Dimensions (Eerdmans paperback, 1963, page 53):
"Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?"
A recent answer:
"We symbolize logical necessity
with the box
and logical possibility
with the diamond
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by _{} ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Thursday, February 17, 2005
Thursday February 17, 2005
"We symbolize logical necessity
with the box
and logical possibility
with the diamond
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by _{} ?
On the Lapis Philosophorum,
the Philosophers' Stone –
"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"
– Many Dimensions,
by Charles Williams, 1931
(Eerdmans paperback,
April 1979, pp. 4344)
"The lapis was thought of as a unity
and therefore often stands for
the prima materia in general."
– Aion, by C. G. Jung, 1951
(Princeton paperback,
1979, p. 236)
"Its discoverer was of the opinion that
he had produced the equivalent of
the primordial protomatter
which exploded into the Universe."
– The Stars My Destination,
by Alfred Bester, 1956
(Vintage hardcover,
July 1996, p. 216)
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
See also
The Diamond Archetype.
For more on modal theology, see
Kurt Gödel's Ontological Argument
and
The Ontological Argument
from Anselm to Gödel.