Sunday, November 22, 2020
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
The arXiv versions —
Monday, May 7, 2018
Stanley Fish in the online New York Times today —
". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.
Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"
The final page, 759, of the Harry Potter saga —
"Talk about magical thinking!" — Fish, ibidem .
See also the above Harry Potter page
in this journal Sunday morning.
Comments Off on Fish Babel
Wednesday, May 2, 2018
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Comments Off on Galois’s Space
Sunday, November 19, 2017
This is a sequel to yesterday's post Cube Space Continued.
Comments Off on Galois Space
Saturday, May 20, 2017
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Comments Off on van Lint and Wilson Meet the Galois Tesseract*
Sunday, August 14, 2016
Continued from earlier posts on Boole vs. Galois.
From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
Comments Off on The Boole-Galois Games
Tuesday, May 31, 2016
A very brief introduction:
Comments Off on Galois Space —
Tuesday, January 12, 2016
The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Comments Off on Harmonic Analysis and Galois Spaces
Tuesday, March 24, 2015
Yesterday's post suggests a review of the following —
|
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)
Pages 8-9:
Substructures of S(5, 8, 24)
An octad is a block of S(5, 8, 24).
Theorem 5.1
Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely
the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.
Proof….
… (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say Dembowski-Wagner or Veblen & Young.
An explicit construction of the vector space is also easy….)
|
Related material: Posts tagged Priority.
Comments Off on Brouwer on the Galois Tesseract
Tuesday, November 25, 2014
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 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 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 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Comments Off on Euclidean-Galois Interplay
Sunday, March 10, 2013
(Continued)
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
Comments Off on Galois Space
Monday, March 4, 2013
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.
Comments Off on Occupy Galois Space
Thursday, February 21, 2013
(Continued)
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."
— Thomas Pynchon
Illustrations—
(Click to enlarge.)
Comments Off on Galois Space
Sunday, July 29, 2012
(Continued)
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space 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 vector-space 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 "Self-Dual 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 Conway-Sloane diagram.
Comments Off on The Galois Tesseract
Thursday, July 12, 2012
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
(Click to enlarge.)
_lines_as_arrays-500w.jpg)
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set 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, 488-499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Comments Off on Galois Space
Tuesday, July 10, 2012
Comments Off on Euclid vs. Galois
Friday, September 9, 2011
(Continued from Abel Prize, August 26)
The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.
The coffee scene from “Bleu”
Related material from this journal:
The Dream of
the Expanded Field
Comments Off on Galois vs. Rubik
Saturday, September 3, 2011
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG’s
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four “special tetrads” within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The “35 structures” of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
Comments Off on The Galois Tesseract
Friday, September 17, 2010
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 co-author, 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 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 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.
Comments Off on The Galois Window
Sunday, November 15, 2020
Comments Off on Map Methods
Friday, September 11, 2020
Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry. See other posts now tagged Boole vs. Galois.
See also “A Four-Color Epic” (April 16, 2020).
Comments Off on Kauffman on Algebra
Tuesday, August 13, 2019
The Matrix of Lévi-Strauss —
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of Lévi-Strauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
Comments Off on Putting the Structure in Structuralism
Sunday, September 9, 2018
"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 Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. 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
Comments Off on Plan 9 Continues.
Thursday, May 10, 2018
* I.e., Hemingway's novel The Garden of Eden.
See also Northrop Frye and "interpenetration"
in this journal and a University of Montana master's
thesis from 1994 on the Hemingway novel,
"And a river went out of Eden," by Howard A. Schmid.
See as well remarks by Stanley Fish quoted here on May 7.
Comments Off on Backstory for Eden*
Nature yesterday —
"To synchronize participant activity with experimental operation,
the Bell tests were scheduled to take place on a single day,
Wednesday 30 November 2016."
— "Challenging local realism with human choices:
The BIG Bell Test Collaboration"
This journal on that date, 30 November 2016 —
Cf. other posts tagged Lumber Room.
Comments Off on The Forbidden Garden
Tuesday, May 8, 2018
From April 2008 —
From the Sketchbook page of next Sunday's New York Times Book Review —
Backstory —
Comments Off on The Museum of Slow Art
The glitter-ball-like image discussed in the previous post
is of an artwork by Olafur Eliasson.
See the kaleidoscopic section of his website.
From that section —
Related art in keeping with the theme of last night's Met Gala —
See also my 2005 webpage Kaleidoscope Puzzle.
Comments Off on Wall
Monday, May 7, 2018
In memory of a French film publicist who worked with Clint Eastwood
in 1971 on the release of "The Beguiled" —
From a New York Times graphic review dated Sept. 16, 2016 —
It's Chapter 1 of George Eliot's "Middlemarch."
Dorothea Brooke, young and brilliant, filled with passion
no one needs, is beguiled by some gemstones . . . .
The characters, moving through the book,
glitter as they turn their different facets toward us . . . .
Cf. a glitter-ball-like image in today's New York Times philosophy column
"The Stone" — a column named for the legendary philosophers' stone.
The publicist, Pierre Rissient, reportedly died early Sunday.
See as well Duelle in this journal.
Comments Off on Glitter Ball for Cannes
(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)
Novelist George Eliot and programming pioneer Ada Lovelace —
For an image that suggests a resurrected multifaceted
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post), behold a photo from today's NY Times philosophy
column "The Stone" that was reproduced here in today's previous post —
For a New York Times view of George Eliot data, see a Log24 post
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."
Comments Off on Data
Sunday, May 6, 2018
Comments Off on The Osterman Omega
For Peculiar Children
Comments Off on Sunday School Lesson Plan …
"But perhaps there’s more to the [Harry] Potter books
than the term 'children’s literature' lets on —
indeed, so much so that the category no longer applies."
— Maria Devlin McNair in the online Boston Globe yesterday
Comments Off on Category
Sunday, April 29, 2018
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 amusement-park 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.
Comments Off on Amusement
Monday, March 12, 2018
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Comments Off on “Quantum Tesseract Theorem?”
Sunday, March 4, 2018
1955 ("Blackboard Jungle") —
1976 —
2009 —
2016 —
Comments Off on The Square Inch Space: A Brief History
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —
- “The digital revolution based on the 2 symbols (0,1)”
- “The algebra of George Boole”
- “Binary codes”
- “Dirac’s spinors, with their up/down dichotomy”
These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
- The 2×2 square
- The 2x2x2 cube
- The 4×4 square
- The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on The Binary Revolution
Thursday, January 25, 2018
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Comments Off on Beware of Analogical Extension
Saturday, September 23, 2017
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Comments Off on The Turn of the Frame
Friday, September 15, 2017
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.
Comments Off on Space Art
Sunday, August 27, 2017
The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Comments Off on Black Well
Saturday, June 3, 2017
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Comments Off on Expanding the Spielraum (Continued*)
Tuesday, May 23, 2017
Comments Off on Pursued by a Biplane
Saturday, May 20, 2017
From a review of the 2016 film "Arrival" —
"A seemingly off-hand 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 squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf 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 picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."
For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —
van Lint and Wilson Meet the Galois Tesseract.
The death process of van Lint occurred on Sept. 28, 2004.
See this journal on that date.
Comments Off on The Ludicrous Extreme
Tuesday, May 2, 2017
Comments Off on Image Albums
Thursday, April 20, 2017
See also “Romancing the Omega” —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —
Comments Off on Stone Logic
Sunday, April 16, 2017
This post’s title is from the tags of the previous post —
The title’s “shift” is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
“Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange“
— io9 , July 29, 2016
” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego Comic-Con . . . .
‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “
[Typo now corrected, except in a comment.]
Comments Off on Art Space Paradigm Shift
Wednesday, October 5, 2016
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Comments Off on Sources
Tuesday, August 16, 2016
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
Boulez and Boole and…
Cross and Boolean lattice.
The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —
Update of 10 am on August 16, 2016 —
See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on Midnight Narrative
Saturday, June 18, 2016
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 length-16 sequence."
— Saying adapted from a young-adult novel.
Comments Off on Midnight in Herald Square
Sunday, May 8, 2016
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
by Sol LeWitt
“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 three-dimensional 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 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)” |
See also the Cullinane models of some small Galois spaces —
Comments Off on The Three Solomons
Friday, May 6, 2016
Thursday, April 14, 2016
On this date in 2005, mathematician Saunders Mac Lane died at 95.
Related material —
Max Planck quotations:
Mac Lane on Boolean algebra:
Mac Lane’s summary chart (note the absence of Galois geometry ):
I disagree with Mac Lane’s assertion that “the finite models of
Boolean algebra are dull.” See Boole vs. Galois in this journal.
Comments Off on One Funeral at a Time
Wednesday, January 13, 2016
(Continued from previous episodes)
Boole and Galois also figure in the mathematics of space —
i.e. , geometry. See Boole + Galois in this journal.
Related material, according to Jung’s notion of synchronicity —
Comments Off on Geometry for Jews
Monday, January 11, 2016
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, 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 young-adult novel
Comments Off on Space Oddity
Monday, December 28, 2015
Combining two headlines from this morning’s
New York Times and Washington Post , we have…
Deceptively Simple Geometries
on a Bold Scale
Voilà —
Click image for details.
More generally, see
Boole vs. Galois.
Comments Off on ART WARS Continues
Friday, December 25, 2015
Related material:
The previous post (Bright Symbol) and
a post from Wednesday,
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.
“An analogy between mathematics and religion is apposite.”
— Harvard Magazine review by Avner Ash of
Mathematics without Apologies
(Princeton University Press, January 18, 2015)
Comments Off on Dark Symbol
Wednesday, December 23, 2015
Bleecker Street logo —
Click image for some background.
Related remarks on mathematics:
Boole vs. Galois
Comments Off on Splitting Apart
Sunday, December 13, 2015
“The colorful story of this undertaking begins with a bang.”
— Martin Gardner on the death of Évariste Galois
Comments Off on The Monster as Big as the Ritz
Monday, November 2, 2015
This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space. From the latter —
” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “
Comments Off on The Devil’s Offer
"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
Comments Off on Colorful Story
Saturday, October 31, 2015
Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —
Paraconsistent Logic
“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”
This journal on the date Friday, April 5, 2013 —
The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .
For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —
Related material by Schöter —
A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)
I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching studies is,
I maintain, not Boolean algebra but rather Galois geometry.
See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.
Comments Off on Raiders of the Lost Crucible
Saturday, October 24, 2015
The following slides are from lectures on “Advanced Boolean Algebra” —
The small Boolean spaces above correspond exactly to some small
Galois spaces. These two names indicate approaches to the spaces
via Boolean algebra and via Galois geometry .
A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —
” ‘All you need to do is give me your soul: give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “
Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977. That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra than about Galois geometry .
For later remarks on diamond theory, see finitegeometry.org/sc.
Comments Off on Two Views of Finite Space
Wednesday, October 21, 2015
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Comments Off on Algebra and Space
Sunday, September 6, 2015
Sarah Larson in the online New Yorker on Sept. 3, 2015,
discussed Google’s new parent company, “Alphabet”—
“… Alphabet takes our most elementally wonderful
general-use word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm.”
Emma Watson in “The Bling Ring”
This journal, also on September 3 —
* Actually, Sarah, that would be “phonemes.”
Comments Off on Elementally, My Dear Watson
Friday, September 4, 2015
Galois via Boole
(Courtesy of Intel)
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Thursday, September 3, 2015
For the title, see posts from August 2007 tagged Gyges.
Related theological remarks:
Boolean spaces (old) vs. Galois spaces (new) in
“The Quality Without a Name”
(a post from August 26, 2015) and the…
Related literature: A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.
Comments Off on Rings of August
Wednesday, August 26, 2015
The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander’s book
The Timeless Way of Building (Oxford University Press, 1979).
A quote from the publisher:
“Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself.”
Three paragraphs from the book (pp. xiii-xiv):
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 —
Comments Off on “The Quality Without a Name”
Friday, August 14, 2015
(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:
Comments Off on Discrete Space
Tuesday, June 9, 2015
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the four-element Galois field.
This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.
* A term from the 1947 film "Nightmare Alley."
Comments Off on Colorful Song
Thursday, March 26, 2015
The incidences of points and planes in the
Möbius 84 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.† In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "Self-Dual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413-455
† The subscripts' usual 1-2-3-4 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16-element Galois field GF(24). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Comments Off on The Möbius Hypercube
Sunday, February 8, 2015
For Autism Sunday —
Mathematician John von Neumann
reportedly died on this date.
“He belonged to that so-called
Hungarian phenomenon….”
— A webpage titled
“Von Neumann, Jewish Catholic”
Illustrations of another Hungarian phenomenon:
Comments Off on Hungarian Phenomenon
Monday, January 5, 2015
Wednesday, March 13, 2013
From a review in the April 2013 issue of
Notices of the American Mathematical Society—
"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."
From the book under review—
"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."
— Lockhart, Paul (2009-04-01).
A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form (p. 92).
Bellevue Literary Press. Kindle Edition.
Related material: Blackboard Jungle in this journal.
See also Galois Space and Solomon's Mines.
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"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
grown-ups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
— The Little Prince
* For the title, see Plato Thanks the Academy (Jan. 3).
Comments Off on Gitterkrieg*
Monday, December 29, 2014
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.
Comments Off on Dodecahedron Model of PG(2,5)
Thursday, December 18, 2014
(Five by Five continued)
As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged Galois-Plane Models.
Comments Off on Platonic Analogy
Wednesday, December 3, 2014
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
"triangle-based pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based 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) —
|
|
Mathematics
The Fano plane block design
|
Magic
The Deathly Hallows symbol—
Two blocks short of a design.
|
|
(Updated at about 7 PM ET on Dec. 3.)
Comments Off on Pyramid Dance
Sunday, November 30, 2014
The Regular Tetrahedron
The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edge-to-edge axes.
(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 16-17.)
The Cube
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 symmetric-difference sum of the
other two members.
(This is the eightfold cube discussed at finitegeometry.org.)
Comments Off on Two Physical Models of the Fano Plane
Wednesday, November 26, 2014
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 16-17 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
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."
The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois 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.
Comments Off on A Tetrahedral Fano-Plane Model
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 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:
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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 in-house 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 I-X.
— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge,
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science , 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf
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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. non-Euclidean 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.
Comments Off on Class Act
Monday, September 22, 2014
Review of an image from a post of May 6, 2009:
Comments Off on Space
Sunday, September 14, 2014
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:
Comments Off on Sensibility
Sunday, August 31, 2014
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordion-pleated. 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
3-dimensional 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.]
Comments Off on Sunday School
Friday, June 13, 2014
"The wind of change is blowing throughout the continent.
Whether we like it or not, this growth of national consciousness
is a political fact."— Prime Minister Harold Macmillan,
South Africa, 1960
"Lord knows when the cold wind blows
it'll turn your head around." — James Taylor
From a Log24 post of August 27, 2011:
For related remarks on "national consciousness," see Frantz Fanon.
Comments Off on It’s 10 PM
Monday, April 28, 2014
Suggested by a Saturday death in Jersey City:
Somewhere, over the gray space…
Comments Off on Hymn
(Orlin Wagner/Associated Press) – A vehicle tops a hill along
U.S. Route 56 as a severe thunderstorm moves through the area
near Baldwin City, Kansas, on Sunday, April 27, 2014.
See a related news story.
Comments Off on Gray Space, by Wagner
Sunday, April 27, 2014
Galois and Abel vs. Rubik
(Continued)
“Abel was done to death by poverty, Galois by stupidity.
In all the history of science there is no completer example
of the triumph of crass stupidity….”
— Eric Temple Bell, Men of Mathematics
Gray Space (Continued)
… For The Church of Plan 9.
Comments Off on Sunday School
Saturday, April 26, 2014
Comments Off on For Two Artists of Norway
Friday, April 25, 2014
Tuesday, February 26, 2013
"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
Related material—
See also Steiner on Galois.
Les Miserables at the Academy Awards
Comments Off on Publication
Tuesday, February 5, 2013
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
Comments Off on Arsenal
Monday, August 13, 2012
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
Comments Off on Raiders of the Lost Tesseract
Tuesday, May 29, 2012
(Continued from May 29, 2002)
May 29, 1832—
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
– The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.
Commentary from Dec. 2011 on Gardner's word "published" —
(Click to enlarge.)
Comments Off on The Shining of May 29
Sunday, October 30, 2011
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."
Comments Off on Sermon
Saturday, August 27, 2011
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
Comments Off on Cosmic Cube*
Saturday, July 23, 2011
Suggested by Peter J. Cameron's weblog post today on Halmos,
by a July 18 post in this journal on the Norwegian mathematician Abel,
by a link in the July 18 post to "Death Proof," and by today's
midday New York Lottery (459 and 7404)—
From July 4, 2004 (7404 interpreted as a date)—
"There are two unfortunate connotations of 'proof' that come from mathe-
matics and make the word inappropriate in discussions of the security of cryp-
tographic systems. The first is the notion of 100% certainty. Most people not
working in a given specialty regard a 'theorem' that is 'proved' as something
that they should accept without question. The second connotation is of an intri-
cate, highly technical sequence of steps. From a psychological and sociological
point of view, a 'proof of a theorem' is an intimidating notion: it is something
that no one outside an elite of narrow specialists is likely to understand in detail
or raise doubts about. That is, a 'proof' is something that a non-specialist does
not expect to really have to read and think about.
The word 'argument,' which we prefer here, has very different connotations."
— "Another Look at 'Provable Security',"
by Neal Koblitz and Alfred J. Menezes, July 4, 2004
(updated on July 16, 2004; October 25, 2004; March 31, 2005; and May 4, 2005)
As for 459, see Post 459 in this journal.
Related material: The Race, Crossing the Bridge, Aristophanic View, and Story Theory.
Comments Off on Death Argument
Monday, July 18, 2011
Continuing yesterday's lottery meditation…
The NY evening numbers yesterday were 244 and 2962.
The latter suggests Post 2962—
There is no Post 244 here, but a search within this journal for 244 yields…
See also Halmos Tombstone and Death Proof.
Comments Off on Voldemort
Thursday, May 26, 2011
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.
—John McKay at sci.math
Related material: Harvard Treasure, Favicon, and Crimson Tide.
Comments Off on For the Class of ’11
Friday, March 4, 2011
Two items from the August 5, 2005, anniversary
of the day Marilyn Monroe was found dead—
1. New Chapter in the Mystery
2. Literary Symbol —
See also related material on Hollywood.
Comments Off on Infinite Jest
Wednesday, October 20, 2010
"Why the Celebration?"
"Martin Gardner passed away on May 22, 2010."
Imaginary movie poster from stoneship.org
Context— The Gardner Tribute.
Comments Off on Celebration of Mind
Monday, September 27, 2010
… 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 square-actor by actor-binary 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
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
Comments Off on The Social Network…
Tuesday, June 22, 2010
"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
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 (1811-1832)
Klein on Galois—
"… in France just about 1830 a new star of undreamt-of 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 film-makers to work together on mathematically-inspired literary works."
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Saturday, June 19, 2010
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.
Four-Part Tesseract Divisions—
The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to four-part 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."
— Jamie James in The Music of the Spheres (1993)
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Another picture related to philosophy and religion—
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
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Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—
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).
Notes:
O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)
C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)
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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—
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… 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 multi-dimensionally* 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 multi-dimensional symbols beyond our grasp.
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Related material:
Imago Creationis—
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—
Frame of Reference
The Diamond Theorem—
Some context by a British mathematician —
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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.
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Comments Off on Imago Creationis
Wednesday, June 2, 2010
"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—
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 pie-eating contest—
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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.
Leonard E. Dickson—
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Tuesday, June 1, 2010
“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.”
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Saturday, April 3, 2010
"Democrats– in conclusion– Democrats in America
were put on earth to do one thing– Drag the
ignorant hillbilly half of this country into the next
century, which in their case is the 19th."
– Bill Maher on March 26
Reply to Maher:
"Hell is other people."
— Jean-Paul Sartre
With a laugh track.
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Wednesday, August 5, 2009
Word and Image
From Hall's obituary:
"Edward T. Hall, a cultural anthropologist
who pioneered the study of nonverbal
communication and interactions between
members of different ethnic groups,
died July 20 at his home in
Santa Fe, N.M. He was 95."
NY Times piece quoted here on
the date of Hall's death:
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"July 20, 1969, was the moment NASA needed, more than anything else in this world, the Word. But that was something NASA's engineers had no specifications for. At this moment, that remains the only solution to recovering NASA's true destiny, which is, of course, to build that bridge to the stars."
— Tom Wolfe, author of The Right Stuff, an account of the Mercury Seven astronauts.
Commentary —
The Word according to St. John:
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From Hall's obituary:
"Mr. Hall first became interested in
space and time as forms of cultural
expression while working on
Navajo and Hopi reservations
in the 1930s."
Log24, July 29:
"We are the key."
— Eye of Cat
Update of about 4:45 PM 8/5:
Paul Newall, "Kieślowski's Three Colours Trilogy"—
"Julie recognises the music of the busker outside playing a recorder as that of her husband's. When she asks him where he heard it, he replies that he makes up all sorts of things. This is an instance of a theory of Kieślowski's that 'different people, in different places, are thinking the same thing but for different reasons.' With regard to music in particular, he held what might be characterised as a Platonic view according to which notes pre-exist and are picked out and assembled by people. That these can accord with one another is a sign of what connects people, or so he believed."
The above photo of Juliette Binoche in Blue accompanying the quotations from Zelazny illustrates Kieślowski's concept, with graphic designs instead of musical notes. Some of the same designs are discussed in Abstraction and the Holocaust (Mark Godfrey, Yale University Press, 2007). (See the Log24 entries of June 11, 2009.)
Related material:
"Jeffrey Overstreet, in his book Through a Screen Darkly, comments extensively on Blue. He says these stones 'are like strands of suspended crystalline tears, pieces of sharp-edged grief that Julie has not been able to express.'….
Throughout the film the color blue crops up, highlighting the mood of Julie's grief. A blue light occurs frequently, when Julie is caught by some fleeting memory. Accompanied by strains of an orchestral composition, possibly her husband's, these blue screen shots hold for several seconds while Julie is clearly processing something. The meaning of this blue light is unexplained. For Overstreet, it is the spirit of reunification of broken things."
— Martin Baggs at Mosaic Movie Connect Group on Sunday, March 15, 2009. (Cf. Log24 on that date.)
For such a spirit, compare Binoche's blue mobile in Blue with Binoche's gathered shards in Bee Season.
Comments Off on Wednesday August 5, 2009
Tuesday, October 31, 2006
"The show has an endgame, end-time 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
It is doubtful that Smith
would consider the
following "found" art an
example of originality.
It nevertheless does
"announce a faith."
"First for yourself"
Today's mid-day
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."
Comments Off on Tuesday October 31, 2006
Friday, June 23, 2006
Binary Geometry
There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).
Examples:
- Charles Sanders Peirce, “The Simplest Mathematics.”
- Donald E. Knuth’s discussion of binary hypercubes in “Boolean Basics,” a draft of section 7.1.1 in The Art of Computer Programming, Volume 4: Combinatorial Algorithms
- My own discussion of a binary hypercube in Geometry of the 4x4x4 Cube
- A more sophisticated example: the geometry of elliptic curves over a binary Galois field. For an excellent introduction, see the Certicom online elliptic curve tutorial. This has an applet illustrating elliptic curves in a space of 256 points (256=16×16, with the x and y variables of a curve each having 16 possible values).
- In summary, apart from the fact that the native language of computers has characteristic 2, “binary” mathematics, i.e. mathematics in characteristic 2, is of special interest both in the study of finite geometry (Finite Geometry of the Square and Cube) and in algebraic geometry (see, for instance, the work of Brian Conrad).
Comments Off on Friday June 23, 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.
"There is such a thing as a tesseract."
— A Wrinkle in Time
Comments Off on Wednesday March 29, 2006
Friday, August 5, 2005
For Sir Alec
From Elegance:
"Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday…."
— Bernard Holland, page C12,
The New York Times,
Monday, May 20, 1996.
Holland was pondering the identity of the Juilliard String Quartet, which had just given a series of concerts celebrating its fiftieth anniversary.
"Elegant"
— Page one,
The New York Times,
Monday, August 7, 2000.
The Times was describing the work of Sir Alec Guinness, who died on 8/5/00.
An example of the Holland name problem:
Monday, August 1, 2005 — Visible Mathematics:
"Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
'"Empty" is a misnomer… the Rub' al-Khali contains many hidden riches.'"
Friday, August 5, 2005 —
Comments Off on Friday August 5, 2005
Abel's Birthday
Click on picture
for details.
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Monday, August 5, 2002
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What is Truth?
In honor of the 200th anniversary of the birth of Niels Henrik Abel, a partial answer:
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Elliptic Curves and Modular Forms
and the introductory work,
Elliptic Curves:
Function Theory, Geometry, Arithmetic,
by Henry McKean and Victor Moll
Comments Off on Monday August 5, 2002
History, Stephen said….
— The Modern Word
— To really know a subject you've got to learn a bit of its history….
— John Baez, August 4, 2002
We both know what memories can bring;
They bring diamonds and rust.
— Joan Baez, April 1975
All sorts of structures that can be defined for finite sets have analogues for the projective geometry of finite fields….
Clearly this pattern is trying to tell us something; the question is what. As always, it pays to focus on the simplest case, since that's where everything starts.
— John Baez, August 4, 2002
In the beginning was the word….
— The Gospel according to Saint John
The anonymous author of John makes liberal use of allegory and double-entendre to illustrate this theme.
— The Gospel of John
Born yesterday: Logician John Venn.
Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 nonoverlapping regions….
— History of Mathematics at St. Andrews
Who would not be rapt by the thought of such marvels?….
— Saint Bonaventure on the Trinity
Comments Off on Monday August 5, 2002
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