Log24

Friday, June 12, 2020

Prelude to Westworld (1984)

Filed under: General — m759 @ 12:45 PM

See also An Epic for Kristen .

Tuesday, October 8, 2019

Also* in 1984

Filed under: General — m759 @ 11:32 AM
 

American Mathematical Monthly , June-July 1984 

MISCELLANEA, 129

Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.)
— Steven H. Cullinane

* See Cube Bricks 1984  in previous post.

Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: , — m759 @ 9:24 PM

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Sunday, July 29, 2018

1984 Continues.

Filed under: General — m759 @ 2:15 PM

"We want every student to have a fulfilling experience
of higher education that enriches their lives and careers."

Sure you do.

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , , — m759 @ 9:11 AM

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984

For further details, see Triangles Are Square.

Saturday, August 12, 2017

Images from 1984

Filed under: General — Tags: — m759 @ 9:00 AM

For the author of a Harvard Crimson  opinion piece yesterday on 1984 ,
two images adapted from a 1984 film —

Mola Ram from 'Temple of Doom'

See also, in this  journal, Hume's phrase "perfect nonentity."

Thursday, March 30, 2017

2010 in 1984

Filed under: General — m759 @ 9:29 PM

Click for a more realistic view of these years.

Saturday, January 14, 2017

1984: A Space Odyssey

Filed under: General,Geometry — m759 @ 2:40 PM

See Eightfold 1984 in this journal.

Related material —

"… the object sets up a kind of
 frame or space or field
 within which there can be epiphany."

"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."

— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989

"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 University Press, Ithaca, NY, 1962

Epiphany 2017 —

Click to enlarge:

Sunday, May 1, 2016

Sunday Appetizer from 1984

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —

“What would an organic Christian Sabbath look like today?”

The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —  

Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich
unter http://​www.​encyclopediaofma​th.​org/​index.​php/​
Cullinane_​diamond_​theorem
und http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."

The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the 
1988 first edition of Sphere Packings, Lattices and Groups
by J. H. Conway and N. J. A. Sloane —

A passage from the 1984 document —

Thursday, March 24, 2016

Truth in 1984

Filed under: General,Geometry — m759 @ 2:36 PM

"The theory of elliptic curves and modular forms is
one subject where the most diverse branches
of mathematics come together: complex analysis,
algebraic geometry, representation theory, number theory."

— Neal Koblitz, first sentence of 
Introduction to Elliptic Curves and Modular Forms,
First Edition, Springer-Verlag, 1984

Related material —

A quote co-authored by Koblitz appears in today's
earlier post The Wolf Gang.

See also The Proof and the Lie.

Maryna Viazovska's course on elliptic curves and modular forms used the Koblitz text.

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , — m759 @ 12:06 PM

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Tuesday, August 14, 2012

Hacking 1984

Filed under: General,Geometry — m759 @ 10:00 PM

Ian Hacking in 1984

"… theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling by using a proof in geometry to argue for the transmigration of souls. As reported by Plato in Meno , the boy who invents a proof of a theorem did not experiment on the physical world, but used only his mind in response to Socratic questions. Hence he must have had inborn knowledge of the proof and he must have got this knowledge in a previous incarnation.

Mathematics has never since been a subject for such philosophical levity."

See also this afternoon's post.

Tuesday, June 22, 2010

1984 Story (continued)

Filed under: General — m759 @ 8:00 PM

http://www.log24.com/log/pix10A/100622-PeterFinch-Network.jpg

"There are many accounts of
moral and political anger in
the philosophical literature."

— J. M. Bernstein in today's NY Times

J.M. Bernstein is University Distinguished Professor
of Philosophy at the New School for Social Research.

He is the author of a work
that Google Books files under
"Communism and Literature"—

The Philosophy of the Novel:
Lukács, Marxism, and the Dialectics of Form

(University of Minnesota Press, 1984)

Monday, June 21, 2010

1984 Story (continued)

Filed under: General,Geometry — m759 @ 7:00 PM

This journal's 11 AM Sunday post was "Lovasz Wins Kyoto Prize." This is now the top item on the American Mathematical Society online home page—

http://www.log24.com/log/pix10A/100621-LovaszAMS-sm.jpg

Click to enlarge.

For more background on Lovasz, see today's
previous Log24 post, Cube Spaces, and also
Cube Space, 1984-2003.

"If the Party could thrust its hand into the past and
say of this or that event, it never happened…."

— George Orwell, 1984

Wednesday, July 1, 2020

Actress Descending a Staircase

Filed under: General — m759 @ 3:25 AM

The above title was suggested by a scene in Body Double  (1984) . . .

Variations, starring Theresa Russell, on related themes —

The De Palma Balcony in Body Double , and “ready for my closeup” —

“Bing bang, I heard the whole gang!”

Summary — 

Saturday, May 23, 2020

Structure for Linguists

Filed under: General — Tags: — m759 @ 11:34 AM

“MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts.”

MIT Linguistics, May 1, 2020

The “deep  structure” above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).

See also . . .

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Sunday, March 1, 2020

Notes Towards the Ministry of Culture

Filed under: General — Tags: — m759 @ 8:22 PM

Friday, February 21, 2020

To and Fro, Back and …

Filed under: General — Tags: , — m759 @ 11:44 PM

Also on January 27, 2017 . . .

For other appearances of John Hurt here,
see 1984 Cubes.

Update of 12:45 AM Feb. 22 —

A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.

Thus the following remarks may be more appropriate:

Not to mention what, why, who, and how.

Sunday, February 16, 2020

Zen and the Art

Filed under: General — m759 @ 5:18 PM

(Continued)

Sunday, February 9, 2020

Hors d’Oeuvre

Filed under: General — Tags: , , , — m759 @ 1:29 PM

From the May Day 2016 link above, in "Sunday Appetizer from 1984"

The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —  

Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich unter

http://​www.​encyclopediaofma​th.​org/
​index.​php/​Cullinane_​diamond_​theorem

und

http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."

From the Gap Dance link above, in "Reading for Devil's Night" —

Das Nichts nichtet.” — Martin Heidegger.

And "Appropriation Appropriates."

Friday, January 10, 2020

Jan. 9 Review

Filed under: General — Tags: , — m759 @ 10:51 AM

“Work as if you were in the
early days of a better nation.”

— God, according to the author of
    1982 Janine

From Carole A. Holdsworth,
"Dulcinea and Pynchon’s V":

Tanner may have stated it best:

“V. is whatever lights you to
 the end of the street:
 she is also the dark annihilation
 waiting at  the end of the street.”

(Tony Tanner, page 36,
 “V. and V-2,” in
 Pynchon: A Collection
 of Critical Essays.

 Ed. Edward Mendelson.
 Englewood Cliffs, N. J.:
 Prentice-Hall, 1978. 16-55).

She’s a mystery
She’s everything
   a woman should be
Woman in black
   got a hold on me

— Foreigner 4

Tuesday, November 12, 2019

Social Logic

Filed under: General — m759 @ 9:16 AM
 

Friday, March 10, 2017

The Transformers

Filed under: General — m759 @ 10:00 PM 

"The transformed urban interior is the spatial organisation of an  achiever, one who has crossed the class divide and who uses space to express his membership of, not aspirations towards, an ascendant class in our society: the class of those people who earn their living by transformation — as opposed to the mere reproduction — of symbols, such as writers, designers, and academics."

— The Social Logic of Space ,
     by Bill Hillier and Julienne Hanson,
     Cambridge University Press, 1984

For another perspective on the achievers, see The Deceivers .

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — m759 @ 11:00 PM

Stevens's Omega and Alpha (see previous post) suggest a review.

Omega — The Berlekamp Garden.  See Misère Play (April 8, 2019).
Alpha  —  The Kinder Garten.  See Eighfold Cube.

Illustrations —

The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.

Froebel's Third Gift: A cube made up of eight subcubes  

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Monday, August 19, 2019

A Couple of Tots

Filed under: General — Tags: — m759 @ 10:15 PM

The title is from the post "Child's Play" of May 21, 2012 . . .

"It seems that only one course is open to the philosopher
who values knowledge and truth above all else. He must
refuse to accept from the champions of the forms the
doctrine that all reality is changeless [and exclusively
immaterial], and he must turn a deaf ear to the other party
who represent reality as everywhere changing [and as only
material]. Like a child begging for 'both', he must declare
that reality or the sum of things is both at once  [το όν τε και
το παν συναμφότερα] (Sophist  246a-249d)."

Related material —

"Schoolgirl Space: 1984 Revisited" (July 9, 2019) and
posts tagged Tetrahedron vs. Square.

Tuesday, August 13, 2019

Putting the Structure  in Structuralism

Filed under: General — m759 @ 8:34 PM

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.

Saturday, July 6, 2019

Mythos and Logos

Filed under: General — m759 @ 8:56 AM

Mythos


Logos

The six square patterns which, applied as above to the faces of a cube,
form "diamond" and "whirl" patterns, appear also in the logo of a coal-
mining company —

 .

Related material —

Tuesday, June 25, 2019

Analogy

Filed under: General — Tags: — m759 @ 6:36 AM

Tuesday, June 18, 2019

Paris Review

Filed under: General — Tags: — m759 @ 3:33 AM

"The loveliness of Paris seems somehow sadly gay." — Song lyric

Stewart also starred in "Equals" (2016). From a synopsis —

"Stewart plays Nia, a writer who works at a company that extols
the virtues of space exploration in a post-apocalyptic society.
She falls in love with the film's main character, Silas (Nicholas Hoult),
an illustrator . . . ."

Space art in The Paris Review

For a different sort of space exploration, see Eightfold 1984.

Sunday, June 2, 2019

The Art of Lying

Filed under: General — m759 @ 2:13 PM

(Continued … See “Is Fiction the Art of Lying?” by Mario Vargas Llosa, 
New York Times  essay of October 7, 1984.)

"A non-fiction writer must have the freedom
to imagine the facts they [sic ] use."

Sure they must.

Saturday, December 1, 2018

In Memoriam

Filed under: General — m759 @ 10:06 AM

From today's print New York Times  obituary for a screenwriter
who reportedly died last Sunday —

“Indiana Jones and the Temple of Doom,”
a 1984 follow-up to “Raiders of the Lost Ark”
made an estimated $333 million worldwide.

Friday, November 16, 2018

Parable of India

Filed under: General — m759 @ 12:42 PM

See too  "When thou seekest me, seek towards India."

Sunday, November 4, 2018

Kristen vs. the Space Witch*

Filed under: General — Tags: — m759 @ 11:59 PM

* We know the former. There is no shortage of candidates for the latter.

Saturday, August 25, 2018

“Waugh, Orwell. Orwell, Waugh.”

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM

Suggested by a review of Curl on Modernism —

http://www.log24.com/log/pix18/180825-Ballard-on-Modernism.gif

Related material —

Waugh + Orwell in this journal and

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — Tags: , — m759 @ 8:56 PM

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Sunday, May 20, 2018

Some Style

Filed under: General,Geometry — Tags: — m759 @ 1:03 AM

Dialogue from the 1984 fourth draft of the script, as found on the Web,
for "Back to the Future" (1985) (apparently some changes were made
in the filming) —

A sort of "flux capacitor" (see previous post) —

The Rolls-Royce Cullinan

 plus "e" for Einstein 

Tuesday, April 17, 2018

A Necessary Possibility*

Filed under: General,Geometry — m759 @ 10:00 AM

"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."

— Rodolphe Gasché, The Tain of the Mirror ,
     Harvard University Press, 1986

IMAGE- Harvard University Press, 1986 - A page on Derrida's 'inscription'

An inscription from 2010 —

An inscription from 1984

American Mathematical Monthly, June-July 1984, p. 382

MISCELLANEA, 129

Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.) 
— Steven H. Cullinane

* See also other Log24 posts mentioning this phrase.

Friday, March 23, 2018

From the Personal to the Platonic

Filed under: General,Geometry — Tags: , — m759 @ 11:01 AM

On the Oslo artist Josefine Lyche —

"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."

Ann Cathrin Andersen
    http://bryggmagasin.no/2017/behind-the-glitter/

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic —

The Diamond Cube.

Compare and contrast the shifting hexagon logo in the Rushkoff novel above 
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).

Monday, February 12, 2018

The Void

Filed under: General — Tags: — m759 @ 11:46 AM

In memory of Professor Donald Lynden-Bell,
Emeritus Professor of Astrophysics at the University of Cambridge

Lynden-Bell with colleagues at Meteor Crater, Arizona, reportedly in 1960 —

Lynden-Bell was one of the subjects of the 2015 film "Star Men."

Related material —

"After peering into the void from a perch 
outside the visitor center, young Henry, 9, 
said he liked the rugged landscape. 
'It’s a good place to film a space movie,' he said.

Funny he should mention that — 
the crater was the setting for the climactic scenes 
of the 1984 sci-fi film 'Starman,' with Jeff Bridges 
and Karen Allen arriving for a rendezvous with 
an alien mother ship."

— Henry Fountain in The New York Times , Jan. 22, 2009

Lynden-Bell reportedly died at 82 on Feb. 5, 2018 (British time).

See as well this  journal on that date.

Friday, November 24, 2017

Scholia

Filed under: General — m759 @ 11:00 PM

From this evening's online New York Times : 

"Eric Salzman, a composer and music critic who
championed a new art form, music theater,
that was neither opera nor stage musical, died
on Nov. 12 at his home in Brooklyn. He was 84."

. . . .

"The first American Music Theater Festival 
took place in the summer of 1984.

Among that first festival’s featured works was 
'Strike Up the Band!,' Mr. Salzman’s 'reconstructed
and adapted' version of a satirical musical
with a score by George and Ira Gershwin
that had not been staged in 50 years. The director
of that production, Frank Corsaro, died 
the day before Mr. Salzman did."

Synchronology check :

"The day before" above was November 11, 2017.

Links from this  journal  on November 11

A Log24 search for Michael Sudduth and an 
October 28, 2017, Facebook post by Sudduth.

Detail of Sudduth's Nov. 11 Facebook home page

Click the above for an enlarged view of the Sudduth profile picture.

Related material :

Harold Schonberg, 1977 review of Corsaro production of Busoni's 'Dr. Faust'

Aooo.

Thursday, November 23, 2017

Lévi-Strauss vs. Propp

Filed under: General,Geometry — Tags: , , — m759 @ 12:25 PM
​Claude Lévi-Strauss

From his Structure and Form:
Reflections on a Work by Vladimir Propp
” *

To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable.

The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same  function reappearing at different  moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties  constituting the fundamental tale would themselves be transformations of each other.

Nothing prevents pushing this reduction even further and analyzing each separate partie  into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc.

In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself  his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns).

These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).

* 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).

See also “Lévi-Strauss” + Formula  in this journal.

Some background related to the previous post

Saturday, November 18, 2017

Cube Space Continued

Filed under: General,Geometry — m759 @ 4:44 AM

James Propp in the current Math Horizons  on the eightfold cube

James Propp on the eightfold cube

For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).

Tuesday, October 31, 2017

One Fell Shmoop

Filed under: General — m759 @ 12:48 PM

https://www.shmoop.com/no-country-for-old-men/coin-symbol.html —

"You know the date on this coin?"

Related material —

This journal on March 7, 2014

From Klein’s 1893  Lectures on Mathematics —

The varieties introduced by Wirtinger may be called 
  Kummer varieties….” — E. Spanier, 1956

From the "varieties introduced by Wirtinger" link above —

 .

Tuesday, September 12, 2017

Think Different

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

The New York Times  online this evening

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags:  — m759 @ 11:00 AM 

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
 — "Block Designs," by Andries E. Brouwer

. . . .

See also 1984 Bricks in this journal.

Chin Music

Filed under: General,Geometry — Tags: — m759 @ 9:45 PM

Related image suggested by "A Line for Frank" (Sept. 30, 2013) —

Sunday, July 23, 2017

Reality Butts

Filed under: General — m759 @ 3:00 PM

Continuing the 1984  theme

More about 1984 from the above May 1, 2016, post

 

Tuesday, June 20, 2017

Epic

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

Continuing the previous post's theme  

Group actions on partitions

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material — Posts now tagged Device Narratives.

Tuesday, May 16, 2017

For Westworld’s Man in Black

Filed under: General — m759 @ 12:00 PM

From University of Chicago Press in 1984:

'Erring,' by Mark C. Taylor, U. of Chicago Press, 1984

"Drawing on Hegel, Nietzsche, Derrida,
and others, Mark Taylor extends—and
goes well beyond—pioneering efforts. . . . "
—G. Douglas Atkins, 
Philosophy and Literature

Update at noon on May 16 —

"Follow the Blood Arroyo to the place
where the snake lays its eggs."

— Westworld, Season 1, Episode 2,
air date October 9, 2016

This suggests a review of Derrida + Serpent 
in this journal.

Tuesday, April 4, 2017

Test

Filed under: General — m759 @ 1:01 PM

See also, in this  journal, St. Cyprian's Day last year.

Wednesday, March 29, 2017

The Crimson Abyss

Filed under: General,Geometry — Tags: , — m759 @ 3:19 PM

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson , March 29

1984

IMAGE- 'Affine Groups on Small Binary Spaces,' illustration

2010

Logo design for Stack Exchange Math by Jin Yang
 

Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29

"… as if into a crimson abyss …." —

Update of 9 PM ET March 29, 2017:

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery —

See also "Hexagram 64 in Context" (Log24, March 16, 2017).

Art Space Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

Another view of the previous post's art space  —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

More generally, see Solomon's Cube in Log24.

See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above 
cube arrangement from "Affine groups on small binary spaces" (1984).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , , — m759 @ 11:45 AM

From Log24, "Cube Bricks 1984" —

An Approach to Symmetric Generation of the Simple Group of Order 168

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  

Friday, March 10, 2017

The Transformers

Filed under: General — m759 @ 10:00 PM

"The transformed urban interior is the spatial organisation of
an achiever, one who has crossed the class divide and who uses
space to express his membership of, not aspirations towards, 
an ascendant class in our society: the class of those people who 
earn their living by transformation— as opposed to the mere
reproduction— of symbols, such as writers, designers, and
academics"

The Social Logic of Space ,
     by Bill Hillier and Julienne Hanson,
     Cambridge University Press, 1984

For another perspective on the achievers, see The Deceivers .

Related material —

Exhibit A:

Exhibit B:

Edwin Schlossberg, 'Still Changes Through Structure' text piece

Exhibit C:

Highbeam Woman

Filed under: General — m759 @ 2:40 AM

See as well the previous post.

Sunday, February 5, 2017

Against Bewitchment

Filed under: General — m759 @ 7:55 PM

A footnote in memory of a preservationist

Title page of a thesis on language by Miles Spencer Kimball from 1984

The previous post's quotation from the Kimball thesis contains
a reference (numbered 23) to the source of Wittgenstein's 
"savages" remarkPhilosophical Investigations , § 194.

Kimball's  remarks quoted in the previous post are from
page 121 of his thesis, under the heading "Wittgenstein's
Battle Against Bewitchment by Language."

From a cinematic example of such bewitchment —

Ein Kampf

Filed under: General — Tags: — m759 @ 1:00 PM

This is from a master's thesis of 1984.

For the source, see "Ein Kampf" in this journal.

An image from the Saturday Night Live  version —

Sunday, December 25, 2016

Last Christmas*

Filed under: General — m759 @ 11:30 PM

From "Bright Symbol," a post of 12 AM
on December 25, 2015 —

From "Dark Symbol," a post of 12 PM
on December 25, 2015 —

* Title suggested by a song released by Epic Records in 1984.

Thursday, November 3, 2016

Triple Cross

Filed under: General,Geometry — Tags: , , — m759 @ 1:30 PM

(Continued See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168
Related material —

Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge  and Nanavira Thera.

See as well Bill Murray's 1984 film "The Razor's Edge"

Movie poster from 1984

"A thin line separates
love from hate,
success from failure,
life from death."

Three other dualities, from Nanavira Thera in 1959 —

"I find that there are, in every situation,
three independent dualities…."

(Click to enlarge.)

Friday, September 9, 2016

Ein Kampf

Filed under: General — m759 @ 1:00 PM

(Continued )

1984 master's thesis (PDF, 8+ MB) —

"Language, Linguistics, and Philosophy:
A Comparison of the Work of Roman Jakobson
and the Later Wittgenstein, with Some Attention
to the Philosophy of Charles Saunders Peirce,"
by Miles Spencer Kimball.

Two pages from that thesis —

Saturday, August 6, 2016

Mystic Correspondence:

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

The Cube and the Hexagram

The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic 
hexagram, or Star of David, with the six cube faces 
mapped to the six hexagram lines and the twelve  
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large). 

Exercise:  Is this noteworthy mapping** of faces to lines, 
edges to points, and vertices to triangles an isolated 
phenomenon, or can it be viewed in a larger context?

* See the discussion at magic-squares.net of
   "perimeter-magic cubes"

** Apparently derived from the Cube + Hexagon figure
    discussed here in various earlier posts. See also
    "Diamonds and Whirls," a note from 1984.

Thursday, July 14, 2016

Meditation from an April 1

Filed under: General,Geometry — Tags: — m759 @ 9:38 AM

Related material from the same day —

See also

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

The above bricks appeared in some earlier Log24 posts.

Tuesday, July 12, 2016

Hymn

Filed under: General — m759 @ 11:29 AM

Yesterday was reportedly the dies natalis  (in the Catholic sense)
of a former president of New York University.

From the conclusion of The Chronicles of Narnia

"The term is over:  the holidays have begun. 
The dream is ended:  this is the morning."

Linda Hamilton's related hymn in the 1984 film "Children of the Corn" —

https://www.youtube.com/watch?v=rpHeTcisyRo .

Wednesday, April 27, 2016

Kubrick’s Rube

Filed under: General — m759 @ 9:23 PM

In memory of a culture jammer *—

* "Mr. Lyons … made a living partly by buying,
reconditioning and selling used cars." —
— Ben Ratliff in The New York Times  this evening.

See also the previous post and, from Feb. 14 in
this  journal, the phrase "more global than local."

Friday, April 8, 2016

Space Cross

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

For George Orwell

Illustration from a book on mathematics —

This illustrates the Galois space  AG(4,2).

For some related spaces, see a note from 1984.

"There is  such a thing as a space cross."
— Saying adapted from a young-adult novel

Monday, April 4, 2016

Cube for Berlin

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

Foreword by Sir Michael Atiyah —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

 Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

"Like Rubik's Cube, The Eight  demands to be pondered."

As does a figure from 1984, Cullinane's Cube —

The Eightfold Cube

For natural group actions on the Cullinane cube, 
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 1984-2003."

(From Anatomy of a Cube, Sept. 18, 2011.)

Monday, March 21, 2016

Trophy

Filed under: General — Tags: — m759 @ 12:00 AM

From the 1984 New Orleans film Tightrope

http://www.log24.com/log/pix11A/110615-EastwoodFootball400w.jpg

This post was suggested by the late Yale literary critic
Geoffrey Hartman, who reportedly died on March 14.

" 'Interpretation is like a football game,' Professor Hartman
wrote in 'The Voice of the Shuttle,' a 1969 essay." 

A 2016 obituary by Margalit Fox

Saturday, March 12, 2016

Masonic Melody

Filed under: General — Tags: , , — m759 @ 1:31 AM

"Button your lip baby
Button your coat
Let's go out dancing
Go for the throat"

Read more: Rolling Stones – Mixed Emotions Lyrics | MetroLyrics 

This melody was suggested by a post of February 25, 2016,
by tonight's previous post "Brick-Perfect," and by
the post "Cube Bricks 1984" of March 4, 2016.

"Only connect." — E. M. Forster.

Thursday, March 10, 2016

The Ghost Machine*

Filed under: General — m759 @ 10:12 AM

"I had joined the White House early in 1984, after three years
writing Dan Rather's radio commentaries."

— Peggy Noonan, "Confessions of a White House Speechwriter,"
a 1989 New York Times  excerpt from her book What I Saw
at the Revolution

* See also What IS the frequency, Kenneth?

Thursday, November 12, 2015

The Unbaked, the Baked, and the Half-Baked

Filed under: General — Tags: — m759 @ 5:07 PM

Consider the trichotomy of the title as applied to the paragraph
by Adam Gopnik in the previous post (The Raw, the Cooked,
and the Spoiled
).

The following quotation seems to place Gopnik's words
among the half -baked.

"L'axe qui relie le cru et le cuit est caractéristique du passage
à la culture; celui qui relie le cru et le pourri, du retour à la nature,
puisque la cuisson accomplit la transformation culturelle du cru
comme la putréfaction en achève la transformation naturelle."

— Claude Lévi-Strauss, Paroles données, p.54, Plon, 1984,
     as quoted in a weblog

See also Lévi-Strauss's bizarre triangle culinaire  (French Wikipedia) —

The source of this structuralist nonsense —
Lévi-Strauss, Claude. 1969. “Le triangle culinaire.”
L’Arc  no. 26: 19-29.

Friday, October 23, 2015

Retro or Not?

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Happy birthday to the late Michael Crichton (Harvard ’64).

See also Diamond Theory Roulette —

Part of the ReCode Project (http://recodeproject.com).
Based on "Diamond Theory" by Steven H. Cullinane,
originally published in "Computer Graphics and Art" 
Vol. 2 No. 1, February 1977.
Copyright (c) 2013 Radames Ajna 
— OSI/MIT license (http://recodeproject/license).

Related remarks on Plato for Harvard’s
Graduate School of Design

See also posts from the above publication date, March 31,
2006, among posts now tagged “The Church in Philadelphia.”

Saturday, May 23, 2015

Art

Filed under: General — Tags: — m759 @ 12:00 PM

"Is Fiction the Art of Lying?" by Mario Vargas Llosa

The above link is to a Google Books Search for references
to a 1984 piece in The New York Times .

To find the Times 's  own version, change "Lying" to "Living."

"We tell ourselves stories in order to live." — Joan Didion

Wednesday, February 11, 2015

Dead Reckoning

Filed under: General — Tags: — m759 @ 5:28 PM

Continued from yesterday evening

IMAGE- Bogart in 'Casablanca' with chessboard

Today's mathematical birthday — 

Claude Chevalley, 11 Feb. 1909 – 28 June 1984.

From MacTutor —

Chevalley's daughter, Catherine Chevalley, wrote about
her father in "Claude Chevalley described by his daughter"
(1988):—

For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged.

The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons.

Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.

Wednesday, November 26, 2014

Class Act

Filed under: General,Geometry — Tags: — m759 @ 7:18 AM

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:

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

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.

Saturday, September 27, 2014

Plan B: Books

Filed under: General — m759 @ 9:48 AM
http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
Starting Out in the Evening

Right: Johnny Depp in
The Ninth Gate

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

“One must proceed cautiously, for this road— of truth and falsehood
in the realm of fiction— is riddled with traps and any enticing oasis
is usually a mirage.”

– “Is Fiction the Art of Lying?” by Mario Vargas Llosa,
New York Times  essay of October 7, 1984

For the title plan, see Sisteen in this journal.

Friday, September 19, 2014

Metaphysics for Gilliam

Filed under: General — m759 @ 9:29 PM

See also…

Related remarks: Diederik Aerts at arXiv.org.

See also Aerts (as above) on the metaphysics of entities  (1984):

Thursday, August 28, 2014

Source of the Finite

Filed under: General,Geometry — Tags: — m759 @ 10:20 AM

"Die Unendlichkeit  ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche  stamme…."

— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: Aubier-Flammarion, 1969), fragment 120, p. 118

Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p.  181.

This quotation was suggested by the Bauhaus-related phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 1984-2003.

For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.

Monday, July 7, 2014

Tricky Task

Filed under: General,Geometry — m759 @ 12:25 PM

Roger Cooke in the Notices of the American
Mathematical Society 
, April 2010 —

Life on the Mathematical Frontier:
Legendary Figures and Their Adventures

“In most cases involving the modern era, there
are enough documents to produce a clear picture
of mathematical developments, and conjectures
for which there is no eyewitness or documentary
evidence are not needed. Even so, legends do
arise. (Who has not heard the ‘explanation’ of
the absence of a Nobel Prize in mathematics?)
The situation is different regarding ancient math-
ematics, however, especially in the period before
Plato’s students began to study geometry. Much
of the prehistory involves allegations about the
mysterious Pythagoreans, and sorting out what is
reliable from what is not is a tricky task.

In this article, I will begin with some modern
anecdotes that have become either legend or
folklore, then work backward in time to take a
more detailed look at Greek mathematics, especially
the Pythagoreans, Plato, and Euclid. I hope at the
very least that the reader finds my examples
amusing, that being one of my goals. If readers
also take away some new insight or mathematical
aphorisms, expressing a sense of the worthiness of
our calling, that would be even better.”

Aphorism:  “Triangles are square.” 

(American Mathematical Monthly , June-July 1984)

Insight:  The Square-Triangle Theorem.

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 AM

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Tuesday, February 25, 2014

Singing Contest

Filed under: General — m759 @ 12:00 AM

"What a lovely singing voice you must have."
— Bill Murray in Ghostbusters  (1984)

Contestant One Ruth Margraff

Contestant Two Sandra Sangiao 

Thursday, January 23, 2014

For the Snow Queen —

Filed under: General — m759 @ 8:11 AM

Dark Epiphanies

Part I: 

Part II:

"A Little Boy and a Little Girl," by Hans Christian Andersen
(second story of the seven that make up The Snow Queen )

Part III:

A former Snow White —

Saturday, January 18, 2014

The Triangle Relativity Problem

Filed under: General,Geometry — m759 @ 5:01 PM

A sequel to last night's post The 4×4 Relativity Problem —

IMAGE- Triangle Coordinatization

In other words, how should the triangle corresponding to
the above square be coordinatized ?

See also a post of July 8, 2012 — "Not Quite Obvious."

Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly  item published
in 1984.

Wednesday, January 8, 2014

Not Subversive, Not Fantasy

Filed under: General,Geometry — m759 @ 2:01 PM

The title refers to that of today's previous post, which linked to
a song from the June 1, 1983, album Synchronicity .
(Cf.  that term in this journal.)

For some work of my own from the following year, 1984, see

IMAGE- Internet Archive, 'Notes on Groups and Geometry, 1978-1986'

as well as the Orwellian dictum Triangles Are Square.

(The cubical figure at left above is from the same month,
if not the same day, as Synchronicity —  June 21, 1983.)

Sunday, January 5, 2014

For Amy…

Filed under: General — m759 @ 5:01 PM

And for Scarlett — A Venus Flytrap

From last evening's  John Fogerty 1984 video —

From this morning's paper —

Click the soup for some backstory.

"Assume a can opener."

Sunday, November 24, 2013

Logic for Jews*

Filed under: General,Geometry — Tags: , — m759 @ 7:20 AM

The search for 1984 at the end of last evening's post
suggests the following Sunday meditation.

My own contribution to this genre—

A triangle-decomposition result from 1984:

American Mathematical Monthly ,  June-July 1984, p. 382

MISCELLANEA, 129

Triangles are square

"Every triangle consists of n  congruent copies of itself"
is true if and only if  is a square. (The proof is trivial.) 
— Steven H. Cullinane

The Orwell slogans are false. My own is not.

* The "for Jews" of the title applies to some readers of Edward Frenkel.

Saturday, November 23, 2013

Light Years Apart?

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a recent attempt to vulgarize the Langlands program:

"Galois’ work is a great example of the power of a mathematical insight…. 

And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related."

— Frenkel, Edward (2013-10-01).
     Love and Math: The Heart of Hidden Reality
     (p. 78, Basic Books, Kindle Edition) 

(Links to related Wikipedia articles have been added.)

 

Wikipedia on the Langlands program

The starting point of the program may be seen as Emil Artin's reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

 

From "An Elementary Introduction to the Langlands Program," by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)

On page 194:

"The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations.

In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called 'right regular' representation of G….

Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described."

Gross ]  K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548.

Mackey ]  G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.

* A link to a related Math Overflow article has been added.

In 2011, Frenkel published a commentary in the A.M.S. Bulletin  
on Gelbart's Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) "light years apart" rhetoric from his new book 
quoted above.

In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.

For some remarks of my own that mention
that year, see a search for 1984 in this journal.

Friday, June 21, 2013

Lexicon

Filed under: General — Tags: — m759 @ 1:00 PM

From the final pages of the new novel
Lexicon , by Max Barry:

“… a fundamental language
of the human mind—
the tongue in which the human animal
speaks to itself at the basest level.
The machine language, in essence….”

“… the questions raised by
this underlying lexicon.
What are its words?
How many are there? ….
Can we learn to speak them?
What does it sound like
when who we are is expressed
in its most fundamental form?
Something to think about.”

       R. Lowell

Related material:

IMAGE- Hypokeimenon in Liddell and Scott's Greek-English Lexicon

“… the clocks were striking thirteen.” — 1984

Thursday, June 6, 2013

The Deep End (continued)

Filed under: General — m759 @ 6:29 PM

Latin Lesson

Details in an etymology linked to here Monday, June 3,
in a post titled The Deep End  

"… mid-15c., from Middle French pensée  … from
  fem. past participle of penser  'to think,' from
  Latin pensare  'consider'…." 

A remembrance of the late, great, Esther Williams,
who died early today:

After marrying Lamas, she retired from public life.
Williams explained in a 1984 interview, "A really terrific guy
comes along and says, 'I wish you'd stay home and be
my wife,' and that's the most logical thing in the world for a Latin.
And I loved being a Latin wife — you get treated very well.
There's a lot of attention in return for that sacrifice."

See, too, the link alea  from yesterday's Stitch.

Review Comment Submitted

Filed under: General,Geometry — m759 @ 2:19 AM

The Mathematical Association of America has a
submit-a-review form that apparently allows readers
to comment on previously reviewed books.

This morning I submitted the following comment on
Alexander Bogomolny's March 16, 2012, review of 
Martin J. Erickson's Beautiful Mathematics :

In section 5.17, pages 106-108, "A Group of Operations,"
Erickson does not acknowledge any source. This section
is based on the Cullinane diamond theorem. See that
theorem (published in an AMS abstract in 1979) at
PlanetMath.org and EncyclopediaOfMath.org, and
elsewhere on the Web. Details of the proof given by
Erickson may be found in "Binary Coordinate Systems,"
a 1984 article on the Web at
http://finitegeometry.org/sc/gen/coord.html.

If and when the comment may be published, I do not know.

Update of about 6:45 PM ET June 7:

The above comment is now online at the MAA review site.

Update of about 7 PM ET July 29:

The MAA review site's web address was changed, and the 
above comment was omitted from the page at the new address.
This has now been corrected.

Tuesday, May 14, 2013

Snakes on a Plane

Filed under: General,Geometry — m759 @ 7:27 AM

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

Monday, April 8, 2013

Magic for Jews

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.

Here is such a look (click to enlarge):

I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram  is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)

The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not  be encouraged to waste their time on this topic.

For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems
."

Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .

Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.

Related material from today's Daily Princetonian :

See also Nash + Princeton in this journal.

Tuesday, February 19, 2013

Configurations

Filed under: General,Geometry — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Monday, February 18, 2013

Permanence

Filed under: General,Geometry — m759 @ 2:00 PM

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Related material

"Here is another way to present the deep question 1984  raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
     The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

— G. H. Hardy, A Mathematician’s Apology

Sunday, February 3, 2013

Trophy

Filed under: General — Tags: , — m759 @ 11:30 PM

From the 1984 New Orleans film Tightrope

http://www.log24.com/log/pix11A/110615-EastwoodFootball400w.jpg

Related material: Walking the Tightrope and Transgressing.

Wednesday, January 2, 2013

PlanetMath link

Filed under: General,Geometry — m759 @ 2:00 PM

Update of May 27, 2013:
The post below is now outdated. See
http://planetmath.org/cullinanediamondtheorem .

__________________________________________________________________

The brief note on the diamond theorem at PlanetMath
disappeared some time ago. Here is a link to its
current URL: http://planetmath.org/?op=getobj;from=lec;id=49.

Update of 3 PM ET Jan. 2, 2013—

Another item recovered from Internet storage:

IMAGE- Miscellanea, 129: 'Triangles are square'- Amer. Math. Monthly, Vol. 91, No. 6, June-July 1984, p. 382

Click on the Monthly  page for some background.

Tuesday, November 6, 2012

Bend Sinister

Filed under: General — m759 @ 6:30 AM

This morning's New York Times  obituaries—

These suggest a look at Solving Nabokov's Lolita Riddle ,
by Joanne Morgan (Sydney: Cosynch Press, 2005).

That book discusses Lolita as a character like Lewis Carroll's Alice.

(The Red Queen and Alice of course correspond to figures in
the first two thumbnails above.)

From the obituary associated with the third thumbnail above:

"Front-page headlines combined concision and dark humor." 

The title of this post, Bend Sinister , is not unlike such a headline.
It is the title of a novel by Nabokov (often compared with Orwell's 1984 )
that is discussed in the Lolita Riddle  book.

Related material— The bend sinister found in Log24 searches
for Hexagram 14 and for the phrase Hands-On

IMAGE- Magician's hands on his wand, viewed as a diagonal of a square

Monday, November 5, 2012

Design Cubes

Filed under: General,Geometry — m759 @ 9:48 PM

Continued from April 2, 2012.

Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties

Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz  cubes (see links below).

Horowitz Design Cubes Package

Horowitz Design Cubes (1971)

1973 Horowitz Design Cubes Patent

Horowitz Biography

Thursday, August 23, 2012

Chapman’s Homer

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

Louis Sahagun in today's Los Angeles Times

The late Professor Marvin W. Meyer

 "was our Indiana Jones,"  said James L. Doti,
president of Chapman University in Orange,
where Meyer held the Griset Chair in Bible
and Christian Studies and was director of
the Albert Schweitzer Institute.

Meyer reportedly died on August 16.

IMAGE- The late Professor Marvin W. Meyer of Chapman University in Orange, CA, with the university's emblem, the eight-pointed star

Thursday, August 16, 2012

Semiotics

m759 @ 4:00 AM

IMAGE- Eight-pointed star formed by the four symmetry axes of the square

"Two clichés make us laugh, but
a hundred clichés move us
because we sense dimly that the clichés
are talking among themselves and
celebrating a reunion."

— Umberto Eco

"'Casablanca': Cult Movies and Intertextual Collage,"
by Umberto Eco in SubStance , Vol. 14, No. 2, Issue 47:
In Search of Eco's Roses  (1985), pp. 3-12.

(This paper was presented at a symposium,
"Semiotics of the Cinema: The State of the Art,"
in Toronto on June 18, 1984.)
Journal article published by U. of Wisconsin Press.
Stable URL: http://www.jstor.org/stable/3685047.

Click image for some related material.

 

 

Thursday, August 16, 2012

Semiotics

Filed under: General — Tags: — m759 @ 4:00 AM

"Two clichés make us laugh, but
a hundred clichés move us
because we sense dimly that the clichés
are talking among themselves and
celebrating a reunion."

Umberto Eco

"'Casablanca': Cult Movies and Intertextual Collage,"
by Umberto Eco in SubStance , Vol. 14, No. 2, Issue 47:
In Search of Eco's Roses  (1985), pp. 3-12.
(This paper was presented at a symposium,
"Semiotics of the Cinema: The State of the Art,"
in Toronto on June 18, 1984.)
Journal article published by U. of Wisconsin Press.
Stable URL: http://www.jstor.org/stable/3685047.

Click image for some related material.

Thursday, August 2, 2012

Logos

Filed under: General,Geometry — m759 @ 12:00 PM

(Continued from December 26th, 2011)

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Some material at math.stackexchange.com related to
yesterday evening's post on Elementary Finite Geometry

Questions on this topic have recently been
discussed at Affine plane of order 4? and at
Turning affine planes into projective planes.

(For a better discussion of the affine plane of order 4,
see Affine Planes and Mutually Orthogonal Latin Squares
at the website of William Cherowitzo, professor at UC Denver.)

Saturday, July 14, 2012

Lemma

Filed under: General,Geometry — m759 @ 2:25 PM

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—

… a "square-triangle" lemma:

   ( t ∈ T , t  is an  -replica )
    if and only if  
n  is a square.

  [I.e., "Every triangle is an -replica"
   is true if and only if n  is a square.]

For definitions, see the 1985 letter in Triangles Are Square.

(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)

A search today for related material yielded the following—

"Suppose that one side of a triangle
has length . Then it can be cut
into n  2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length  have lengths 1."

This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite  obvious.

At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.

Sunday, July 8, 2012

Not Quite Obvious

Filed under: General,Geometry — m759 @ 11:00 AM

"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."

— Steven H. Cullinane, letter in the
American Mathematical Monthly 
1985 June-July issue

If the ancient Greeks had not been distracted by
investigations of triangular  (as opposed to square )
numbers, they might have done something with this fact.

A search for occurrences of the phrase

"n2 [i.e., n 2 ] congruent triangles" 

indicates only fairly recent (i.e., later than 1984) results.*

Some related material, updated this morning—

This suggests a problem
 

What mappings of a square  array of n 2 points to
a triangular  array of n 2 points are "natural"?

http://www.log24.com/log/pix12B/120708-SquareAndTriangle.jpg

In the figure above, whether
the 322,560 natural permutations
of the square's 16 points
map in any natural way to
  permutations of the triangle's 16 points
is not immediately apparent.

 

* Update of July 15, 2012 (11:07 PM ET)—

Theorem on " rep-" (Golomb's terminology)
triangles from a 1982 book—

IMAGE- Theorem (12.3) on Golomb and 'rep-k^2' triangles in book published in 1982-- 'Transformation Geometry,' by George Edward Martin

Friday, June 22, 2012

Bowling in Diagon Alley

Filed under: General,Geometry — Tags: — m759 @ 8:28 AM

IMAGE- Josefine Lyche bowling, from her Facebook page

Josefine Lyche bowling (Facebook, June 12, 2012)

"Where Does Math Come From?"

A professor of philosophy in 1984 on Socrates's geometric proof in Plato's Meno  dialogue—

"These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling…."

— Ian Hacking in The New York Review of Books , Feb. 16, 1984

The same professor introducing a new edition of Kuhn's Structure of Scientific Revolutions

"Paradigms Regained" (Los Angeles Review of Books , April 18, 2012)—

"That is the structure of scientific revolutions: normal science with a paradigm and a dedication to solving puzzles; followed by serious anomalies, which lead to a crisis; and finally resolution of the crisis by a new paradigm. Another famous word does not occur in the section titles: incommensurability. This is the idea that, in the course of a revolution and paradigm shift, the new ideas and assertions cannot be strictly compared to the old ones."

The Meno  proof involves inscribing diagonals  in squares. It is therefore related, albeit indirectly, to the classic Greek discovery that the diagonals of a square are incommensurable  with its sides. Hence the following discussion of incommensurability seems relevant.

IMAGE- Von Fritz in 1945 on incommensurability and the tetractys (10 as a triangular number)

See also von Fritz and incommensurability in The New York Times  (March 8, 2011).

For mathematical remarks related to the 10-dot triangular array of von Fritz, diagonals, and bowling, see this  journal on Nov. 8, 2011— "Stoned."

Thursday, March 22, 2012

Square-Triangle Theorem continued

Filed under: General,Geometry — m759 @ 6:00 AM

Last night's post described a book by Alexander Soifer
on questions closely related to— and possibly
suggested by— a Miscellanea  item and a letter to
the editor
in the American Mathematical Monthly ,
June-July issues of 1984 and 1985.

Further search yields a series of three papers by
Michael Beeson on the same questions. These papers are
more mathematically  presentable than Soifer's book.

Triangle Tiling I 

http://www.michaelbeeson.com/research/papers/TriangleTiling1.pdf

       March 2, 2012

Triangle Tiling II 

http://www.michaelbeeson.com/research/papers/TriangleTiling2.pdf

       February 18, 2012

Triangle Tiling III 

http://www.michaelbeeson.com/research/papers/TriangleTiling3.pdf

       March 11, 2012 

These three recent preprints replace some 2010 drafts not now available.
Here are the abstracts of those drafts—

"Tiling triangle ABC with congruent triangles similar to ABC"
 (March 13, 2010),

"Tiling a triangle with congruent triangles"
(July 1, 2010).

Beeson, like Soifer, omits any reference to the "Triangles are square" item
of 1984 and the followup letter of 1985 in the Monthly .

Thursday, January 12, 2012

Triangles Are Square

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

Coming across John H. Conway's 1991*
pinwheel  triangle decomposition this morning—

http://www.log24.com/log/pix12/120112-ConwayTriangleDecomposition.jpg

— suggested a review of a triangle decomposition result from 1984:

IMAGE- Triangle and square, each with 16 parts

Figure A

(Click the below image to enlarge.)

IMAGE- 'Triangles Are Square,' by Steven H. Cullinane (American Mathematical Monthly, 1985)

The above 1985 note immediately suggests a problem—

What mappings of a square  with c 2 congruent parts
to a triangle  with c 2 congruent parts are "natural"?**

(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)

* Communicated to Charles Radin in January 1991. The Conway
  decomposition may, of course, have been discovered much earlier.

** Update of Jan. 18, 2012— For a trial solution to the inverse
    problem, see the "Triangles are Square" page at finitegeometry.org.

Monday, December 5, 2011

The Shining (Norwegian Version)

Filed under: General,Geometry — m759 @ 4:01 AM

A check tonight of Norwegian artist Josefine Lyche's recent activities
shows she has added a video to her web page that has for some time
contained a wall piece based on the 2×2 case of the diamond theorem

http://www.log24.com/log/pix11C/111205-Lyche-DiamondTheoremPage.jpg

The video (top left in screenshot above) is a tasteless New-Age discourse
that sounds frighteningly like the teachings of the late Heaven's Gate cult.

Investigating the source of the video on vimeo.com, I found the account of one "Jo Lyxe,"
who joined vimeo in September 2011. This is apparently a variant of Josefine Lyche's name.

The account has three videos—

  1. "High on RAM (OverLoad)"– Fluid running through a computer's innards
  2. "Death 2 Everyone"– A mystic vision of the afterlife
  3. "Realization of the Ultimate Reality (Beyond Form)"– The Blue Star video above

Lyche has elsewhere discussed her New-Age interests, so the contents of the videos
were not too surprising… except for one thing. Vimeo.com states that all three videos
were uploaded "2 months ago"— apparently when "Lyxe" first set up an account.*

I do not know, or particularly care, where she got the Blue Star video, but the other
videos interested me considerably when I found them tonight… since they are
drawn from films I discussed in this journal much more recently than "2 months ago."

"High on RAM (OverLoad)" is taken from the 1984 film "Electric Dreams" that I came across
and discussed here yesterday afternoon, well before  re-encountering it again tonight.

http://www.log24.com/log/pix11C/111205-Lyxe-HighOnRam.jpg

http://www.log24.com/log/pix11C/111205-ElectricDreamsTrailer.jpg

And "Death 2 Everyone" (whose title** is perhaps a philosophical statement about inevitable mortality
rather than a mad terrorist curse) is taken from the 1983 Natalie Wood film "Brainstorm."

http://www.log24.com/log/pix11C/111205-Lyxe-Death2.jpg

http://www.log24.com/log/pix11C/111205-Brainstorm-FreakyPart.jpg

"Brainstorm" was also discussed here recently… on November 18th, in a post suggested by the
reopening of the investigation into Wood's death.

I had no inkling that these "Jo Lyxe" videos existed until tonight.

The overlapping content of Lyche's mental ramblings and my own seems rather surprising.
Perhaps it is a Norwegian mind-meld, perhaps just a coincidence of interests.

* Update: Google searches by the titles  on Dec. 5 show that all three "Lyxe" videos
                 were uploaded on September 20 and 21, 2011.

** Update: A search shows a track with this title on a Glasgow band's 1994 album.

Sunday, September 18, 2011

Anatomy of a Cube

Filed under: General,Geometry — m759 @ 12:00 PM

R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following

"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955

http://www.log24.com/log/pix11B/110918-SprottAndCube.jpg

The figure by Cullinane included above shows a way to visualize Sprott's remarks.

For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."

Saturday, June 25, 2011

The Fano Entity

Filed under: General,Geometry — m759 @ 2:02 AM

The New York Times  at 9 PM ET June 23, 2011

ROBERT FANO: I’m trying to think briefly how to put it.

GINO FANO: "On the Fundamental Postulates"—

"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."

"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."

Also from the Times  article quoted above…

"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”

— Tom Van Vleck

Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—

http://www.log24.com/log/pix11A/110624-1984-Bricks-Sm.jpg

The art is also related to the mathematics of Gino Fano.

For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."

Wednesday, June 15, 2011

Mischief

Filed under: General — m759 @ 11:32 AM

The New York Times  today on a new show by tightrope artist Philippe Petit—

“He comes out of that really wonderful European tradition of street performance— it blends a boundary of what’s art and what’s life,” said Jay Wegman, the director of the Abrons Arts Center, who offered Mr. Petit the three-night run. “He’s also kind of mischievous, not in a threatening or evil way, but in a child’s way of teasing and having fun.”

For a much darker approach to street performance that also involves mischief and blended boundaries, see "Tightrope" (1984)—

http://www.log24.com/log/pix11A/110615-EastwoodFootball400w.jpg

Background: Men in Feminism , edited by Alice Jardine and
published by Taylor & Francis in 1987, "Walking the Tightrope
of Feminism and Male Desire," by Judith Mayne, page 64

See also yesterday's Another Opening and Football in this journal.

Thursday, May 26, 2011

Prime Cubes

Filed under: General,Geometry — m759 @ 11:09 AM

The title refers not to numbers  of the form p 3, p  prime, but to geometric  cubes with p 3 subcubes.

Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation  groups of degree  (not order ) p 3.

IMAGE- From preface to Larry C. Grove, 'Classical Groups and Geometric Algebra

For the case p =2, see The Eightfold Cube.

For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")

For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—

"Actions of GL(2,p )  on a p ×p  coordinate-array
have the same sorts of symmetries,
where p  is any odd prime."

* Group Actions, 1984-2009

Sunday, April 10, 2011

Finishing Up at Noon

Filed under: General — Tags: — m759 @ 12:00 PM

From last October—

Friday, October 8, 2010

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

A stitch in time…

http://www.log24.com/log/pix11/110410-BeastFavicon.jpg

Related material—

    This journal on April 8
http://www.log24.com/log/pix11/110408-HopkinsAsExorcist.jpg

See also "Putting Mental Health on the Map at Harvard"—

Harvard Crimson , Friday, April 8, 2011, 2:09 AM—

They're outside the Science Center with their signs, their cheer, and their smiles. They've been introducing themselves over House lists, and they want you to ask questions. They're here for you. They're the Student Mental Heath Liaisons.

Harvard's SMHL crewthey pronounce it smilehave recently launched a new website and recruited more members in their effort to foster an informed and understanding environment on campus….

Mental Health Services, SMHL said, are not meant for "students who are really 'crazy.'" Everyone is entitled to a little help smiling.

http://www.log24.com/log/pix11/110410-DrLecter.jpg

Friday, April 8, 2011

Windows

Filed under: General — m759 @ 12:00 PM

Roberta Smith in today's New York Times

"… the argument that painting may ultimately be about
little more than the communication of some quality of
light and space, however abstract or indirect."

— Review of "Rooms With a View" at the Met

Lowry —

http://www.log24.com/log/pix10B/101101-LowryWindow.jpg

Malcolm Lowry, author of Under the Volcano

Hollywood —

http://www.log24.com/log/pix11/110408-HopkinsAsExorcist.jpg

Related material —

Friday, October 8, 2010

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Tuesday, April 5, 2011

For Ned*

Filed under: General — m759 @ 12:00 PM

(A sequel to last night's "For Taylor")

On Joan Tewkesbury, who wrote the script for the 1975 film "Nashville"—

She urges writers to continue to generate new ideas
and new material. "Keep writing. The hardest thing
is to sell one script and not have another to follow it with."

One script— Yesterday's link titled "An Ordinary Evening in Tennessee"

Another— "A Point of Central Arrival"

Related material from last October—

Friday, October 8, 2010

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

* A former governor of Tennessee who died at 80 yesterday in Nashville

Saturday, January 8, 2011

True Grid (continued)

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

Sunday December 10, 2006

 

 m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
– Thomas Pynchon   

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Friday, January 7, 2011

Coxeter and the Aleph

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

In a nutshell —

Epigraph to "The Aleph," a 1945 story by Borges:

O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2

http://www.log24.com/log/pix11/110107-BorgesElAleph.jpg

The story in book form, 1949

A 2006 biography of geometer H.S.M. Coxeter:

http://www.log24.com/log/pix11/110107-KingOfInfiniteSpace-Sm.jpg

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Related material: Group Actions, 1984-2009.

Friday, October 8, 2010

Starting Out in the Evening

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

… and Finishing Up at Noon

This post was suggested by last evening’s post on mathematics and narrative
and by Michiko Kakutani on Vargas Llosa in this morning’s New York Times.

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
Starting Out in the Evening

Right: Johnny Depp in
The Ninth Gate

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

“One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage.”

— “Is Fiction the Art of Lying?”* by Mario Vargas Llosa, New York Times  essay of October 7, 1984

My own adventures in that realm— as reader, not author— may illustrate Llosa’s remark.

A nearby stack of paperbacks I haven’t touched for some months (in order from bottom to top)—

  1. Pale Rider by Alan Dean Foster
  2. Franny and Zooey by J. D. Salinger
  3. The Hobbit by J. R. R. Tolkien
  4. Le Petit Prince by Antoine de Saint Exupéry
  5. Literary Reflections by James A. Michener
  6. The Ninth Configuration by William Peter Blatty
  7. A Streetcar Named Desire by Tennessee Williams
  8. Nine Stories by J. D. Salinger
  9. A Midsummer Night’s Dream by William Shakespeare
  10. The Tempest by William Shakespeare
  11. Being There by Jerzy Kosinski
  12. What Dreams May Come by Richard Matheson
  13. Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig
  14. A Gathering of Spies by John Altman
  15. Selected Poems by Robinson Jeffers
  16. Hook— Tinkerbell’s Challenge by Tristar Pictures
  17. Rising Sun by Michael Crichton
  18. Changewar by Fritz Leiber
  19. The Painted Word by Tom Wolfe
  20. The Hustler by Walter Tevis
  21. The Natural by Bernard Malamud
  22. Truly Tasteless Jokes by Blanche Knott
  23. The Man Who Was Thursday by G. K. Chesterton
  24. Under the Volcano by Malcolm Lowry

What moral Vargas Llosa might draw from the above stack I do not know.

Generally, I prefer the sorts of books in a different nearby stack. See Sisteen, from May 25. That post the fanciful reader may view as related to number 16 in the above list. The reader may also relate numbers 24 and 22 above (an odd couple) to By Chance, from Thursday, July 22.

* The Web version’s title has a misprint— “living” instead of “lying.”

Saturday, August 14, 2010

Iconic Notation

Filed under: General — m759 @ 12:12 AM

Continued from Friday the 13th

(Click to enlarge.)

http://www.log24.com/log/pix10B/100814-DBsm.jpg

Related material—

The image “http://www.log24.com/log/pix07A/070814-timejoin15.jpg” cannot be displayed, because it contains errors.
Cover art by Barclay Shaw reprinted from an earlier (1984) edition

IMAGE- Variations on Hexagram 14

A question from Ivan Illich
(founder of CIDOC, the Center for Intercultural Documentation,
in Cuernavaca, Mexico)—

"Who can be served by bridges to nowhere?"

For more about nowhere, see Utopia. See also http://outis.blogspot.com.

Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

Example 1— The 2×2×2 Cube—

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
 

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
  3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 
note by Cullinane

Pegg gives no reference to the 1985 work on group actions.
 

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Sunday, June 20, 2010

Lovasz Wins Kyoto Prize

Filed under: General,Geometry — m759 @ 11:00 AM

From a June 18 press release

KYOTO, Japan, Jun 18, 2010 (BUSINESS WIRE) — The non-profit Inamori Foundation (President: Dr. Kazuo Inamori) today announced that Dr. Laszlo Lovasz will receive its 26th annual Kyoto Prize in Basic Sciences, which for 2010 focuses on the field of Mathematical Sciences. Dr. Lovasz, 62, a citizen of both Hungary and the United States, will receive the award for his outstanding contributions to the advancement of both the academic and technological possibilities of the mathematical sciences.

Dr. Lovasz currently serves as both director of the Mathematical Institute at Eotvos Lorand University in Budapest and as president of the International Mathematics Union. Among many positions held throughout his distinguished career, Dr. Lovasz also served as a senior research member at Microsoft Research Center and as a professor of computer science at Yale University.

Related material: Cube Space, 1984-2003.

See also "Kyoto Prize" in this journal—

The Kyoto Prize is "administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”

— – Montreal Gazette, June 20, 2008

http://www.log24.com/log/pix10A/100620-KyoceraLogo.gif

 

Wittgenstein and Fly from Fly-Bottle

Fly from Fly Bottle

Wednesday, February 24, 2010

Transvections

Filed under: General,Geometry — m759 @ 4:24 PM

A topic related to A Simple Reflection Group of Order 168

Transvection groups over GF(2). See, for instance…

  1. Binary Coordinate Systems, by Steven H. Cullinane, 1984
     
  2. Classification of the Finite N-Generator Transvection Groups Over Z2, by Jizhu Nan and Jing Zhao, 2009, Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010), 185–202
     
  3. Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"

Wednesday, November 4, 2009

Sinner or Saint?

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

As noted here yesterday, Claude Levi-Strauss may have died on Devil's Night, on Halloween, or on All Saints' Day. He was apparently a myth-transformer to the end.

The Independent says today he died on Sunday, All Saints' Day. Its eulogy, by Adam Kuper, is well-written, noting that linguist Roman Jakobson was a source of Levi-Strauss's theory of oppositions in myth, and observing that

"… binary oppositions tend to accumulate to form structures…."

Yes, they do. Examples:

I. The structures in the Diamond Puzzle

Adam and God (Sistine Chapel), with Jungian Self-Symbol and Ojo de Dios (The Diamond Puzzle)

Click on image for Jungian background.

II: The structure on a recent cover of Semiotica

http://www.log24.com/log/pix09A/091103-SemioticaSm.jpg

Click to enlarge.

The Semiotica article by mathematical linguist Solomon Marcus is a defense of the Levi-Strauss canonic formula mentioned here yesterday.

It is available online for $40.

A less expensive, and possibly more informative, look at oppositions in linguistics is available for free online in a 1984 master's thesis (pdf, 8+ mb)–

"Language, Linguistics, and Philosophy: A Comparison of the Work of Roman Jakobson and the Later Wittgenstein, with Some Attention to the Philosophy of Charles Saunders Peirce," by Miles Spencer Kimball.

Wednesday, August 19, 2009

Wednesday August 19, 2009

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

Group Actions, 1984-2009

From a 1984 book review:

"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

— Jonathan L. Alperin,
   review of books on group theory,
   Bulletin (New Series) of the American
   Mathematical Society
10 (1984) 121, doi:
   10.1090/S0273-0979-1984-15210-8
 

A more specific example:


Actions of GL(2,3) on a 3x3 coordinate-array

The same example
at Wolfram.com:

Ed Pegg Jr.'s program at Wolfram.com to display a large number of actions of small linear groups over finite fields

Caption from Wolfram.com:
 
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."

Citation data from Wolfram.com:

"GL(2,p) and GL(3,3) Acting on Points"
 from The Wolfram Demonstrations Project,
 http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
 Contributed by: Ed Pegg Jr"

As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:

"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."

Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
 

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