Log24

Friday, October 19, 2018

Kaleidoscope and Old Lace

Filed under: General — m759 @ 12:31 PM

See posts now tagged "Kaleidoscope Society" and, more generally,
a search in this journal for "Kaleidoscope."

Related material —

Photo caption in a news story today:

"Father Gary Thomas attends the premiere of Warner Brothers’
'The Rite' at Grauman’s Chinese Theatre, in Los Angeles,
on January 26, 2011. Thomas is holding a special Mass
on Thursday and Saturday [Oct. 18 and 20] to counter
a planned hex on Supreme Court Justice Brett Kavanaugh."

See as well posts tagged "Rubik Exorcism."

IMAGE- Anthony Hopkins exorcises a Rubik cube

Monday, August 6, 2018

The Girl with Kaleidoscope Eyes

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

http://www.log24.com/log/pix18/180806-Lexicon-image-search.jpg

“All right, Jessshica. It’s time to open the boxsssschhh.”

“Gahh,” she said. She began to walk toward the box, but her heart failed her and she retreated back to the chair. “Fuck. Fuck.” Something mechanical purred. The seam she had found cracked open and the top of the box began to rise. She squeezed shut her eyes and groped her way into a corner, curling up against the concrete and plugging her ears with her fingers. That song she’d heard the busker playing on the train platform with Eliot, “Lucy in the Sky with Diamonds”; she used to sing that. Back in San Francisco, before she learned card tricks. It was how she’d met Benny: He played guitar. Lucy was the best earner, Benny said, so that was mainly what she sang. She must have sung it five times an hour, day after day. At first she liked it but then it was like an infection, and there was nothing she could do and nowhere she could go without it running across her brain or humming on her lips, and God knew she tried; she was smashing herself with sex and drugs but the song began to find its way even there. One day, Benny played the opening chord and she just couldn’t do it. She could not sing that fucking song. Not again. She broke down, because she was only fifteen, and Benny took her behind the mall and told her it would be okay. But she had to sing. It was the biggest earner. She kind of lost it and then so did Benny and that was the first time he hit her. She ran away for a while. But she came back to him, because she had nothing else, and it seemed okay. It seemed like they had a truce: She would not complain about her bruised face and he would not ask her to sing “Lucy.” She had been all right with this. She had thought that was a pretty good deal.

Now there was something coming out of a box, and she reached for the most virulent meme she knew. “Lucy in the sky!” she sang. “With diamonds!”

•   •   •

Barry, Max. Lexicon: A Novel  (pp. 247-248).
Penguin Publishing Group. Kindle Edition.

Related material from Log24 on All Hallows' Eve 2013

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Tuesday, March 27, 2012

Finnegans Kaleidoscope

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

IMAGE- Philip Kitcher and David Albert read Finnegans Wake

In appreciation of their essays in last
Sunday’s New York Times Book Review ,
a link for David Albert and Philip Kitcher

Finnegans Kaleidoscope.

Sunday, September 2, 2018

Picture Yourself …

Filed under: General — Tags: , — m759 @ 10:12 PM

http://www.log24.com/log/pix18/180902-Bird_Box-boat.jpg

http://www.log24.com/log/pix18/180902-Kaleidoscope_Eyes-post-180806.jpg

Saturday, June 16, 2018

For June 16

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

"But perhaps the desire for story
is what gets us into trouble to begin with."

Sarah Marshall on June 5, 2018

"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence  is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."

— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018

Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce

Tuesday, May 8, 2018

Wall

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

The glitter-ball-like image discussed in the previous post
is of an artwork by Olafur Eliasson.

See the kaleidoscopic  section of his website.

From that section —

Eliasson, 'When Love Is Not Enough' wall, 2007

Related art in keeping with the theme of last night's Met Gala —

See also my 2005 webpage Kaleidoscope Puzzle.

Saturday, April 7, 2018

Word and Image: Backstory for James Spader

Filed under: General,Geometry — m759 @ 2:14 PM

Actor James Spader in a 2014 interview

". . . my father taught English. My mother taught art . . . ."

Detail of part of a text by Magritte (1929) that appeared  
without attribution in the online New York Times  today —

See also, from a search for the phrase "Word and Image"
in this  journal —

The Philosophers' Stone as originally
illustrated in The New York Times 

http://www.log24.com/log/pix10A/100518-TheStoneNYT.jpg .

Related images —

Juliette Binoche in 'Blue'  The
 24 2x2 Cullinane Kaleidoscope animated images

See as well a Log24 search for "Philosophers' Stone"
and remarks related to the Magritte pictures above 
in the post Story of March 13, 2014.

Thursday, December 28, 2017

Memorandum of Misunderstanding

Filed under: General — m759 @ 11:40 AM

Harrison Ford in "Blade Runner 2049" —

Click the above quote for a scholium.

See also the previous post and . . .

Tuesday, October 21, 2014

Tools

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

(Night at the Museum continues.)

"Strategies for making or acquiring tools

While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"

— Jamie Cohen-Cole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science  vol. 40, no. 4 (2007): 567-597.

Obituary of a co-founder, in 1960, of the Center for Cognitive Studies at Harvard:

"Disciplinary Exchange" —

In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —

The Kaleidoscope Puzzle,  The Diamond 16 Puzzle
The 2x2x2 Cube, and The 4x4x4 Cube

"Intellectual Economies" —

In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —

This illustrates not geometry, but rather creative capitalism.

Related material from the date of the above Harvard death:  Art Wars.

Friday, February 28, 2014

Code

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

From Northrop Frye's The Great Code: The Bible and Literature , Ch. 3: Metaphor I —

"In the preceding chapter we considered words in sequence, where they form narratives and provide the basis for a literary theory of myth. Reading words in sequence, however, is the first of two critical operations. Once a verbal structure is read, and reread often enough to be possessed, it 'freezes.' It turns into a unity in which all parts exist at once, without regard to the specific movement of the narrative. We may compare it to the study of a music score, where we can turn to any part without regard to sequential performance. The term 'structure,' which we have used so often, is a metaphor from architecture, and may be misleading when we are speaking of narrative, which is not a simultaneous structure but a movement in time. The term 'structure' comes into its proper context in the second stage, which is where all discussion of 'spatial form' and kindred critical topics take their origin."

Related material: 

"The Great Code does not end with a triumphant conclusion or the apocalypse that readers may feel is owed them or even with a clear summary of Frye’s position, but instead trails off with a series of verbal winks and nudges. This is not so great a fault as it would be in another book, because long before this it has been obvious that the forward motion of Frye’s exposition was illusory, and that in fact the book was devoted to a constant re-examination of the same basic data from various closely related perspectives: in short, the method of the kaleidoscope. Each shake of the machine produces a new symmetry, each symmetry as beautiful as the last, and none of them in any sense exclusive of the others. And there is always room for one more shake."

— Charles Wheeler, "Professor Frye and the Bible," South Atlantic Quarterly  82 (Spring 1983), pp. 154-164, reprinted in a collection of reviews of the book.
 

For code  in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.

For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
Kaleidoscope Puzzle.

Tuesday, November 12, 2013

Soundtrack

Filed under: General,Geometry — Tags: , — m759 @ 6:45 AM

IMAGE- 'Devil Music' from 'Kaleidoscopes- Selected Writings of H.S.M. Coxeter'

"DEVIL – MUSIC

20 pages of incidental music written at school
for G. K. Chesterton's play MAGIC

by D. Coxeter."

See also

Related material —  Chesterton + Magic in this journal.

Wednesday, November 6, 2013

Bullshit Studies

Filed under: General — m759 @ 3:12 PM

The essay excerpted in last night's post on structuralism
is of value as part of a sustained attack by the late
Robert de Marrais on the damned nonsense of the late
French literary theorist Jacques Derrida—

Catastrophes, Kaleidoscopes, String Quartets:
Deploying the Glass Bead Game

Part I:  Ministrations Concerning Silliness, or:
Is “Interdisciplinary Thought” an Oxymoron?

Part II:  Canonical Collage-oscopes, or:
Claude in Jacques’ Trap?  Not What It Sounds Like!

Part III:  Grooving on the Sly with Klein Groups

Part IV:  Claude’s Kaleidoscope . . . and Carl’s

Part V:  Spelling the Tree, from Aleph to Tav
(While  Not Forgetting to Shin)

The response of de Marrais to Derrida's oeuvre  nicely
exemplifies the maxim of Norman Mailer that

"At times, bullshit can only be countered
with superior bullshit."

Thursday, October 31, 2013

Interpenetrative Ogdoad

Filed under: General,Geometry — m759 @ 2:21 PM

The title is from an essay by James C. Nohrnberg

(Click to enlarge.)

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Tuesday, October 1, 2013

Frame Tale

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

From an academic's website:

IMAGE- Remarks by Paul Hertz, alias Ignotus the Mage

For Josefine Lyche and Ignotus the Mage,
as well as Rose the Hat and other Zingari shoolerim —

Sabbatha hanti, lodsam hanti, cahanna risone hanti :
words that had been old when the True Knot moved
across Europe in wagons, selling peat turves and trinkets.
They had probably been old when Babylon was young.
The girl was powerful, but the True was all-powerful,
and Rose anticipated no real problem.

— King, Stephen (2013-09-24).
     Doctor Sleep: A Novel
     (pp. 278-279). Scribner. Kindle Edition. 

From a post of November 10, 2008:

Twenty-four Variations on a Theme of Plato

Twenty-four Variations on a Theme of Plato,
a version by Barry Sharples based on the earlier
kaleidoscope puzzle  version of Steven H. Cullinane

The King and the Corpse  —

"The king asked, in compensation for his toils
during this strangest of all the nights he had
ever known, that the twenty-four riddle tales
told him by the specter, together with the story
of the night itself, should be made known
over the whole earth and remain eternally
famous among men."

Frame Tale: 

Finnegans Wake  —

"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 PM

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
[Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Thursday, July 5, 2012

Claves Regni Caelorum

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

Or: Night of Lunacy

From 9 PM Monday

IMAGE- Page 304 of Heidegger's 'Existence and Being' - Heidegger's essay 'Hölderlin and the Essence of Poetry,' tr. by Douglas Scott, publ. by Henry Regnery Company, Chicago, in 1949
Note that the last line, together with the page number, forms
a sort of key

The rest  of the story—

IMAGE- Heidegger quote continued, ending with reference to Hölderlin's 'night of lunacy'

For one reinterpretation of the page number 304, see a link— 
Sermon— from Tuesday's post Diamond Speech.

The linked-to sermon itself has a link, based on a rereading
of 304 as 3/04, to a post of March 4, 2004, with…

WW and ZZ

as rendered by figures from the Kaleidoscope Puzzle

     .

Yesterday morning the same letter-combinations occurred
in a presentation at CERN of a newly discovered particle—

IMAGE- 'High mass: WW, ZZ'

(Click for context.)

Since the particle under discussion may turn out to be the
God  particle, it seems fitting to interpret WW and ZZ as part
of an imagined requiem  High Mass.

Ron Howard, director of a film about CERN and the God particle,
may regard this imaginary Mass as performed for the late
Andy Griffith, who played Howard's father in a television series.

Others may prefer to regard the imaginary Mass as performed 
for the late John E. Brooks, S. J., who served as president of
The College of the Holy Cross, Worcester, Mass., for 24 years.

Griffith died Tuesday. Brooks died Monday.

For some background on the Holy Cross, see posts of
Sept. 14 (Holy Cross Day) and Sept. 15, 2010—

  1. Language Game,
  2. Wittgenstein, 1935, and
  3. Holy Cross Day Revisited.

For more lunacy, see…

Continue a search for thirty-three and three
— Katherine Neville, The Eight

Sunday, September 18, 2011

What Rough Beast

Filed under: General — m759 @ 9:00 PM

Lurching Toward Decision

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

"Suskind… nails, I think, Obama's intellectual blind spot. Indeed, Obama himself nails it, telling Suskind that he was too inclined to search for 'the perfect technical answer' to the myriad of complex issues coming at him."

Frank Rich on Ron Suskind's new book about the White House, Confidence Men

Very distantly related material—

From "Confidence Game," an Oct. 12, 2008, post in this journal, a quasi-European perspective—

Juliette Binoche in 'Blue'  Animated 2x2 kaleidoscope figures from Diamond Theory

Kaleidoscope turning…
Shifting pattern
within unalterable structure…

– Roger Zelazny, Eye of Cat   

See also …

Gravity’s Rainbow , Penguin Classics, 1995, page 742:

"… knowing his Tarot, we would expect to look among the Humility, among the gray and preterite souls, to look for him adrift in the hostile light of the sky, the darkness of the sea….

Now there’s only a long cat’s-eye of bleak sunset left over the plain tonight, bright gray against a purple ceiling of clouds, with an iris of

   742"

Wednesday, April 6, 2011

De Marrais Memorial

Filed under: General,Geometry — m759 @ 7:11 PM

In memory of Robert de Marrais, an excerpt from an obituary at Legacy.com—

(Click to enlarge.)

http://www.log24.com/log/pix11/110406-deMarraisObit-Sm.jpg

Robert “Bob” Paul de Marrais died April 4, 2011 in Boston, Mass. One measure of a life is those that grieve our absence. Bob is dearly missed. He is survived by his 92 year old mother Yvette (nee Pétronille) in NY, his brother John A. in NY, his Aunt Mae in NJ; three children Luc, Sylvie, and Nathalie in Mass, and his devoted wife Dali (nee Zangurashvili) from Georgia of the ex-Soviet-Union. Bob was born Nov. 30, 1948, grew up in Cresskill, NJ, made life-long friends during some of his happiest days at MIT in Mass., and did not wander far from there for the rest of his life. He had a lifelong interest in history, his French heritage, music, history of science, and multidimensional algebras. His wife, friends Izzy and Mitch, brother John (and wife Caroline), little nephew Louis J., and two of his own children got to say goodbye. He found the energy to reward us with a smile. Bob has now joined his loving dad Louis J., Uncle Jack, Aunt Ginny, Uncle Gil, et. al.

For some details of de Marrais's life, see a separate biography from Legacy.com.

Related material—  A search for "deMarrais" in this journal. (The name often occurs only within links.)
Cached copies of the 5-part "Kaleidoscopes" work by de Marrais referred to in the search can be found here.

A more personal note, from a quotation linked to here on the date of de Marrais's death

… and who shall ever tell the sorrow of being on this earth,
lying, on quilts, on the grass, in a summer evening, among the sounds of the night.

May God bless my people, my uncle, my aunt, my mother, my good father,
oh, remember them kindly in their time of trouble;
and in the hour of their taking away.

After a little I am taken in and put to bed.

— James Agee, "Knoxville: Summer of 1915"

Wednesday, January 26, 2011

Early Nothing

Filed under: General,Geometry — Tags: — m759 @ 11:29 AM

(Continued from yesterday)

Today's New York Times  obituaries —

http://www.log24.com/log/pix11/110126-BruceGordon.jpg

From Wes Clark's site Web Noir

Scenes from "The Set-Up," a 1949 noir classic by Robert Wise

http://www.log24.com/log/pix11/110126-ParadiseCity.jpg

From Bruce Gordon's obituary in today's New York Times

"Mr. Gordon appeared on Broadway many times. He was in the original cast of the hit comedy 'Arsenic and Old Lace,' which opened in 1941 and starred Boris Karloff. Uncharacteristically, given his later résumé, Mr. Gordon played a policeman." —Margalit Fox

Related material —

http://www.log24.com/log/pix10B/101221-BrewsterSociety.jpg

(See Savage Solstice in this journal on December 21st, 2010.)

Tuesday, December 21, 2010

Savage Solstice

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

In memory of kaleidoscope enthusiast Cozy Baker, who died at 86, according to Saturday's Washington Post , on October 19th.

http://www.log24.com/log/pix10B/101221-BrewsterSociety.jpg

This journal on that date — Savage Logic and Savage Logic continued.

See this journal on All Saints' Day 2006 for some background to those posts—

“Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought. Both anecdotal and geometric, it builds coherent structures out of ‘the odds and ends left over from psychological or historical process.’

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore. (How, precisely, they have come into being in the first place is one of the points on which Levi-Strauss is not too explicit, referring to them vaguely as the ‘residue of events… fossil remains of the history of an individual or a society.’) Such images are inevitably embodied in larger structures– in myths, ceremonies, folk taxonomies, and so on– for, as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular. But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort. Quoting Franz Boas that ‘it would seem that mythological worlds have been built up, only to be shattered again, and that new worlds were built from the fragments,’ Levi-Strauss generalizes this permutational view of thinking to savage thought in general.”

– Clifford Geertz, “The Cerebral Savage: the Structural Anthropology of Claude Levi-Strauss,” in Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.

Related material  —

http://www.log24.com/log/pix10B/101221-TristesTropiques.jpg

See also "Levi-Strauss" in this journal and "At Play in the Field."

Thursday, October 21, 2010

St. Ursula’s Day

Filed under: General,Geometry — Tags: — m759 @ 4:07 PM

Mathematics and Narrative continued

A search for Ursula in this journal yields a story…

“The main character is a slave woman who discovers new patterns in the mosaics.”

Other such stories: Plato’s Meno  and Changing Woman

Changing Woman:

Kaleidoscope turning…

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Shifting pattern within
unalterable structure…”

— Roger Zelazny, Eye of Cat  

Philosophical postscript—

“That Lévi-Strauss should have been able to transmute the romantic passion of Tristes Tropiques  into the hypermodern intellectualism of La Pensée Sauvage  is surely a startling achievement. But there remain the questions one cannot help but ask. Is this transmutation science or alchemy? Is the ‘very simple transformation’ which produced a general theory out of a personal disappointment real or a sleight of hand? Is it a genuine demolition of the walls which seem to separate mind from mind by showing that the walls are surface structures only, or is it an elaborately disguised evasion necessitated by a failure to breach them when they were directly encountered? Is Lévi-Strauss writing, as he seems to be claiming in the confident pages of La Pensée Sauvage,  a prolegomenon to all future anthropology? Or is he, like some uprooted neolithic intelligence cast away on a reservation, shuffling the debris of old traditions in a vain attempt to revivify a primitive faith whose moral beauty is still apparent but from which both relevance and credibility have long since departed?”

— Clifford Geertz, conclusion of “The Cerebral Savage: On the Work of Claude Lévi-Strauss

Tuesday, October 19, 2010

Savage Logic continued…

Filed under: General,Geometry — m759 @ 9:36 AM

CHAPTER V

THE KALEIDOSCOPE

"This is an account of the discrete groups generated by reflections…."

Regular Polytopes , by H.S.M. Coxeter (unabridged and corrected 1973 Dover reprint of the 1963 Macmillan second edition)

"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields…. Let V  be an n-dimensional vector space over a field F, and let G ≤ Gln (F) be a finite group…. An element of finite order in Gl(V ) is a reflection if its fixed point space in V  is a hyperplane, called the reflecting hyperplane. There are two types of reflections: the diagonalizable reflections in Gl(V ) have a single nonidentity eigenvalue which is a root of unity; the nondiagonalizable reflections in Gl(V ) are called transvections and have determinant 1 (note that they can only occur if the characteristic of F is positive)…. A reflection group is a finite group G  generated by reflections."

— Julia Hartmann and Anne V. Shepler, "Reflection Groups and Differential Forms," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955-971

"… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field. The reflections in Gl(V ) not only include diagonalizable reflections (with a single nonidentity eigenvalue), but also transvections, reflections with determinant 1 which can not be diagonalized. The transvections in Gl(V ) prevent one from developing a theory of reflection groups mirroring that for Coxeter groups or complex reflection groups."

— Julia Hartmann and Anne V. Shepler, "Jacobians of Reflection Groups," Transactions of the American Mathematical Society , Vol. 360, No. 1 (2008), pp. 123-133 (Pdf available at CiteSeer.)

See also A Simple Reflection Group of Order 168 and this morning's Savage Logic.

Savage Logic…

Filed under: General,Geometry — Tags: — m759 @ 2:22 AM

and the New York Lottery

IMAGE-- NY Lottery Oct. 18, 2010-- Midday 069, Evening 359

A search in this journal for yesterday's evening number in the New York Lottery, 359, leads to…

The Cerebral Savage: 
On the Work of Claude Lévi-Strauss

by Clifford Geertz

Shown below is 359, the final page of Chapter 13 in
The Interpretation of Cultures: Selected Essays by Clifford Geertz,
New York, 1973: Basic Books, pp. 345-359 —

http://www.log24.com/log/pix10B/101019-Geertz359.gif

This page number 359 also appears in this journal in an excerpt from Dan Brown's novel Angels & Demons

See this journal's entries for March 1-15, 2009, especially…

Sunday, March 15, 2009  5:24 PM

Philosophy and Poetry:

The Origin of Change

A note on the figure
from this morning's sermon:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

Sunday, March 15, 2009  11:00 AM

Ides of March Sermon:

Angels, Demons,
"Symbology"

"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…

'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:

"Perfer et obdura: multo graviora tulisti."

 "Hold out and persist:
  you have got through
  far more difficult situations."

 (Tristia, Liber  V, Elegia  XI, verse 7).'"
 

This journal
on 9 March:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Note the color-interchange
symmetry
of each symbol
under 180-degree rotation.

Related material:
The Illuminati Diamond:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

The symmetry of the yin-yang symbol, of the diamond-theorem symbol, and of Brown's Illuminati Diamond is also apparent in yesterday's midday New York lottery number (see above).

"Savage logic works like a kaleidoscope…." — Clifford Geertz on Lévi-Strauss

Monday, August 23, 2010

Diamond Puzzle Downloads

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

The Diamond 16 Puzzle and the Kaleidoscope Puzzle can now be downloaded in the normal way from a browser, with the save-as web-page-complete option, and have their JavaScript still work— if  the files are saved with the name indicated in the instructions on the puzzles' web pages. (There was a problem with file names in the JavaScript that has been fixed.)

The JavaScript pages Design Cube 2x2x2 and Design Cube 4x4x4 have not been changed. To download these, it is necessary to…

  1. Do a web-page-complete save to get an image-files folder, then
  2. do an HTML-only save to the image-files folder  to put an unaltered copy of the the web page there, then
  3. rename the image-files folder to unlink it from the altered HTML page downloaded in step 1, then
  4. delete the altered HTML page downloaded in step 1.

The result is a folder containing both image files and the HTML page, just as it is on the Web.

Tuesday, May 18, 2010

Stone Junction*

Filed under: General,Geometry — m759 @ 8:00 AM

The Philosophers' Stone
according to
  The New York Times

http://www.log24.com/log/pix10A/100518-TheStoneNYT.jpg

Related material
from French cinema—

"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa."

See also

   Word and Image

Juliette Binoche in 'Blue'  The
 24 2x2 Cullinane Kaleidoscope animated images

* The title is a reference to Jim Dodge's 1989 novel Stone Junction: An Alchemical Potboiler.

Friday, September 11, 2009

Friday September 11, 2009

Filed under: General,Geometry — Tags: — m759 @ 2:56 AM
Theology
 in memory of
physicist Aage Bohr,
who died at 87 on
Tuesday, Sept. 8, 2009

Swarthmore physics honors thesis, 136 pp., 2007–

Abstract:

"Quantum mechanics, which has no completely accepted interpretation but many seemingly strange physical results, has been interpreted in a number of bizarre and fascinating ways over the years. The two interpretations examined in this paper, [Aage] Bohr and [Ole] Ulfbeck's 'Genuine Fortuitousness' and Stuckey, Silberstein, and Cifone's 'Relational Blockworld,' seem to be two such strange interpretations; Genuine Fortuitousness posits that causality is not fundamental to the universe, and Relational Blockworld suggests that time does not act as we perceive it to act. In this paper, I analyze these two interpretations…."

Footnote 55, page 114:
 
"Thus far, I have been speaking in fairly abstract terms, which can sometimes be unhelpful on the issue of explaining anything about the structure of space-time. I want to pause for a moment to suggest a new potential view of the blockworld within a 'genuinely fortuitous' universe in more visual terms. Instead of the 'static spacetime jewel' of blockworld that is often invoked by eternalists to help their readers conceptualize of what a blockworld would 'look like' from the outside, now imagine that a picture on a slide is being projected onto the surface of this space-time jewel."

Interpolated figure
from Log24:

 

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Cf. August 5, 2009.


From the perspective of one inside the jewel, one might ask 'Why is this section blue while this section is black?,' and from within the jewel, one could not formulate an answer since one could not see the entire picture projected on the jewel; however, from outside the jewel, an observer (some analogue of Newton's God, perhaps, looking down on his 'sensorium' from the 5th dimension) could easily see the pattern and understand that all of the 'genuinely fortuitous' events inside the space-time jewel are, in fact, completely determined by the pattern in the projector."

— "Genuine Fortuitousness, Relational Blockworld, Realism, and Time" (pdf), by Daniel J. Peterson, Honors Thesis, Swarthmore College, December 13, 2007

Wednesday, August 5, 2009

Wednesday August 5, 2009

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

 

Word and Image

NYT obituary summaries for Charles Gwathmey and Edward Hall, morning of Aug. 5, 2009

From Hall's obituary
:

"Edward T. Hall, a cultural anthropologist
who pioneered the study of nonverbal
 communication and interactions between
members of different ethnic groups,
 died July 20 at his home in
 Santa Fe, N.M. He was 95."

NY Times piece quoted here on
 the date of Hall's death:
 

"July 20, 1969, was the moment NASA needed, more than anything else in this world, the Word. But that was something NASA's engineers had no specifications for. At this moment, that remains the only solution to recovering NASA's true destiny, which is, of course, to build that bridge to the stars."

— Tom Wolfe, author of The Right Stuff, an account of the Mercury Seven astronauts.

Commentary
The Word according to St. John:

Jill St. John, star of 'Diamonds are Forever'

 

From Hall's obituary:

"Mr. Hall first became interested in
space and time as forms of cultural
 expression while working on
Navajo and Hopi reservations
 in the 1930s."

Log24, July 29
:

Changing Woman:

"Kaleidoscope turning…

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Shifting pattern within   
unalterable structure…"
— Roger Zelazny,  
Eye of Cat  

"We are the key."
Eye of Cat  

Update of about 4:45 PM 8/5:

Paul Newall, "Kieślowski's Three Colours Trilogy"

"Julie recognises the music of the busker outside playing a recorder as that of her husband's. When she asks him where he heard it, he replies that he makes up all sorts of things. This is an instance of a theory of Kieślowski's that 'different people, in different places, are thinking the same thing but for different reasons.' With regard to music in particular, he held what might be characterised as a Platonic view according to which notes pre-exist and are picked out and assembled by people. That these can accord with one another is a sign of what connects people, or so he believed."

The above photo of Juliette Binoche in Blue accompanying the quotations from Zelazny illustrates Kieślowski's concept, with graphic designs instead of musical notes. Some of the same designs are discussed in Abstraction and the Holocaust (Mark Godfrey, Yale University Press, 2007). (See the Log24 entries of June 11, 2009.)

Related material:

"Jeffrey Overstreet, in his book Through a Screen Darkly, comments extensively on Blue. He says these stones 'are like strands of suspended crystalline tears, pieces of sharp-edged grief that Julie has not been able to express.'….

Throughout the film the color blue crops up, highlighting the mood of Julie's grief. A blue light occurs frequently, when Julie is caught by some fleeting memory. Accompanied by strains of an orchestral composition, possibly her husband's, these blue screen shots hold for several seconds while Julie is clearly processing something. The meaning of this blue light is unexplained. For Overstreet, it is the spirit of reunification of broken things."

Martin Baggs at Mosaic Movie Connect Group on Sunday, March 15, 2009. (Cf. Log24 on that date.)

For such a spirit, compare Binoche's blue mobile in Blue with Binoche's gathered shards in Bee Season.

Wednesday, July 29, 2009

Wednesday July 29, 2009

Filed under: General,Geometry — m759 @ 12:21 PM
Kaleideion

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

Related material:

“A great deal has been made of the fact that Forbidden Planet is essentially William Shakespeare’s The Tempest (1611) in an science-fiction setting. It is this that transforms Forbidden Planet into far more than a mere pulp science-fiction story” — Richard Scheib

Dialogue from Forbidden Planet


“… Which makes it a gilt-edged priority that one of us gets into that Krell lab and takes that brain boost.”

Dialogue from another story —

“They thought they were doing a linear magnification, sort of putting me through a  magnifying glass.”

“Sizewise?”

“Brainwise, but what they did was multiply me by myself into a quadratic.”

Psychoshop, by Bester and Zelazny, 1998 paperback, p. 7

“… which would produce a special being– by means of that ‘cloned quadratic crap.’ [P. 75] The proper term sounds something like ‘Kaleideion‘….”

“So Adam is a Kaleideion?”

She shook her head.

“Not a Kaleideion. The Kaleideion….”

Psychoshop, 1998 paperback, p. 85


See also

Changing Woman:

Kaleidoscope turning…

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Shifting pattern within   
unalterable structure…”
— Roger Zelazny, Eye of Cat  

“When life itself seems lunatic,
who knows where madness lies?”

— For the source, see 
Joyce’s Nightmare Continues.

Monday, November 10, 2008

Monday November 10, 2008

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

Frame Tales

From June 30

("Will this be on the test?")

Frame Tale One:

Summer Reading

The King and the Corpse: Tales of the Soul's Conquest of Evil

Subtitle:
Tales of the Soul's
Conquest of Evil

Frame Tale Two:

Barry Sharples
on his version of the
  Kaleidoscope Puzzle

Background:

"A possible origin of this puzzle is found in a dialogue
 between Socrates and Meno written by the Greek philosopher,
 Plato, where a square is drawn inside
a square such that
the blue square is twice the area  of the yellow square.

Plato's Diamond

Colouring the triangles produces a starting pattern
which is a one-diamond figure made up of four tiles
and there are 24 different possible arrangements."

Twenty-four Variations on a Theme of Plato

The King and the Corpse  —

"The king asked, in compensation for his toils during this strangest
of all the nights he had ever known, that the twenty-four riddle tales
told him by the specter, together with the story of the night itself,
should be made known over the whole earth
and remain eternally famous among men."

Frame Tale Three:

Finnegans Wake

"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."

Friday, November 7, 2008

Friday November 7, 2008

Filed under: General,Geometry — m759 @ 2:22 PM
The Sincerest Form
of Flattery

At a British puzzle website today I found this, titled “Tiles Puzzle by Steven H. Cullinane”–

http://www.log24.com/log/pix08A/081107-Tilespuzzle.jpg

The version there states that

“there are 322,560 patterns made by swapping rows, swapping columns and swapping the four 2×2 quadrants!”

Actually, only 840 patterns can be made in this version. These are formed by 322,560 permitted permutations of the 16 tiles. This is also true in my Kaleidoscope Puzzle. For a display of all 322,560 permutations as pairs of (orthogonal) patterns, see the Diamond 16 Puzzle.

Update of Nov. 10, 2008: The error has been corrected.

Sunday, October 12, 2008

Sunday October 12, 2008

Filed under: General,Geometry — m759 @ 3:28 PM
Confidence Game
 
Paul Newman and Robert Redford in 'The Sting'

The Winners:

European leaders in Paris agree on plan to aid banks

Related material:
Dec. 16, 2003

Moulin Bleu

Juliette Binoche in 'Blue'  Animated 2x2 kaleidoscope figures from Diamond Theory

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat   

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
Wechsler Cubes

 "Confusion is nothing new."
— Song lyric, Cyndi Lauper  

Part I:
Magister Ludi

Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–

"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."

Part II:
A Bulky Memorandum

From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–

A description of the work of Bill Wechsler, a fictional artist:

"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has

— End of page 168 —

opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.

The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."

Part III:
Wechsler Cubes

(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
   David Wechsler) —

From 2002:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.


Part IV:
A Magic Gallery
 
Log24, March 4, 2004
 

ZZ
WW

Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:


Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte

Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen


 Numbers and Names,
Wording and Words


With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes


English translation
by Catherine Schelbert



A related poem:

Alphabets
by Hermann Hesse

From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone's command;
it is a game whose rules
are nice and neat.

But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens'
feet upon the snow.
He'd hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He'd see the way love burns
and anguish trembles,
He'd wonder, laugh,
shake with fear and weep
because beyond this cipher's
cross-barred keep
he'd see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
prisoner-fashion.
He'd think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He'll sigh, and smile,
and feel all right again.

— Hermann Hesse (1943),
"Buchstaben," from
Das Glasperlenspiel,
translated by
Richard and Clara Winston

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 AM
MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Thursday, May 8, 2008

Thursday May 8, 2008

Filed under: General — m759 @ 4:48 PM
Synchronicity,
Part Deux

Footprints at Log24 on the afternoon of May 8, 2008, including two from France

From
“On the Holy Trinity,”
the entry in the 3:20 PM
French footprint:

“…while the scientist sees
everything that happens
in one point of space,
the poet feels
everything that happens
in one point of time…
all forming an
instantaneous and transparent
organism of events….”

Vladimir Nabokov

From
“Angel in the Details,”
 the entry in the 3:59 PM
French footprint:

“I dwell in Possibility –
A fairer House than Prose”

Emily Dickinson

These, along with this afternoon’s
earlier entry, suggest a review
of a third Log24 item, Windmills,
with an actress from France as…

Changing Woman:

Kaleidoscope turning…

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Shifting pattern
within unalterable structure…”
— Roger Zelazny, Eye of Cat  

“When life itself seems lunatic,
who knows where madness lies?”

— For the source, see 
Joyce’s Nightmare Continues.

Thursday, March 6, 2008

Thursday March 6, 2008

Filed under: General,Geometry — m759 @ 12:00 PM
This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA’s Icosahedron Society.

Royal Road

“The historical road
from the Platonic solids
to the finite simple groups
is well known.”

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that “there is no royal road to geometry.” The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, “Map-Colouring Problems,” in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. “the king of geometry.”

See also Log24, May 21, 2007.

A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M24.

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”

— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M24):

“Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code.” [Here I is the identity matrix and J is the matrix of all 1’s.]

Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube

and
Jewel in the Crown

“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?'”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
“story theory” of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.


* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Thursday, June 14, 2007

Thursday June 14, 2007

Filed under: General,Geometry — m759 @ 1:06 AM
Scholarly Notes

In memory of
Rudolf Arnheim,
who died on
Saturday, June 9

“Originally trained in Gestalt psychology, with its emphasis on the perception of forms as organized wholes, he was one of the first investigators to apply its principles to the study of art of all kinds.” —Today’s New York Times

From the Wikipedia article on Gestalt psychology prior to its modification on May 31, 2007:

“Emergence, reification, multistability, and invariance are not separable modules to be modeled individually, but they are different aspects of a single unified dynamic mechanism.

For a mathematical example of such a mechanism using the cubes of psychologists’ block design tests, see Block Designs in Art and Mathematics and The Kaleidoscope Puzzle.”

The second paragraph of the above passage refers to my own work.

Some Gestalt-related work of Arnheim:

The image “http://www.log24.com/log/pix07/070614-Arnheim.gif” cannot be displayed, because it contains errors.

—  From p. 242 of
  “Perceptual Analysis of a
  Symbol of Interaction,”
  pp. 222-244 in
  Toward a Psychology of Art:
  Collected Essays
,
  Univ. of Calif. Press, 1966

Time of this entry:
1:06:18 AM ET.

Thursday, May 31, 2007

Thursday May 31, 2007

Filed under: General,Geometry — m759 @ 8:06 PM

Blitz by anonymous
New Delhi user

From Wikipedia on 31 May, 2007:

Shown below is a list of 25 alterations to Wikipedia math articles made today by user 122.163.102.246.

All of the alterations involve removal of links placed by user Cullinane (myself).

The 122.163… IP address is from an internet service provider in New Delhi, India.

The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews. (See User talk: Cullinane.)

Related material:

Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163…
at Talk: Four color theorem.

May 31, 2007, alterations by
user 122.163.102.246:

  1. 17:17 Orthogonality (rm spam)
  2. 17:16 Symmetry group (rm spam)
  3. 17:14 Boolean algebra (rm spam)
  4. 17:12 Permutation (rm spam)
  5. 17:10 Boolean logic (rm spam)
  6. 17:08 Gestalt psychology (rm spam)
  7. 17:05 Tesseract (rm spam)
  8. 17:02 Square (geometry) (rm spam)
  9. 17:00 Fano plane (rm spam)
  10. 16:55 Binary Golay code (rm spam)
  11. 16:53 Finite group (rm spam)
  12. 16:52 Quaternion group (rm spam)
  13. 16:50 Logical connective (rm spam)
  14. 16:48 Mathieu group (rm spam)
  15. 16:45 Tutte–Coxeter graph (rm spam)
  16. 16:42 Steiner system (rm spam)
  17. 16:40 Kaleidoscope (rm spam)
  18. 16:38 Efforts to Create A Glass Bead Game (rm spam)
  19. 16:36 Block design (rm spam)
  20. 16:35 Walsh function (rm spam)
  21. 16:24 Latin square (rm spam)
  22. 16:21 Finite geometry (rm spam)
  23. 16:17 PSL(2,7) (rm spam)
  24. 16:14 Translation plane (rm spam)
  25. 16:13 Block design test (rm spam)

The deletions should please Charles Matthews and fans of Ashay Dharwadker’s work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.

There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.

Cullinane 23:28, 31 May 2007 (UTC)

Saturday, March 24, 2007

Saturday March 24, 2007

Filed under: General,Geometry — m759 @ 2:09 AM
Savage Scrutiny

“They sang desiring an object that was near,
In face of which desire no longer moved,
Nor made of itself that which it could not find…
Three times the concentred self takes hold, three times
The thrice concentred self, having possessed

The object, grips it in savage scrutiny,
Once to make captive, once to subjugate
Or yield to subjugation, once to proclaim
The meaning of the capture, this hard prize,
Fully made, fully apparent, fully found.”

— “Credences of Summer,” VII,
    by Wallace Stevens, from
    Transport to Summer (1947)

Clifford Geertz on Levi-Strauss, from The Cerebral Savage:

“Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns…. “

Related material:

The kaleidoscope puzzle and “Claude Levi-Strauss and the Aesthetic Object,” a videotaped interview with Dr. Boris Wiseman.

Wednesday, November 1, 2006

Wednesday November 1, 2006

Filed under: General,Geometry — Tags: — m759 @ 8:24 AM

The image “http://www.log24.com/log/pix06B/061101-Geertz2.jpg” cannot be displayed, because it contains errors.

Clifford Geertz 

Professor Emeritus,
Institute for Advanced Study

Savage Logic

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought. Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore. (How, precisely, they have come into being in the first place is one of the points on which Levi-Strauss is not too explicit, referring to them vaguely as the 'residue of events… fossil remains of the history of an individual or a society.') Such images are inevitably embodied in larger structures– in myths, ceremonies, folk taxonomies, and so on– for, as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular. But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort. Quoting Franz Boas that 'it would seem that mythological worlds have been built up, only to be shattered again, and that new worlds were built from the fragments,' Levi-Strauss generalizes this permutational view of thinking to savage thought in general."

— Clifford Geertz, "The Cerebral Savage: the Structural Anthropology of Claude Levi-Strauss," in Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.

Today's New York Times
reports that
Geertz died on Monday,
October 30, 2006.

Related material:

Kaleidoscope Puzzle,

Being Pascal Sauvage,

and Up the River:
 

 

 
The Necessity For Story

by Frederick Zackel

While it's a story that's never been written, a suggested title– Indiana Jones Sails Up The River Of Death–

The image “http://www.log24.com/log/pix04B/041016-Poster2.jpg” cannot be displayed, because it contains errors.

shows how readily we as individuals or we as a culture can automatically visualize a basic story motif. We may each see the particular elements of the story differently, but almost instantaneously we catch its drift.

The hero sails up the river of death to discover what lies within his own heart: i.e., how much moral and physical strength he has.

Indiana Jones sails up the River of Death.

We are following Indiana Jones up the River of Death. We're going to visit with Colonel Kurtz. (You may not want to get off the boat.)

No, I am not mixing up metaphors.

These are the Story.

 

Amen.

Wednesday, May 10, 2006

Wednesday May 10, 2006

Filed under: General,Geometry — m759 @ 4:29 PM
My Space

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“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”

— Robert A. Heinlein,
The Number of the Beast

The image “http://www.log24.com/theory/images/Screenshot-64cube.gif” cannot be displayed, because it contains errors.

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

And so…

Compare and contrast:

Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.

Related material:

A more extensive excerpt from
The Number of the Beast, and

Story Theory and
the Number of the Beast.

Tuesday, November 15, 2005

Tuesday November 15, 2005

Filed under: General,Geometry — m759 @ 11:07 AM
Windmills
 
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Upper part of above picture–

From today’s New York Times,
Seeing Mountains in
Starry Clouds of Creation.

Lower part of above picture–
Pilgrimage to Spider Rock:

“This magical place, according to Navajo Legend, was the home of Spider Woman, who gave the gift of weaving to the Dineh’ People.  Today’s Navajos trace the excellence of their finest textiles to this time of legends, when their patron, Changing Woman, met Spider Woman, the first Weaver.”

Vine Deloria Jr.,
 
Evolution, Creationism,
and Other Modern Myths:

“The continuing struggle between evolutionists and creationists, a hot political topic for the past four decades, took a new turn in the summer of 1999 when the Kansas Board of Education voted to omit the mention of evolution in its newly approved curriculum, setting off outraged cries of foul by the scientific establishment.  Don Quixotes on both sides mounted their chargers and went searching for windmills.”

Related material–

A figure from
last night’s entry,
Spider Woman:

Fritz Leiber's 'Spider' symbol

From Sunday, the day
of Vine Deloria’s death,
a picture that might be
called Changing Woman:

  

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat     

See also the windmill figure

The image “http://www.log24.com/theory/images/Whirl3.gif” cannot be displayed, because it contains errors.

in Time and Eternity
(Log 24, Feb. 1, 2003)

and

a review
of Fritz Leiber’s
The Big Time,

The image “http://www.log24.com/log/pix05B/051115-BigTimePic.jpg” cannot be displayed, because it contains errors.

a story that works.”

Tuesday November 15, 2005

Filed under: General,Geometry — m759 @ 2:56 AM
Spider Woman

    “Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, ‘Look, Buster, do you want to live?’….
    Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question.

Fritz Leiber's 'Spider' symbol

Bordered version
of the sigil

The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar.  An X superimposed on a plus sign.  It looked permanent.”

— Fritz Leiber, “Damnation Morning

For Vine Deloria Jr., who died at 72 on Sunday, Nov. 13, 2005:

        Things forgotten are shadows.
        The shadows will be as real
        as wind and rain and song and light,
        there in the old place.
        Spider Woman atop your rock,
        I would greet you,
        but I am going the other way.
        Only a fool would pursue a Navajo
        into the Canyon of Death.

— Roger Zelazny, Eye of Cat

Related material:
from a Log24 entry
on the morning of
Deloria’s death–

Kaleidoscope turning…
Shifting pattern
within unalterable structure…

— Roger Zelazny, Eye of Cat

  

Sunday, November 13, 2005

Sunday November 13, 2005

Filed under: General,Geometry — m759 @ 6:40 AM
Structure

“Sunrise–
Hast thou a Flag for me?”
— Emily Dickinson

From a
Beethoven’s Birthday entry:

  

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat

Related material:

Blue
(below),

The image “http://www.log24.com/log/pix05B/051113-Blue.jpg” cannot be displayed, because it contains errors.

Bee Season
(below),

The image “http://www.log24.com/log/pix05B/051113-Scope1.jpg” cannot be displayed, because it contains errors.

Halloween Meditations,
Aquarius Jazz,
We Are the Key,
and
Jazz on St. Lucia’s Day.

“Y’know, I never imagined
the competition version involved
so many tricky permutations.”

— David Brin, Glory Season

Saturday, August 13, 2005

Saturday August 13, 2005

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

Kaleidoscope, continued:

Austere Geometry

From Noel Gray, The Kaleidoscope: Shake, Rattle, and Roll:

“… what we will be considering is how the ongoing production of meaning can generate a tremor in the stability of the initial theoretical frame of this instrument; a frame informed by geometry’s long tradition of privileging the conceptual ground over and above its visual manifestation.  And to consider also how the possibility of a seemingly unproblematic correspondence between the ground and its extrapolation, between geometric theory and its applied images, is intimately dependent upon the control of the truth status ascribed to the image by the generative theory.  This status in traditional geometry has been consistently understood as that of the graphic ancilla– a maieutic force, in the Socratic sense of that term– an ancilla to lawful principles; principles that have, traditionally speaking, their primary expression in the purity of geometric idealities.*  It follows that the possibility of installing a tremor in this tradition by understanding the kaleidoscope’s images as announcing more than the mere subordination to geometry’s theory– yet an announcement that is still in a sense able to leave in place this self-same tradition– such a possibility must duly excite our attention and interest.

* I refer here to Plato’s utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave.”

See also

Noel Gray, Ph.D. thesis, U. of Sydney, Dept. of Art History and Theory, 1994:

“The Image of Geometry: Persistence qua Austerity– Cacography and The Truth to Space.”

Saturday August 13, 2005

Filed under: General,Geometry — m759 @ 12:04 PM

Kaleidoscope, continued:

In Derrida’s Defense

The previous entry quoted an attack on Jacques Derrida for ignoring the “kaleidoscope” metaphor of Claude Levi-Strauss.  Here is a quote by Derrida himself:

“The time for reflection is also the chance for turning back on the very conditions of reflection, in all the senses of that word, as if with the help of an optical device one could finally see sight, could not only view the natural landscape, the city, the bridge and the abyss, but could view viewing. (1983:19)

— Derrida, J. (1983) ‘The Principle of Reason: The University in the Eyes of its Pupils’, Diacritics 13.3: 3-20.”

The above quotation comes from Simon Wortham,  who thinks the “optical device” of Derrida is a mirror.  The same quotation appears in Desiring Dualisms at thispublicaddress.com, where the “optical device” is interpreted as a kaleidoscope.

Derrida’s “optical device” may (for university pupils desperately seeking an essay topic) be compared with Joyce’s “collideorscape.”  For a different connection with Derrida, see The ‘Collideorscape’ as Différance.

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: General,Geometry — m759 @ 8:16 AM

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

Tuesday, August 9, 2005

Tuesday August 9, 2005

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

Kaleidoscope

A new web page simplifies the Diamond 16 Puzzle and relates the resulting “kaleidoscope” to Hesse’s Bead Game.

Friday, April 30, 2004

Friday April 30, 2004

Filed under: General,Geometry — m759 @ 5:24 PM

Notes

  

On “Notes Toward a Supreme Fiction,” by Wallace Stevens:

“This third section continues its play of opposing forces, introducing in the second canto a ‘blue woman,’ arguably a goddess- or muse-figure, who stands apart from images of fecundity and sexuality….”

Michael Bryson 

From a Beethoven’s Birthday entry:

Moulin Bleu

  

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat   

See, too, Blue Matrices, and
a link for Beethoven’s birthday:

Song for the
Unification of Europe
(Blue 1)

From today’s news:

PRAGUE, Czech Republic (AP) – Ushering in a bold new era, hundreds of thousands of people packed streets and city squares across Europe on Friday for festivals and fireworks marking the European Union’s historic enlargement to 25 countries from 15.

The expanded EU, which takes in a broad swath of the former Soviet bloc – a region separated for decades from the West by barbed wire and Cold War ideology – was widening to 450 million citizens at midnight (6 p.m.EDT) to create a collective superpower rivalling the United States.

“All these worlds are yours
except Europa.
Attempt no landing there.”

Tuesday, December 16, 2003

Tuesday December 16, 2003

Filed under: General,Geometry — Tags: , — m759 @ 2:24 PM

Moulin Bleu

  

Kaleidoscope turning…
Shifting pattern
within unalterable structure…

— Roger Zelazny, Eye of Cat   

See, too, Blue Matrices, and
a link for Beethoven's birthday:

Song for the
Unification of Europe
(Blue 1)

Tuesday December 16, 2003

Filed under: General — m759 @ 1:08 AM

Jazz on St. Lucia’s Day

December 13, Saturday, was
the feast day of St. Lucia.


Lauryn Hill
at St. Lucia

Log24 entry for December 13:

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
Was it a mistake?
There is pain with the power…
Time’s friction at the edges…
Center loosens, forms again elsewhere…

— Roger Zelazny, Eye of Cat

Washington Post, Names and Faces,
Tuesday, Dec. 16, 2003

“A Christmas concert at the Vatican may not be the best place to criticize the Catholic Church for the sexual abuse scandals that have plagued it for the past few years. Or maybe it’s the perfect place.

Musician Lauryn Hill did just that while performing there Saturday night. The Grammy winner read a statement during the concert that scolded the church and its leaders….

La Repubblica newspaper quoted her as saying, ‘I realize some of you may be offended by what I’m saying, but what do you say to the families who were betrayed by the people in whom they believed?’ …

The Vatican said Sunday it had no comment.”

 

“Now you has jazz.”
High Society, 1956   

Related entries:
9/28/03, 8/29/02.

Saturday, December 13, 2003

Saturday December 13, 2003

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

We Are the Key:
The Shining of December 13

For James and Lucia Joyce

In the Orbit of Genius —
TIME, Dec. 1, 2003
:

“Once, when her mother asked if Joyce should visit her in the sanatorium, Lucia said, ‘Tell him I am a crossword puzzle, and if he does not mind seeing a crossword puzzle, he is to come out.’ “

Compare and contrast
with Finnegans Wake

From Roger Zelazny’s Eye of Cat:

“A massive, jaguarlike form with a single, gleaming eye landed on the vehicle’s hood forward and to the front.  It was visible for but an instant, and then it sprang away. The car tipped, its air cushion awry, and it was already turning onto its side before he left the trail.  He fought with the wheel and the attitude control, already knowing that it was too late.  There came a strong shock accompanied by a crunching noise, and he felt himself thrown forward.

DEADLY, DEADLY, DEADLY…
Kaleidoscope turning… Shifting  pattern within unalterable structure… Was it a mistake? There is pain with the power…  Time’s friction at the edges…  Center loosens, forms again elsewhere…  Unalterable?  But – Turn outward.  Here songs of self erode the will till actions lie stillborn upon night’s counterpane.  But – Again the movement…  Will it hold beyond a catch of moment?  To fragment…  Not kaleidoscope.  No center.  But again… To form it will.  To will it form.  Structure… Pain…  Deadly, deadly…  And lovely.  Like a sleek, small dog… A plastic statue… The notes of an organ, the first slug of gin on an empty stomach… We settle again, farther than ever before… Center. The light!… It is difficult being a god. The pain. The beauty. The terror of selfless –  Act!  Yes. Center, center, center… Here? Deadly…

necess yet again from bridge of brainbow oyotecraven stare decesis on landaway necessity timeslast the arnings ent and tided turn yet beastfall nor mindstorms neither in their canceling sarved cut the line that binds ecessity towarn and findaway twill open pandorapack wishdearth amen amenusensis opend the mand of min apend the pain of durthwursht vernichtung desiree tolight and eadly dth cessity sesame

We are the key.”

Wednesday, July 23, 2003

Wednesday July 23, 2003

Filed under: General,Geometry — Tags: — m759 @ 4:17 PM

Being Pascal Sauvage

Pascal

"Voilà ce que je sais par une longue expérience de toutes sortes de livres et de personnes. Et sur cela je fais le même jugement de ceux qui disent que les géomètres ne leur donnent rien de nouveau par ces règles, parce qu' ils les avaient en effet, mais confondues parmi une multitude d' autres inutiles ou fausses dont ils ne pouvaient pas les discerner, que de ceux qui cherchant un diamant de grand prix

Diamant

parmi un grand nombre de faux, mais qu' ils n' en sauraient pas distinguer, se vanteraient, en les tenant tous ensemble, de posséder le véritable aussi bien que celui qui, sans s' arrêter à ce vil amas, porte la main sur la pierre choisie que l' on recherche, et pour laquelle on ne jetait pas tout le reste."

— Blaise Pascal, De l'Esprit Géométrique

La Pensée Sauvage

"….the crowning image of the kaleido­scope, lavishly analogized to the mythwork in a three-hundred-word iconic apotheosis that served to put the wraps on the sustained personification of “la pensée sauvage” in the figure of the bricoleur, in an argument developed across two chapters and some twenty pages in his [Claude Lévi-Strauss's] most famous book…."

— Robert de Marrais in
Catastrophes, Kaleidoscopes,
String Quartets:
Deploying the Glass Bead Game


Pascal Sauvage

Chiasmus

For more on pensée sauvage, see

"Claude Lévi-Strauss,

Chiasmus

and the Ethnographic Journey."

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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