Log24

Monday, January 1, 2018

Diamond Theory 1976

Filed under: Geometry — m759 @ 8:26 PM

The first 12 pages of my 1976 preprint "Diamond Theory" are 
now scanned and uploaded.  See a slideshow.

For downloading, all 12 pages are combined in a PDF.

Sunday, February 2, 2014

Diamond Theory Roulette

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

ReCode Project program from Radamés Ajna of São Paulo —

At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants
. Note the resulting image's ordinary
or color-interchange symmetry.

Wednesday, November 28, 2012

Diamond Theory

Filed under: Uncategorized — Tags: — m759 @ 2:18 AM

A pdf of a 1977 three-page article with this title
has been added at finitegeometry.org/sc.

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: Uncategorized — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Thursday, October 14, 2010

Diamond Theory and Magic Squares

Filed under: Uncategorized — Tags: , — m759 @ 6:19 PM

"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."

— e. e. cummings, 1944

For one such specimen, see The Matrix of Abraham
a 5×5 square that is hypermagical… indeed, diabolical.

Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—

  1. Finite Geometry of the Square and Cube
  2. Clifford Pickover on a 4×4 square
  3. Christopher J. Henrich on the geometry of 4×4 magic squares
    (without any mention of  [1] above or related work dating back to 1976)

" … listen: there's a hell
of a good universe next door; let's go"

— e. e. cummings

Happy birthday, e. e.

Monday, February 5, 2018

Stranger Things than Pulp Fiction

Filed under: Uncategorized — m759 @ 12:30 PM

Diamond Theory cover, said to resemble Proginoskes in 'A Wind in the Door'

Click on the image for a
relevant Wallace Stevens poem.

A new Facebook page will describe
some background for the above image.

Wednesday, January 24, 2018

The Pentagram Papers

Filed under: Uncategorized — m759 @ 12:40 PM

(Continued)

From a Log24 post of March 4, 2008 —

SINGER, ISAAC:
"Are Children the Ultimate Literary Critics?"
— Top of the News 29 (Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments
'slice of life' and chaos in children's literature. Maintains that
children like good plots, logic, and clarity, and that they
have a concern for 'so-called eternal questions.'"

— An Annotated Listing of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to
her youngest brother, Charles Wallace, and to their father,
who were deep in concentration, bent over the model
they were building of a tesseract: the square squared,
and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time," see A Fold in TimeTime Fold,
and Diamond Theory in 1937

A Swiftly Tilting Planet  is a fantasy for children 
set partly in Vespugia, a fictional country bordered by
Chile and Argentina.

Ibid.

The pen's point:

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

Note the figure on the cover of National Review  above —

A related figure from Pentagram Design

See, more generally,  Isaac Singer  in this  journal.

Monday, December 18, 2017

Mathematics and Art

Filed under: Geometry — m759 @ 5:09 PM

From the American Mathematical Society homepage today —

From concinnitasproject.org

"Concinnitas  is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (1404-1472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."

The favicon of the Concinnitas Project —

The structure of the Concinnitas favicon —

This structure is from page 15 of
"Diamond Theory," a 1976 preprint —

 .

Thursday, August 31, 2017

A Conway-Norton-Ryba Theorem

Filed under: Uncategorized — m759 @ 1:40 PM

In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton,  and Alex Ryba present the following
result on order-four magic squares —

A monograph published in 1976, "Diamond Theory," deals with 
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16).  These squares have remarkable, if not 
"magic," symmetry properties.  See excerpts in a 1977 article.

See also Magic Square and Diamond Theorem in this  journal.

Tuesday, May 2, 2017

Image Albums

Filed under: Uncategorized — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Thursday, December 8, 2016

Finite Groups and Their Geometric Representations

Filed under: Uncategorized — Tags: — m759 @ 8:06 AM

The title is that of a presentation by Arnold Emch
at the 1928 International Congress of Mathematicians:

See also yesterday's "Emch as a Forerunner of S(5, 8, 24)."

Related material: Diamond Theory in 1937.

Further remarks:  Christmas 2013 and the fact that
759 × 322,560 = the order of the large Mathieu group  M24 .

Friday, November 25, 2016

Priority

Filed under: Uncategorized — Tags: , — m759 @ 12:00 AM

Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:

Hudson in 1905 —

Golomb in 1967 —

It is also likely that some figures illustrating Walsh functions  as
two-color square arrays were published prior to 1976.

Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.

Monday, October 10, 2016

Mono Type 1, by Sultan (1966)

Filed under: Uncategorized — m759 @ 12:06 PM

"Sultan" was a pseudonym of Peter Lindbergh, now a 
well-known fashion photographer. Click image for the source.

Related art — Diamond Theory Roullete, by Radames Ajna,
2013 (Processing  code at ReCode Project based on
"Diamond Theory" by Steven H. Cullinane, 1977).

Saturday, October 8, 2016

Unity of Opposites: Plato and Beyond

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

The "unity" of the title was suggested by this morning's update
at the end of yesterday's post Paz.

For the Plato of the title, see the Sept. 27, 2016, post

Chomsky and Lévi-Strauss in China
Or:  Philosophy for Jews

For glyphs representing the "unity of opposites" of the title,
see a webpage linked to here on Groundhog Day 2014

The above image is related to Jung's remarks on Coincidentia
Oppositorum
 
. (See also coincidentia in this journal.)

A different Jung, in a new video with analogues of the rapidly
flashing images in Ajna's webpage "Diamond Theory Roullete" —

The above video promotes Google's new open-source "Noto" font

Sunday, May 29, 2016

The Ideogram Principle …

Filed under: Uncategorized — Tags: — m759 @ 4:23 PM

According to McLuhan

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)

A relation in four terms

A : B  ::  C : D   as   Model : Crutch  ::  Metaphor : Ornament —

See also Dueling Formulas and Symmetry.

Wednesday, May 25, 2016

Framework

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

Wednesday, May 4, 2016

Golomb and Symmetry

Filed under: Uncategorized — m759 @ 12:00 PM

From the webpage Diamond Theory Bibliography

Golomb, Solomon W. 
Shift register sequences  (Revised edition)
Aegean Park Press, Laguna Hills, CA, 1982
   The fifteen "stencils" in Golomb's Fig. VIII-8, page 219,
   are the same as the fifteen affine hyperplanes that
   account for patterns' symmetry in diamond theory.
   This figure occurs in a discussion of Rademacher-
   Walsh functions.

Elsewhere

Friday, December 25, 2015

At Play in the Fields

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

See Fields of Force  and recent posts.

From PR Newswire  in July 2011 —

Campus Crusade for Christ Adopts New Name: Cru
60-year-old Int'l Ministry Aims to Increase
Relevance and Global Effectiveness

Related material:

Yin + Yang —

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

Tuesday, December 8, 2015

Conceptual Art

Filed under: Uncategorized — Tags: — m759 @ 12:06 PM

A December 7th  New York Times  column:

A current exhibition by Joseph Kosuth in Oslo:

From the two texts by Mondrian at the right hand of Kosuth —

"The positive and negative states of being bring about action."

"Through its pure relationships, purely abstract art
can approach the expression of the universal …."

These texts may be viewed as glosses on the following image —

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

Click image for related posts.

Friday, November 27, 2015

Once Upon a Matrix

Filed under: Uncategorized — m759 @ 10:20 PM

Or:  The Strife of Luminosity and Obscurity

(Continued from "Once Upon a Time," November 25, 2015)

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


Wednesday, November 25, 2015

Once Upon a Time

Filed under: Uncategorized — m759 @ 5:31 PM

This post's title was suggested by the previous post
and by today's news of a notable sale of a one-copy
record album, "Once Upon a Time in Shaolin."

See as well posts from Tuesday, March 11, 2014,
the day Emma Watson unveiled a new trailer

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

Saturday, October 24, 2015

Two Views of Finite Space

Filed under: Uncategorized — Tags: — m759 @ 10:00 AM

The following slides are from lectures on "Advanced Boolean Algebra" —

The small Boolean  spaces above correspond exactly to some small
Galois  spaces. These two names indicate approaches to the spaces
via Boolean algebra  and via Galois geometry .

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

" 'All you need to do is give me your soul:  give up geometry 
and you will have this marvellous machine.' (Nowadays you
can think of it as a computer!) "

Related material — The article "Diamond Theory" in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977.  That
article, despite the word "computer" in the journal's title, was
much less about Boolean algebra  than about Galois geometry 

For later remarks on diamond theory, see finitegeometry.org/sc.

Friday, October 23, 2015

Retro or Not?

Filed under: Uncategorized — 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."

Monday, September 28, 2015

Hypercube Structure

Filed under: Uncategorized — m759 @ 1:01 AM

Click to enlarge:

Two views of tesseracts as 4D vector spaces over GF(2)

For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .

For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.

For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.

Sunday, July 26, 2015

Sunday Sermon

Filed under: Uncategorized — m759 @ 10:20 AM

"Little emblems of eternity"
— Phrase by Oliver Sacks in today's
New York Times  Sunday Review

Some other emblems —

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

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

Click an emblem for
some background.

Thursday, July 2, 2015

Deepening the Spielraum

Filed under: Uncategorized — Tags: — m759 @ 2:00 AM

(A sequel to Expanding the Spielraum (Feb. 3, 2015))

"Knowledge, wisdom even, lies in depth, not extension."

Tim Parks in The New York Review of Books ,
     5 PM ET on June 26, 2015

See also Log24 posts on the following figure —

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

Sunday, November 30, 2014

Agents of a Great Despair

Filed under: Uncategorized — Tags: — m759 @ 6:00 PM

Or:  Concepts of Space

1976 according to Cullinane:

1976 according to Plotnick:

“Irony and ridicule are entertaining and effective, and . . .
at the same time they are the agents of a great despair
and stasis in U.S. culture.”  — David Foster Wallace,
as quoted by Adam Kirsch today at Salon

Thursday, October 30, 2014

Mimicry

Filed under: Uncategorized — m759 @ 5:09 PM

This journal Tuesday,  Oct. 28, 2014, at 5 PM ET:

“What is a tai chi master, and what is it that he unfolds?”

From an earlier post, Hamlet’s father’s ghost
on “the fretful porpentine”:

Hamlet , Act 1, Scene 5 —

Ghost:

“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combinèd locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood.”

Galway Kinnell:

“I roll
this way and that in the great bed, under
the quilt
that mimics this country of broken farms and woods”

— “The Porcupine”

For quilt-block designs that do not mimic farms or woods,
see the cover of Diamond Theory .  See also the quotations
from Wallace Stevens linked to in the last line of yesterday’s
post in memory of Kinnell.

“… a bee for the remembering of happiness” — Wallace Stevens

Tuesday, October 28, 2014

Raiders of the Lost Symbol

Filed under: Uncategorized — m759 @ 5:00 PM

A print copy of next Sunday’s New York Times Book Review
arrived in today’s mail. From the front-page review:

Marcel Theroux on The Book of Strange New Things ,
a novel by Michel Faber —

“… taking a standard science fiction premise and
unfolding it with the patience and focus of a
tai chi master, until it reveals unexpected
connections, ironies and emotions.”

What is a tai chi master, and what is it that he unfolds?

Perhaps the taijitu  symbol and related material will help.

The Origin of Change

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”

Monday, October 13, 2014

Raiders of the Lost Theorem

Filed under: Uncategorized — Tags: — m759 @ 12:05 PM

(Continued from Nov. 16, 2013.)

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric  or, less formally, geomagic  is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles  or pieces ) may appear in the cells instead of numbers.”

** See some geometric  matrices by Cullinane in a March 2001 webpage.

Earlier actual specimens — see Diamond Theory  excerpts published in
February 1977 and a brief description of the original 1976 monograph:

“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”

— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society

The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.

Monday, August 4, 2014

A Wrinkle in Space

Filed under: Uncategorized — Tags: , — m759 @ 10:30 AM

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

Tuesday, July 15, 2014

Photo Opportunity

Filed under: Uncategorized — Tags: — m759 @ 2:02 PM

"I need a photo opportunity, I want a shot at redemption.
Don't want to end up a cartoon in a cartoon graveyard."
– Paul Simon

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt  went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less.

 

— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel   (Knopf, 1951)

For background on the planes illustrated above,
see Diamond theory in 1937.

Thursday, June 26, 2014

Study This Example

Filed under: Uncategorized — Tags: — m759 @ 10:09 AM

The authors of the following offer an introduction to symmetry
in quilt blocks.  They assume, perhaps rightly, that their audience
is intellectually impaired:

“A quilt block is made of 16 smaller squares.
Each small square consists of two triangles.”

Study this example of definition.
(It applies quite precisely to the sorts of square patterns
discussed in the 1976 monograph Diamond Theory , but
has little relevance for quilt blocks in general.)

Some background for those who are not  intellectually impaired:
Robinson’s book Definition in this journal and at Amazon.

Friday, May 16, 2014

Way to Go

Filed under: Uncategorized — m759 @ 3:17 PM

Or: Death Edit

IMAGE- On Elaine Sturtevant, an artist who reportedly died on May 7, 2014

Log24 on the reported date of Sturtevant’s death:

Conceptual Art

Filed under: Uncategorized — m759 @ 2:01 AM

Yesterday’s online New York Times  has the following quote:

“The idea becomes a machine that makes the art.”
— Sol LeWitt

For instance, some conceptual art not  by LeWitt:

Diamond Theory Roulette (Feb. 2, 2014).

Tuesday, May 13, 2014

An Artist’s Memorial

Filed under: Uncategorized — m759 @ 8:00 PM

See the above weblog post honoring a Swiss artist‘s
“wit, his perception, his genius, his horizon,
his determination, his humour, his friendship,
and his immeasurable kindness.”

Not a bad sendoff. Contrast with events at Harvard
on the date of the artist’s death.

Related material:  An album cover, and …

See also this  journal in September 2008.

As far as Swiss art goes, I personally prefer the work of, say,
Karl Gerstner and Paul Talman.

Saturday, May 10, 2014

Test Patterns

Filed under: Uncategorized — m759 @ 11:00 AM

 Raven’s Progressive Matrices  intelligence test—
IMAGE- Raven's Progressive Matrices problem based on triangular half- and quarter-diamonds

Wechsler Adult Intelligence Scale  test—  

Related art —  (Click images for further details.)

Patterns suggesting those of the Raven test:

Patterns suggesting those of the Wechsler test:

The latter patterns were derived from the former.

Wednesday, May 7, 2014

Conceptual Art

Filed under: Uncategorized — m759 @ 2:01 AM

Yesterday’s online New York Times  has the following quote:

“The idea becomes a machine that makes the art.”
— Sol LeWitt

For instance, some conceptual art not  by LeWitt:

Diamond Theory Roulette (Feb. 2, 2014).

Thursday, April 17, 2014

Thursday with the Nashes

Filed under: Uncategorized — m759 @ 3:00 PM

“For every kind of vampire, there is a kind of cross.” — Gravity’s Rainbow

“I don’t write exclusively on Jewish themes or about Jewish characters.
My collection of short stories, Strange Attractors , contained nine pieces,
five of which were, to some degree, Jewish, and this ratio has provided me
with a precise mathematical answer (for me, still the best kind of answer)
to the question of whether I am a Jewish writer. I am five-ninths a Jewish writer.”

— Rebecca Goldstein, “Against Logic

Midrashim for Rebecca: 

The Diamond Theory vs.  the Story Theory (of truth)

Story Theory and the Number of the Beast

The Palm Sunday post “Gray Space”

For those who prefer the diamond theory of truth,
a “precise mathematical” view of a Gray code —

IMAGE- Six-bit binary and Gray codes

For those who prefer the story theory of truth,
Thursday with the Nashes —

The actors who portrayed Mr. and Mrs. John Nash in
‘A Beautiful Mind’ now portray Mr. and Mrs. Noah…

IMAGE- At UMC.org, the actors who portrayed Mr. and Mrs. John Nash in 'A Beautiful Mind' now portray Mr. and Mrs. Noah.

Tuesday, March 11, 2014

Depth

Filed under: Uncategorized — Tags: , — m759 @ 11:16 AM

"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."

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

Part I:  An Inch Deep

IMAGE- Catch-phrase 'a mile wide and an inch deep' in mathematics education

Part II:  An Inch Wide

See a search for "square inch space" in this journal.

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

 

See also recent posts with the tag depth.

Friday, February 28, 2014

Code

Filed under: Uncategorized — 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.

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: Uncategorized — Tags: , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Wednesday, December 25, 2013

Rotating the Facets

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

Previous post

"… her mind rotated the facts…."

Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:

IMAGE- Count rotational symmetries by rotating facets. Illustrated with 'Plato's Dice.'

"I've heard of affairs that are strictly Platonic"

Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

 See Arnold Emch, "Triple and Multiple Systems, Their Geometric 
 Configurations and Groups
," Trans. Amer. Math. Soc.  31 (1929),
 No. 1, 25–42. 

 On page 42, Emch describes the above method of rotating a
 hypercube's 8 facets (i.e., three-dimensional cubes) to count
 rotational symmetries —

See also Diamond Theory in 1937.

Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .

Tuesday, December 10, 2013

Wittgenstein’s Tesseract

Filed under: Uncategorized — m759 @ 5:14 PM

See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)

See also the tag Diamond Theory tesseract in this journal.

Monday, October 21, 2013

Edifice Complex

Filed under: Uncategorized — m759 @ 8:00 PM

New! Improved!

"Euclid's edifice loomed in my consciousness 
as a marvel among sciences, unique in its
clarity and unquestionable validity." 
—Richard J. Trudeau in
   The Non-Euclidean Revolution  (First published in 1986)

Readers of this journal will be aware that Springer's new page
advertising Trudeau's book, pictured above, is a bait-and-switch
operation. In the chapter advertised, Trudeau promotes what he
calls "the Diamond Theory of Truth" as a setup for his real goal,
which he calls "the Story Theory of Truth."

For an earlier use of the phrase "Diamond Theory" in
connection with geometry, see a publication from 1977.

Saturday, September 21, 2013

Mathematics and Narrative (continued)

Filed under: Uncategorized — Tags: , — m759 @ 1:00 AM

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — Tags: , — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Monday, August 12, 2013

Form

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

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest 
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Tuesday, July 16, 2013

Space Itself

Filed under: Uncategorized — Tags: — m759 @ 10:18 AM

"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space
but in space itself?"

Megan Garber in The AtlanticAug. 16, 2012

One approach:

"There is  such a thing as a tesseract" and
Diamond Theory in 1937.

See, too, Baez in this journal.

Tuesday, July 9, 2013

Vril Chick

Filed under: Uncategorized — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Sunday, June 30, 2013

Book Award

Filed under: Uncategorized — m759 @ 5:01 PM
 

"What on earth is
a 'concrete universal'?
"

— Said to be an annotation
(undated) by Robert M. Pirsig
of A History of Philosophy ,
by Frederick Copleston,
Society of Jesus.

In the spirit of the late Thomas Guinzburg

See also "Concrete Universal" in this journal.

Related material— From a Bloomsday reply
to a Diamond Theory  reader's comment, an excerpt—

The reader's comment suggests the following passages from
the book by Stirling quoted above—

 

Here Stirling plays a role analogous to that of Professor Irwin Corey
accepting the National Book Award for Gravity's Rainbow  in 1974.

Tuesday, June 18, 2013

Multispeech

Filed under: Uncategorized — Tags: — m759 @ 4:48 PM

(Continued)

For those who prefer Trudeau's
"Story Theory" of truth to his "Diamond Theory"

IMAGE- Janet Maslin's review of Max Barry's novel 'Lexicon'

Related material: Click images below for the original posts.

See as well the novel  "Lexicon" at Amazon.com 
and the word  "lexicon" in this journal.

Sunday, June 16, 2013

Mathematical Review

Filed under: Uncategorized — m759 @ 10:00 PM

From a weblog post on June 11, 2013, by one Pete Trbovich:

Diamond Theory

Here again, I don't think Steven Cullinane is really unhinged per se. At the very least, his geometric study is fun to play with, particularly when you find this toy. And I'm not really sure that anything he says is wrong per se. But you might find yourself asking "So what?" or more to the point, "Why is this supposed to be the central theory to explaining life, the universe, and everything?"

It isn't  supposed to be such a theory.
I do not know why Trbovich thinks it is 

— Steven H. Cullinane

Update of 11 PM June 16:

For one such central theory of everything, see
the I Ching .  Diamond theory is, unlike that
Chinese classic, pure mathematics, but the larger
of the binary-coordinate structures  it is based on
are clearly isomorphic, simply as structures , to
the I Ching 's 
64 hexagrams.

Make of this what you will.

Monday, June 10, 2013

Galois Coordinates

Filed under: Uncategorized — Tags: , — m759 @ 10:30 PM

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material: 

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
   "Galois coordinates" may have some implications 
   for the naive form of the relativity problem
   related to square and cubical arrays.

Tuesday, May 28, 2013

Codes

Filed under: Uncategorized — Tags: , , , — m759 @ 12:00 PM

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Tuesday, April 30, 2013

Logline

Filed under: Uncategorized — m759 @ 9:29 AM

Found this morning in a search:

logline  is a one-sentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It's the short blurb in TV guides that tells you what a movie
is about and helps you decide if you're interested 

The search was suggested by a screenwriting weblog post,
"Loglines: WHAT are you doing?".

What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?

The screenwriting post was found in an earlier search for
the highlighted phrase.

The screenwriting post was dated December 15, 2009.

What I am doing now  is checking for synchronicity.

This  weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .

I guess the script I'm summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.

For that group in action, see the Diamond 16 Puzzle.

The "super overarching" phrase was used to describe
this same group in a different context:

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

This is from "Mathieu Moonshine," a webpage by Anne Taormina.

A logline summarizing my  approach to that group:

Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs— 
found, for instance, in quilts.

The story thus summarized is perhaps not destined for movie greatness.

Wednesday, February 13, 2013

Form:

Filed under: Uncategorized — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Thursday, January 17, 2013

Brazil Revisited

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

Yesterday's post Treasure Hunt, on a Brazilian weblog,
suggests a review of Brazil  in this journal.  The post
most relevant to yesterday's remarks is from
August 15, 2003, with a link, now broken, to the work
of Brazilian artist Nicole Sigaud* that also uses the
four half-square tiles used in 1704 by Sebastien Truchet 
and somewhat later by myself in Diamond Theory 
(see a 1977 version).

A more recent link that works:

http://vismath9.tripod.com/sigaud/e-index.html

ANACOM PROJECT

 

APPLICATIONS
HISTORY
THE FONT
ALGORITHMS
FAMILY I
FAMILY 2
EXAMPLES
EXAMPLES II
DOWNLOADS
INTERACTIVE PROGRAM (JAVASCRIPT)
 
VisMathHOME

 

© 1997 – 2002 Nicole Sigaud

* Sigaud shares the interests of her fellow Brazilian
   whose weblog was the subject of yesterday's
   Treasure Hunt.—

   "For many years I have dedicated myself to the study
    of medieval magic, demonology, Kabbalah, Astrology,
    Alchemy, Tarot and divination in general."

     — Nicole Sigaud (translated by Google) in a self-profile: 
     http://www.recantodasletras.com.br/autor.php?id=78359.

    I do not share the interest of these authors in such matters,
    except as they are reflected in the works of authors like
    Charles Williams and Umberto Eco.

Wednesday, January 16, 2013

Treasure Hunt

Filed under: Uncategorized — Tags: — m759 @ 3:17 PM

The Mathematical Association of America (MAA)
newsmagazine Focus  for December 2012/January 2013: 

The Babylonian tablet on the cover illustrates the
"Mathematical Treasures" article.

A search for related material yields a Babylonian tablet
reproduced in a Brazilian weblog on July 4, 2012:

In that weblog on the same day, July 4, 2012,
another post quotes at length my Diamond Theory page,
starting with the following image from that page—

IMAGE- Plato's Diamond

That Brazilian post recommends use of geometry together
with Tarot and astrology. I do not concur with this 
recommendation, but still appreciate the mention.

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: Uncategorized — Tags: , — m759 @ 10:18 AM

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Monday, December 24, 2012

Eternal Recreation

Filed under: Uncategorized — Tags: , , — m759 @ 3:17 AM

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

“Only gradually did I discover what the mandala really is:
‘Formation, Transformation, Eternal Mind’s eternal recreation.’*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions.”

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

                   … Gestaltung, Umgestaltung, 
  Des ewigen Sinnes ewige Unterhaltung….

Jung’s “Formation, Transformation” quote is from the realm of
the Mothers (Faust Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

“In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust’s next encounter with
the feminine, but it’s obviously of a very different kind than his
relationship with Gretchen.”

The phrase “Gestaltung, Umgestaltung ” suggests a more mathematical
approach to the Unterhaltung . Hence

Part I: Mothers

“The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time.”

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity 
,
Johns Hopkins Press, 1969, page 37

Part II: Matrices

        

Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Sunday, December 9, 2012

Deep Structure

Filed under: Uncategorized — Tags: , — m759 @ 10:18 AM

The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.

It still applies, however, to the 1976 mathematics, diamond theory  ,
underlying the formal patterns discussed in a Royal Society paper
this year.

A review of deep structure, from the Wikipedia article Cartesian linguistics

[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .]

Deep structure vs. surface structure

"Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not.

Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39).

Summary of Port Royal Grammar

The Port Royal Grammar is an often cited reference in Cartesian Linguistics  and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42).

The corresponding concepts from diamond theory are

"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns

"A base system that generates deep structures"—
Group actions on square arrays for instance, on the 4×4 square

"A transformational system"— The decomposition theorem 
that maps deep structure into surface structure (and vice-versa)

Saturday, December 8, 2012

It’s 10 PM

Filed under: Uncategorized — Tags: — m759 @ 10:00 PM

Do you know where the mushrooms are?

IMAGE- Cover image for a free mixtape, 'Lawrence Class - The Diamond Theory,' that contains images from Steven H. Cullinane's 'Diamond Theory.'

Above: Image from Log24 on Dec. 4th, 2012, at 4:23 PM ET.

See also… on that date at that time …
The American College of Neuropsychopharmacology… (click to enlarge)

Defining the Contest…

Filed under: Uncategorized — Tags: , , , , — m759 @ 5:48 AM

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

"Happy Birthday, Noam Chomsky." by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

See Meno Diamond in this journal. For instance, from 
the Feast of Saint Nicholas (Dec. 6th) this year—

The Meno Embedding

http://www.log24.com/log/pix10B/101128-TheEmbedding.gif

For related truths about geometry, see the diamond theorem.

For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).

See esp. the Sept. 11 post,  on a Royal Society paper from July 2012
claiming that

"With the results presented here, we have taken the first steps
in decoding the uniquely human  fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "

The sorts of patterns discussed in the 2012 paper —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Wednesday, December 5, 2012

Arte Programmata*

Filed under: Uncategorized — m759 @ 9:30 PM

The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced 
according to strict rules, and were in this sense 
"programmed," but were drawn by hand.

Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not  in
Diamond Theory).

See Isaac Gierard's program at GitHub

https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde

As the suffix indicates, this program is in the
Processing Development Environment language.

It produces the following sketch:

IMAGE- Sketch programmed by Isaac Gierard to mimic some of the images of 'Diamond Theory' (© 1976 by Steven H. Cullinane).

The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the 
overall effect of the sketch is pleasing.

For some background for the program, see The ReCode Project.

It is good to learn that the Processing language is well-adapted to making the 
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory  in "Diamond Theory."

For some related remarks, see Theory (Sept. 30, 2012).

* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."

Tuesday, December 4, 2012

McKenna Theory

Filed under: Uncategorized — m759 @ 4:23 PM

A 1976 monograph:

IMAGE- 'Diamond Theory,' © 1976 by Steven H. Cullinane

A 2012 mixtape cover:

IMAGE- Cover image for a free mixtape, 'Lawrence Class - The Diamond Theory,' that contains images from Steven H. Cullinane's 'Diamond Theory.'

A new "Diamond Theory" image found on the Web
today links my work to the "Stoned Ape Theory"
of human evolution due to Terence McKenna

This link is via a picture, apparently copied from deviantart.com,
of two apes contemplating some psychedelic mushrooms.
The picture is titled "Stoned Ape Theory." The mushrooms in
the picture are apparently taken from an image at DrugNet.net:
 

Actually, the mathematical work called "diamond theory"
has nothing whatever to do with psychedelic experiences,
although some of the illustrations may appeal to McKenna fans.

Thursday, November 29, 2012

Conceptual Art

Filed under: Uncategorized — m759 @ 12:09 PM

Quotes from the Bremen site
http://dada.compart-bremen.de/ 
 

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

" 'compArt | center of excellence digital art' is a project
at the University of Bremen, Germany. It is dedicated
to research and development in computing, design,
and teaching. It is supported by Rudolf Augstein Stiftung,
the University of Bremen, and Karin und Uwe Hollweg Stiftung."

See also Stiftung in this journal.

Sunday, November 18, 2012

Sermon

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

Happy birthday to

IMAGE- Margaret Atwood, Kim Wilde, Peta Wilson

Today's sermon, by Marie-Louise von Franz

Number and Time, by Marie-Louise von Franz

For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.

Another modern physicist, Niels Bohr, died
on this date in 1962

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

The circle above is marked with a version
of the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

For the square, see the diamond theorem.

"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"

Thursday, November 1, 2012

Theories of Truth

Filed under: Uncategorized — Tags: — m759 @ 7:20 PM

A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The Non-Euclidean Revolution

The Story Theory of Truth:

"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."

New Yorker  profile of tesseract
     author Madeleine L'Engle

The Diamond Theory of Truth:

(Click image for some background.)

Spaces as Hypercubes

See also the links on a webpage at finitegeometry.org.

Monday, August 13, 2012

Raiders of the Lost Tesseract

Filed under: Uncategorized — m759 @ 3:33 PM

(An episode of Mathematics and Narrative )

A report on the August 9th opening of Sondheim's Into the Woods

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning  "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

Compare and contrast…

1.  The following item from Walpurgisnacht 2012

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

2.  The six partitions of a tesseract's 16 vertices 
       into four parallel faces in Diamond Theory in 1937

Friday, March 2, 2012

Douat Facsimile

Filed under: Uncategorized — Tags: — m759 @ 5:14 PM

Title of a treatise by Dominique Douat

"Méthode pour faire une infinité de desseins différens avec des carreaux mi-partis de deux couleurs par une ligne diagonale : ou observations du Père Dominique Doüat Religieux Carmes de la Province de Toulouse sur un mémoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Révérend Père Sébastien Truchet religieux du même ordre, Académicien honoraire  " (Paris, 1722)

"The earliest (and perhaps the rarest) treatise on the theory of design"

— E. H. Gombrich, 1979, in The Sense of Order

A facsimile version (excerpts, 108 pp., Feb. 5, 2010) of this treatise is available from

http://jacques-andre.fr/ed/ in a 23.1 MB pdf.

Sample page—

For a treatise on the finite geometry underlying such designs (based on a monograph I wrote in 1976, before I had heard of Douat or his predecessor Truchet), see Diamond Theory.

Saturday, February 18, 2012

Symmetry

Filed under: Uncategorized — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

IMAGE- Brading and Castellani, 'Symmetries in Physics'- Four main sections include 'Continuous Symmetries' and 'Discrete Symmetries.'

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics 
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Thursday, January 19, 2012

Mathematical Imagery

Filed under: Uncategorized — m759 @ 10:28 PM

From the Crafoord Prize website

Related meta -mathematical image from Diamond Theory

Mathematical  image related to combinatorics—

See also permutahedron in this journal.

Tuesday, January 3, 2012

Theorum

Filed under: Uncategorized — Tags: — m759 @ 7:48 AM

In memory of artist Ronald Searle

IMAGE- Ronald Searle, 'Pythagoras puzzled by one of my theorums,' from 'Down with Skool'

Searle reportedly died at 91 on December 30th.

From Log24 on that date

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012

Theorum

 

From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth  to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just  a theory" – and the often cited response of "but gravity is also just  a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

See also

 

Some backgound— In this journal, "Diamond Theory of Truth."

Friday, October 28, 2011

The Soul’s Code

Filed under: Uncategorized — m759 @ 7:20 AM

http://www.log24.com/log/pix11C/111028-NYT-JamesHillman-360w.jpg

James Hillman reportedly died on Thursday, October 27, 2011.

For some commentary, see Wednesday's link to 779

http://www.log24.com/log/pix11C/111028-SoulsCode.JPG

Daimon
  Theory

Diamond Theory

Sunday, September 18, 2011

What Rough Beast

Filed under: Uncategorized — 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, August 10, 2011

Objectivity

Filed under: Uncategorized — m759 @ 12:25 PM

From math16.com

Quotations on Realism
and the Problem of Universals:

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268

"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.

"People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only 'truths' strictly worthy of the name. Such truths I will call 'diamonds'; they are highly desirable but hard to find….The happy metaphor is Morris Kline's in Mathematics in Western Culture (Oxford, 1953), p. 430."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 114 and 117

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call 'models of reality'). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture."
— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 256 and 259

Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is non-Euclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000

The correct statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)— 

From The Search for Invariants (June 20, 2011):

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

http://www.log24.com/log/pix11B/110810-MajaLovrenovBio.jpg

Related material from Sunday's New York Times  travel section—

"Exhibit A is certainly Ljubljana…."

Sunday, June 19, 2011

Abracadabra (continued)

Filed under: Uncategorized — m759 @ 12:00 AM

Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator  of R.T. Curtis.

A search tonight for related material on the Web yielded…

(Click to enlarge.)

IMAGE- Art by Steven H. Cullinane displayed as his own in Steve Richards's Piracy Project contribution

   Weblog post by Steve Richards titled "The Search for Invariants:
   The Diamond Theory of Truth, the Miracle Octad Generator
   and Metalibrarianship." The artwork is by Steven H. Cullinane.
   Richards has omitted Cullinane's name and retitled the artwork.

The author of the post is an artist who seems to be interested in the occult.

His post continues with photos of pages, some from my own work (as above), some not.

My own work does not  deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.

The artist's post concludes with the following (note also the beginning of the preceding  post)—

http://www.log24.com/log/pix11A/110619-MOGsteverichards.jpg

"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.

Saturday, May 28, 2011

Savage Detectives

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

IMAGE- Rubeus Hagrid and Jorn Barger


IMAGE- Cover of 'The Savage and Beautiful Country'

   Alan McGlashan

From Savage Logic

Sunday, March 15, 2009  5:24 PM

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"

Friday, April 22, 2011

Romancing the Hyperspace

Filed under: Uncategorized — m759 @ 7:59 PM

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Image-- examples from Galois affine geometry

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"–

"… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry."

Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory."

Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds.

* "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

Saturday, February 5, 2011

Zen and the Art of Philosophy

Filed under: Uncategorized — m759 @ 12:00 PM

Wallace Stevens Concordance

An Ordinary Evening in New Haven
line 540 (xxx.18): In which hundreds of eyes, in one mind, see at once.

The cover art of a 1976 monograph, "Diamond Theory," was described in this morning's post.

As Madeleine L'Engle noted in 1976, the cover art resembles the character Proginoskes in her novel A Wind in the Door.

A search today for Proginoskes yields a description by Brendan Kidwell

http://www.log24.com/log/pix11/110205-KidwellProginoskesArt.png

A link at Kidwell's site leads to a weblog by Jeff Atwood, a founder of Stack Overflow, a programmers' question-and-answer site.
(Stack Overflow is said to have inspired the similar site for mathematicians, Math Overflow.)

Yesterday Atwood discussed technical writing.

This suggests a look at Robert M. Pirsig on that subject in his 1974 philosophical novel Zen and the Art of Motorcycle Maintenance.

(See also a document on Pirsig's technical-writing background.)

Pirsig describes his novel as "a sort of Chautauqua."

This, together with the Stevens and Proginoskes quotes above, leads back to the Log24 Feb. 1 post The Search.

An image from that post (click to enlarge)—

http://www.log24.com/log/pix11/110201-TwoViews-300w.jpg

Here the apparently fragmented nature of the set of
images imagined as rising above the podium of the
Hall of Philosophy at Chautauqua rather naturally
echoes Stevens's "hundreds of eyes" remark.

Saturday, January 22, 2011

High School Squares*

Filed under: Uncategorized — Tags: — m759 @ 1:20 AM

The following is from the weblog of a high school mathematics teacher—

http://www.log24.com/log/pix11/110121-LatinSquares4x4.jpg

This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory

http://www.log24.com/log/pix11/110122-DiamondTheoryCover.jpg

Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.

For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem.  The result is 24 structural diagrams—

    Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares

Some of the squares are structurally congruent under the group of 8 symmetries of the square.

This can be seen in the following regrouping—

   Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares, with Congruent Squares Adjacent

      (Image corrected on Jan. 25, 2011– "seven" replaced "eight.")

* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.

Monday, December 27, 2010

Church Diamond

Filed under: Uncategorized — m759 @ 3:09 PM

IMAGE- The diamond property

Also known, roughly speaking, as confluence  or the Church-Rosser property.

From "NYU Lambda Seminar, Week 2" —

[See also the parent page Seminar in Semantics / Philosophy of Language or:
 What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn't Know To Ask)
]

A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn't matter which order the sub-expressions are evaluated in.

A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn't terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.

But the untyped lambda calculus enjoys some compensation for this weakness. It's Turing complete! It can represent any computation we know how to describe. (That's the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don't know how to rigorously define "any computation we know how to describe.") And in fact, it's been proven that you can't have both. If a computational system is Turing complete, it cannot be strongly normalizing.

There is no connection, apart from the common reference to an elementary geometric shape, between the use of "diamond" in the above Church-Rosser sense and the use of "diamond" in the mathematics of (Cullinane's) Diamond Theory.

Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.

Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —

 Link One — "Insane Symmetry"  (Click image for further details)—

http://www.log24.com/log/pix10B/101227-InsaneSymmetry.jpg

See also the quilt symmetry in this  journal on Christmas Day.

Link Two — Divine Symmetry

(George Steiner on the Name in this journal on Dec. 31 last year ("All about Eve")) —

"The links are direct between the tautology out of the Burning Bush, that 'I am' which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That 'I am' has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes 'a dead letter.'"

George Steiner, Grammars of Creation

(See also, from Hanukkah this year,  A Geometric Merkabah and The Dreidel is Cast.)

Link Three – Spanning the Arc —

Part A — Architect Louis Sullivan on "span" (see also Kindergarten at Stonehenge)

Part B — "Span" in category theory at nLab —

http://www.log24.com/log/pix10B/101227-nLabSpanImage.jpg

Also from nLab — Completing Spans to Diamonds

"It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property)."

The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).

They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).

For more on what the above San Francisco art curator is pleased to call "insane symmetry," see this journal on Christmas Day.

For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times  "The Stone" essay "Span: A Remembrance," from December 22—

“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”

Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”

See also the link from Christmas Day to remarks on Euclid and "architectonic" in Mere Geometry.

Tuesday, October 19, 2010

Savage Logic…

Filed under: Uncategorized — 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

Tuesday, September 7, 2010

Burning Patrick —

Filed under: Uncategorized — m759 @ 11:15 AM

Notes on Mathematics and Narrative

Background—

  1. The Burning Man in Bester's classic The Stars My Destination,
  2. The not-so-classic Hitler Plans Burning Man, and
  3. The cult film The Wicker Man

Commentary on The Wicker Man

Originally The Wicker Man  was not well-received by critics in the UK. It was considered
to be bizarre, disturbing, and uncomfortable, with the hasty editing making the story confusing
and out of order…. Today this movie is considered a cult classic and has been called
the “Citizen Kane  of horror films” by some reviewers. How did this film become a cult classic?

Real estate motto— Location, Location, Location.

Illustration— The fire leap scene from Wicker Man, filmed at Castle Kennedy

http://www.log24.com/log/pix10B/100907-WickerManFireLeapScene.jpg

From August 27

In today's New York Times, Michiko Kakutani reviews a summer thriller
by Kevin Guilfoile.  The Thousand  is in the manner of Dan Brown's
2003 The Da Vinci Code  or of Katherine Neville's 1988 The Eight .

From the review—

What connects these disparate events, it turns out, is a sinister organization
called the Thousand, made up of followers of the ancient Greek mathematician
and philosopher Pythagoras (yes, the same Pythagoras associated with
the triangle theorem that we learned in school).

As Mr. Guilfoile describes it, this organization is part Skull and Bones,
part Masonic lodge, part something much more twisted and nefarious….

The plot involves, in part,

… an eccentric artist’s mysterious masterwork, made up of thousands of
individually painted tiles that may cohere into an important message….

Not unlike the tiles in the Diamond Theory cover (see yesterday's post)
or, more aptly, the entries in this journal.
http://www.log24.com/log/pix10B/100827-GuilfoileTiles2.jpg

A brief prequel to the above dialogue—

http://www.log24.com/log/pix10B/100907-PatrickBlackburn-TheThousand.jpg

In lieu of songs, here is a passage by Patrick Blackburn
more relevant to the art of The Thousand

http://www.log24.com/log/pix10B/100907-PatrickBlackburn.jpg

See also the pagan fire leaping in Dancing at Lughnasa.

Friday, August 27, 2010

Mathematics and Narrative continued…

Filed under: Uncategorized — m759 @ 5:01 PM

Narrative Sequence

In today's New York Times, Michiko Kakutani reviews a summer thriller by Kevin Guilfoile.  The Thousand  is in the manner of Dan Brown's 2003 The Da Vinci Code  or of Katherine Neville's 1988 The Eight .

From the review—

What connects these disparate events, it turns out, is a sinister organization called the Thousand, made up of followers of the ancient Greek mathematician and philosopher Pythagoras (yes, the same Pythagoras associated with the triangle theorem that we learned in school).

As Mr. Guilfoile describes it, this organization is part Skull and Bones, part Masonic lodge, part something much more twisted and nefarious….

The plot involves, in part,

… an eccentric artist’s mysterious masterwork, made up of thousands of individually painted tiles that may cohere into an important message….

Not unlike the tiles in the Diamond Theory cover (see yesterday's post) or, more aptly, the entries in this journal.

http://www.log24.com/log/pix10B/100827-GuilfoileTiles2.jpg

Thursday, August 26, 2010

Home from Home continued

Filed under: Uncategorized — m759 @ 2:02 PM

Or— Childhood's Rear End

This post was suggested by…

  1. Today's New York Times
    "For many artists Electric Lady has become a home away from home…. For Jimmy Page the personal imprimaturs of Hendrix and Mr. Kramer made all the difference when Led Zeppelin mixed parts of 'Houses of the Holy' there in 1972."
  2. The album cover pictures for "Houses of the Holy"
  3. Boleskine House, home to Aleister Crowley and (occasionally) to Jimmy Page.

Related material:

The Zeppelin album cover, featuring rear views of nude children, was shot at the Giant's Causeway.

From a page at led-zeppelin.org—

http://www.log24.com/log/pix10B/100826-Causeway.jpg

See also Richard Rorty on Heidegger

Safranski, the author of ''Schopenhauer and the Wild Years of Philosophy,'' never steps back and pronounces judgment on Heidegger, but something can be inferred from the German title of his book: ''Ein Meister aus Deutschland'' (''A Master From Germany''). Heidegger was, undeniably, a master, and was very German indeed. But Safranski's spine-chilling allusion is to Paul Celan's best-known poem, ''Death Fugue.'' In Michael Hamburger's translation, its last lines are:

death is a master from Germany his eyes are blue
he strikes you with leaden bullets his aim is true
a man lives in the house your golden hair Margarete
he sets his pack on us he grants us a grave in the air
he plays with the serpents and daydreams death is a master from Germany

your golden hair Margarete
your ashen hair Shulamith.

No one familiar with Heidegger's work can read Celan's poem without recalling Heidegger's famous dictum: ''Language is the house of Being. In its home man dwells.'' Nobody who makes this association can reread the poem without having the images of Hitler and Heidegger — two men who played with serpents and daydreamed — blend into each other. Heidegger's books will be read for centuries to come, but the smell of smoke from the crematories — the ''grave in the air'' — will linger on their pages.

Heidegger is the antithesis of the sort of philosopher (John Stuart Mill, William James, Isaiah Berlin) who assumes that nothing ultimately matters except human happiness. For him, human suffering is irrelevant: philosophy is far above such banalities. He saw the history of the West not in terms of increasing freedom or of decreasing misery, but as a poem. ''Being's poem,'' he once wrote, ''just begun, is man.''

For Heidegger, history is a sequence of ''words of Being'' — the words of the great philosophers who gave successive historical epochs their self-image, and thereby built successive ''houses of Being.'' The history of the West, which Heidegger also called the history of Being, is a narrative of the changes in human beings' image of themselves, their sense of what ultimately matters. The philosopher's task, he said, is to ''preserve the force of the most elementary words'' — to prevent the words of the great, houses-of-Being-building thinkers of the past from being banalized.

Related musical meditations—

Shine On (Saturday, April 21, 2007), Shine On, Part II, and Built (Sunday, April 22, 2007).

Related pictorial meditations—

http://www.log24.com/log/pix10B/100826-CameronBlog.jpg

The Giant's Causeway at Peter J. Cameron's weblog

and the cover illustration for Diamond Theory (1976)—

http://www.log24.com/log/pix10B/100826-CoverArt.jpg

The connection between these two images is the following from Cameron's weblog today

… as we saw, there are two different Latin squares of order 4;
one, but not the other, can be extended to a complete set
of 3 MOLS [mutually orthogonal Latin squares].

The underlying structures of the square pictures in the Diamond Theory cover are those of the two different Latin squares of order 4 mentioned by Cameron.

Connection with childhood—

The children's book A Wind in the Door, by Madeleine L'Engle. See math16.com. L'Engle's fantasies about children differ from those of Arthur C. Clarke and Led Zeppelin.

Thursday, July 1, 2010

Darkness at Seven

Filed under: Uncategorized — m759 @ 7:00 PM

Hoax and Hype 
Four Years Ago Today—

Image-- Fanfiction-- Harry Potter and Plato's Diamond

There is Plato's diamond—

Image-- Plato's Diamond

and there is diamond theory

Google Search result for 'Diamond Theory'

… but there is no "Plato's Diamond Theory."

See, however, today's noon entry, "Plato's Code."

"You gotta be true to your code…" —Sinatra

Tuesday, May 4, 2010

Mathematics and Narrative, continued

Filed under: Uncategorized — Tags: — m759 @ 8:28 PM

Romancing the
Non-Euclidean Hyperspace

Backstory
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937

The cast of 1937's 'King Solomon's Mines' goes back to the future

For the 1937 grid, see Diamond Theory, 1937.

The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.

For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.

Monday, May 3, 2010

An Ordinary Evening

Filed under: Uncategorized — Tags: — m759 @ 8:00 PM

“…geometrically organized, with the parts labeled”

— Ursula K. Le Guin on what she calls “the Euclidean utopia

“There is such a thing as a tesseract.”

Madeleine L’Engle

Related material– Diamond Theory, 1937

Thursday, April 22, 2010

Mere Geometry

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

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

Stanford Encyclopedia of Philosophy

Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”

A non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–

“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”

Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”

Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.

* “Non-Euclidean” here means merely “other than  Euclidean.” No violation of Euclid’s parallel postulate is implied.

Tuesday, March 16, 2010

Variations on a Theme

Filed under: Uncategorized — m759 @ 2:29 PM

Today's previous entry was "Gameplayers of the Academy."

More on this theme–

David Corfield in the March 2010
European Mathematical Society newsletter

    "Staying on the theme of games, the mathematician
Alexandre Borovik* once told me he thinks of mathematics
as a Massively-Multiplayer Online Role-Playing Game. If
so, it would show up very clearly the difference between
internal and external viewpoints. Inside the game people
are asking each other whether they were right about
something they encountered in it– 'When you entered
the dungeon did you see that dragon in the fireplace or
did I imagine it?' But someone observing them from the
outside wants to shout: 'You’re not dealing with anything
real. You’ve just got a silly virtual reality helmet on.' External
nominalists say the same thing, if more politely, to
mathematical practitioners. But in an important way the
analogy breaks down. Even if the players interact with
the game to change its functioning in unforeseen ways,
there were the original programmers who set the bounds
for what is possible by the choices they made. When they
release the next version of the game they will have made
changes to allow new things to happen. In the case of
mathematics, it’s the players themselves who make these
choices. There’s no further layer outside.
    What can we do then instead to pin down internal reality?"

*See previous references to Borovik in this journal.

Related material:

The Diamond Theory vs. the Story Theory of Truth,

Infantilizing the Audience, and

It's Still the Same Old Story…God of War III

Thursday, February 18, 2010

Theories: An Outline

Filed under: Uncategorized — Tags: , — m759 @ 10:31 AM

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

— Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)

Sunday, December 20, 2009

The Test

Filed under: Uncategorized — m759 @ 11:00 AM

Dies Natalis of
Emil Artin

From the September 1953 Bulletin of the American Mathematical Society

Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–

"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."

Art Versus Chaos

http://www.log24.com/log/pix09A/091220-ForakisHypercube.jpg
From an exhibit,
"Reimagining Space
"

The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."

For related mathematical details see
Diamond Theory in 1937.

"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
Fritz Leiber, 1959

See also the Log24 entry for
Nov. 26,  2009, the date that
Forakis died.

"There is such a thing
as a tesseract."
Madeleine L'Engle, 1962

Wednesday, November 11, 2009

Triptych

Filed under: Uncategorized — Tags: , — m759 @ 10:31 AM

Triptych: 'Look at the Birdie,' 'A Wind in the Door,' and 'Diamond Theory'

Related material:

"Harrowing cuteness,"* The Eden Express, and a search on "harrowing" in this journal

* Perhaps a typo, but still a memorable phrase.

Sunday, November 8, 2009

H is for Hogwarts, continued

Filed under: Uncategorized — Tags: — m759 @ 9:48 AM

A Sequel to Koestler's
The Call Girls

Gilles Deleuze, Negotiations 1972-1990,
Columbia University Press paperback, 1997, p. 137–

"Academics' lives are seldom interesting."

But then there is Matt Lee of the University of Greenwich.

See his weblog subtitled "notes and thoughts on philosophy"… particularly his post "Diamond time, daimon time," of August 20, 2009.

See also my own post of August 20, 2009– "Sophists"– and my earlier post "Daimon Theory" of March 12, 2003:


Daimon Theory


Diamond Theory

More about Lee:

"Chaos majik is a form of modern witchcraft."

More about magick:

Noetic Symbology
(Log24 on October 25, 2009)

Some Related Log24 Posts

Sunday, September 27, 2009

Sunday September 27, 2009

Filed under: Uncategorized — m759 @ 3:00 AM
A Pleasantly
Discursive Treatment

In memory of Unitarian
minister Forrest Church,
 dead at 61 on Thursday:

NY Times Sept. 27, 2009, obituaries, featuring Unitarian minister Forrest Church

Unitarian Universalist Origins: Our Historic Faith

“In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history.”

Gravity’s Rainbow–

“For every kind of vampire, there is a kind of cross.”

Unitarian minister Richard Trudeau

“… I called the belief that

(1) Diamonds– informative, certain truths about the world– exist

the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that

(2) The theorems of Euclidean geometry are diamonds….

As the news about non-Euclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.

Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.

Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The Non-Euclidean Revolution, p. 255

H. S. M. Coxeter, 1987, introduction to Trudeau’s book

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…

A kind
 of cross:

Diamond formed by four diagonally-divided two-color squares

See both
Theme and
Variations
and some more
poetic remarks,

Mirror-Play
 of the Fourfold.

* As recent Log24 entries have pointed out, diamond theory (in the original 1976 sense) is a type of non-Euclidean geometry, since finite geometry is not Euclidean geometry– and is, therefore, non-Euclidean, in the strictest sense (though not according to popular usage).

Thursday, May 28, 2009

Thursday May 28, 2009

Filed under: Uncategorized — m759 @ 9:00 PM
Spelling

At right below, an image from the opening of Fox Studios Australia in Sydney on November 7, 1999.  The Fox ceremonies included, notably, Kylie Minogue singing “Diamonds are a Girl’s Best Friend.”

Red Windmill

Windmill image from diamond theory

Kylie Minogue

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

“There comes a time when you have
  learned enough to decide whether
  the way of the Craft is for you….

 First you will need to 
   prepare your sacred space….

 Calling the Corners (or Quarters)
  is something you will always do.”

— “Becoming a Witch” webpage

In related news:

CBS Evening News-- 'New York's Newest  Ballpark'

Happy birthday, Kylie.

Friday, April 10, 2009

Friday April 10, 2009

Filed under: Uncategorized — m759 @ 8:00 AM

Pilate Goes
to Kindergarten

“There is a pleasantly discursive
 treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”

— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
 remarks on the “Story Theory
 of truth as opposed to the
Diamond Theory” of truth in
 The Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.

Simplifying:

“Can a finite geometry describe physical space?”

Simplifying further:

“Yes. VideThe Eightfold Cube.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube

Sunday, March 29, 2009

Sunday March 29, 2009

Filed under: Uncategorized — Tags: — m759 @ 7:48 PM

Getting All
the Meaning In

Webpage heading for the
2009 meeting of the
American Comparative
Literature Association:

ACLA 2009 web page heading with map and alphabetic symbols

The mysterious symbols on
the above map suggest the
following reflections:


From A Cure of the Mind: The Poetics of Wallace Stevens, by Theodore Sampson, published by Black Rose Books Ltd., 2000–

Page x:

"… if what he calls 'the spirit's alchemicana' (CP [Collected Poems] 471) addresses itself to the irrational element in poetry, to what extent is such an element dominant in his theory and practice of poetry, and therefore in what way is Stevens' intricate verbal music dependent on his irrational use of language– a 'pure rhetoric of a language without words?' (CP 374)?"

Related material:

 

From "'When Novelists Become Cubists:' The Prose Ideograms of Guy Davenport," by Andre Furlani:

Laurence Zachar argues that Davenport's writing is situated "aux frontieres intergeneriques" where manifold modes are brought into concord: "L'etonnant chez Davenport est la facon don't ce materiau qui parait l'incarnation meme du chaos– hermetique, enigmatique, obscur, avec son tropplein de references– se revele en fait etre construit, ordonne, structure. Plus l'on s'y plonge, et plus l'on distingue de cohesion dans le texte." 'What astonishes in Davenport is the way in which material that seems the very incarnation of chaos– hermetic, enigmatic, obscure, with its proliferation of allusions– in fact reveals itself to be constructed, organized, structured. The more one immerses oneself in them the more one discerns the texts' cohesion.' (62).

Davenport also works along the intergeneric border between text and graphic, for he illustrates many of his texts. (1) "The prime use of words is for imagery: my writing is drawing," he states in an interview (Hoeppfner 123). Visual imagery is not subordinated to writing in Davenport, who draws on the assemblage practice of superimposing image and writing. "I trust the image; my business is to get it onto the page," he writes in the essay "Ernst Machs Max Ernst." "A page, which I think of as a picture, is essentially a texture of images. […] The text of a story is therefore a continuous graph, kin to the imagist poem, to a collage (Ernst, Willi Baumeister, El Lissitzky), a page of Pound, a Brakhage film" (Geography 374-75).

Note:

(1.) Davenport is an illustrator of books (such as Hugh Kenner's The Stoic Comedians and The Counterfeiters) and journals (such as The Kenyon Review, Parnassus, and Paideuma). His art is the subject of Erik Anderson Reece's monograph, A Balance of Quinces, which reveals the inseparable relationship between Davenport's literary and pictorial work.

References:

Davenport, Guy. The Geography of the Imagination. San Francisco: North Point Press, 1981. Rpt. New York: Pantheon, 1992.

Hoepffner, Bernard. "Pleasant Hill: An Interview with Guy Davenport." Conjunctions 24 (1995): 118-24.

Reece, Erik Anderson. A Balance of Quinces: The Paintings and Drawings of Guy Davenport. New York: New Directions, 1996.

Zachar, Laurence. "Guy Davenport: Une Mosaique du genres." Recherches Anglaises et Nord-Americaines 21 (1994): 51-63.

"… when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in."

— Guy Davenport, The Geography of the Imagination

See also last night's
commentary on the
 following symbols:

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

Saturday, March 28, 2009

Saturday March 28, 2009

Filed under: Uncategorized — m759 @ 11:07 PM

The Rest
of the Story

Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.

The rest of the story:
 

The Revelation Game
(continued from 7/26, 2008)

 
Lotteries
on Reba's
birthday,
2009
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
No belief,
no revelation

726
Revelation
without belief

378
Evening
(Belief)
Belief without
revelation

006
Belief and
revelation

091

Interpretations of the evening numbers–

The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.
 

Will: Do you like apples?     
Clark: Yeah.                       
Will: Well, I got her number.
 How do you like them apples?

— "Good Will Hunting

Kate Beckinsale in 'Underworld: Evolution'

The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–

Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–
 

Page 91 of 'The Innermost Kernel' by Suzanne Gieser, Springer 2005

The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."
 

A particularly
marked circle
 from March 15:

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

The circle above is
marked with a version of
the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

"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"

The square above is marked
with a graphic design
related to the four-diamond
figure of Jung's Aion.

Tuesday, March 17, 2009

Tuesday March 17, 2009

Filed under: Uncategorized — m759 @ 11:07 AM
Deep Structures

The traditional 'Square of Opposition'

The Square of Oppositon
at Stanford Encylopedia of Philosophy


The Square of Opposition diagram in its earliest known form

The Square of Opposition
in its original form

"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."

Edward Buckner at The Logic Museum

From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:
 

The Semiotic Square of Greimas

The Semiotic Square

"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."

* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter

Another version of the semiotic square:
 

Rosalind Krauss's version of the semiotic square, which she calls the Klein group

Krauss says that her figure "is, of course, a Klein Group."

Here is a more explicit figure representing the Klein group:

The Klein Four-Group, illustration by Steven H. Cullinane

There is also the logical
    diamond of opposition

The Diamond of Opposition (figure from Wikipedia)

A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
 a Marxist literary theorist:

"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–

The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.)

Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s1) for example is understood in relation to its contrary, "death" (s2). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (-S), i.e., "not-life" (-s1) and "not-death" (-s2). We would therefore be left with the following semiotic square (Fig. 1):

A semiotic 'diamond of opposition'

As Jameson explains in the Foreword to Greimas' On Meaning, "-s1 and -s2"—which in this example are taken up by "not-death" and "not-life"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, not-life would include more than merely death and not-death more than life.

* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976.

"The Game in the Ship cannot be approached as a job, a vocation, a career, or a recreation. To the contrary, it is Life and Death itself at work there. In the Inner Game, we call the Game Dhum Welur, the Mind of God."

The Gameplayers of Zan, by M.A. Foster

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
 Gravity's Rainbow

Crosses used by semioticians
to baffle their opponents
are illustrated above.

Some other kinds of crosses,
and another kind of opponent:

Monday, July 11, 2005

Logos
for St. Benedict's Day

Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.

Logo of Conference of Catholic Bishops     Logo of Stormfront website

Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth.  Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion.  I personally prefer the "diamond theory" of truth, represented by the logo below.

Illustration of the 2x2 case of the diamond theorem

See also the previous entry
(below) and the entries
  of 7/11, 2003.
 

Sunday, July 10, 2005

Mathematics
and Narrative

 
Click on the title
for a narrative about

Nikolaos K. Artemiadis

Nikolaos K. Artemiadis,
 (co-) author of

Artemiadis's 'History of Mathematics,' published by the American Mathematical Society
 

From Artemiadis's website:
1986: Elected Regular Member
of the Academy of Athens
1999: Vice President
of the Academy of Athens
2000: President
of the Academy of Athens
Seal of the American Mathematical Society with picture of Plato's Academy

"First of all, I'd like to
   thank the Academy…"

— Remark attributed to Plato

Sunday, March 15, 2009

Sunday March 15, 2009

Filed under: Uncategorized — Tags: — m759 @ 5:24 PM

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

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

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

Dan Brown's novel Angels & Demons introduced in the year 2000 the fictional academic discipline of "symbology" and a fictional Harvard professor of that discipline, Robert Langdon (named after ambigram* artist John Langdon).

Fictional Harvard professor of symbology Robert Langdon, as portrayed by Tom Hanks

Tom Hanks as Robert Langdon


A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)

"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus

* "…ambigrams were common in symbology…." —Angels & Demons
 

Monday, March 9, 2009

Monday March 9, 2009

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

Humorism

'The Manchurian Candidate' campaign button

"Always with a
little humor."
Dr. Yen Lo  

Diamond diagram of the four humors, the four qualities, the four elements, the four seasons, and four colors

From Temperament: A Brief Survey

For other interpretations
of the above shape, see
The Illuminati Diamond.

from Jung's Aion:

"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung, Collected Works, Vol. 9, Part II, paragraph 352

As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison
 
Click on image
for a related puzzle.
For a solution, see
 The Diamond Theorem.

A related note on
"Angels & Demons"
director Ron Howard:

Director Ron Howard with illustration of the fictional discipline 'symbology'
 
Click image for details.

Friday, January 30, 2009

Friday January 30, 2009

Filed under: Uncategorized — m759 @ 11:07 AM
Two-Part Invention

This journal on
October 8, 2008,
at noon:

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

— H. S. M. Coxeter, introduction to Richard J. Trudeau’s remarks on the “story theory” of truth as opposed to the “diamond theory” of truth in The Non-Euclidean Revolution

Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees.

Insidehighered.com on
the same day, October 8, 2008,
at 12:45 PM EDT

“Future readers may consider Updike our era’s Mozart; Mozart was once written off as a too-prolific composer of ‘charming nothings,’ and some speak of Updike that way.”

— Comment by BPJ

“Birthday, death-day–
 what day is not both?”
John Updike

Updike died on January 27.
On the same date,
Mozart was born.

Requiem

Mr. Best entered,
tall, young, mild, light.
He bore in his hand
with grace a notebook,
new, large, clean, bright.

— James Joyce, Ulysses,
Shakespeare and Company,
Paris, 1922, page 178

Related material:

Dec. 5, 2004 and

Inscribed carpenter's square

Jan. 27-29, 2009

Sunday, December 14, 2008

Sunday December 14, 2008

Filed under: Uncategorized — m759 @ 4:00 PM
Epigraphs

The New York Times of Sunday, May 6, 2007, on a writer of pulp fiction:

His early novels, written in two weeks or less, were published in double-decker Ace paperbacks that included two books in one, with a lurid cover for each. “If the Holy Bible was printed as an Ace Double,” an editor once remarked, “it would be cut down to two 20,000-word halves with the Old Testament retitled as ‘Master of Chaos’ and the New Testament as ‘The Thing With Three Souls.'”

Epigraph for Part One:

Ours is a very gutsy religion, Cullinane.

James A. Michener

Lurid cover:
The Pussycat

The Pussycat of the film 'The Owl and the Pussycat,' starring Barbra Streisand


Epigraph for Part Two:

Beware lest you believe that you can comprehend the Incomprehensible….

Saint Bonaventure

Lurid cover:
The Owl

Diamond Theory cover, said to resemble Proginoskes in 'A Wind in the Door'

Click on the image for a
relevant Wallace Stevens poem.

Thursday, November 6, 2008

Thursday November 6, 2008

Filed under: Uncategorized — m759 @ 10:07 AM
Death of a Classmate

Michael Crichton,
Harvard College, 1964

Authors Michael Crichton and David Foster Wallace in NY Times obituaries, Thursday, Nov.  6, 2008

Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries

The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.

“Crystal”

Crystal from 'Diamond Theory'

“Dragon”

(from Crichton’s
Jurassic Park)–


Dragon Curve from 'Jurassic Park'

For the mathematics
(dyadic harmonic analysis)
relating these two figures,
see Crystal and Dragon.

Some philosophical
remarks related to
the Harvard background
  that Crichton and I share–

Hitler’s Still Point

and
The Crimson Passion.

Sunday, October 12, 2008

Sunday October 12, 2008

Filed under: Uncategorized — 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   

Wednesday, October 8, 2008

Wednesday October 8, 2008

Filed under: Uncategorized — m759 @ 12:00 PM

Serious Numbers

A Yom Kippur
Meditation

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution

Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–

"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"

(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)

Richard J. Trudeau on the 'Story Theory' of truth

Excerpt from
The Non-Euclidean Revolution

What does this have to do with numbers?

Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.

Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.

For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: Uncategorized — Tags: — m759 @ 3:00 PM
Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

Lucia Joyce

 

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.

 

Log 24, Sept. 3, 2003:
 
Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year's entry on this date: 

 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

 

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and


FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles
.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.
 

Friday, July 11, 2008

Friday July 11, 2008

Filed under: Uncategorized — m759 @ 1:00 PM
AND MORE LOGOS:

“Serious numbers will
always be heard.”
Paul Simon  

http://www.log24.com/log/pix08/080711-DowLg.jpg

http://www.log24.com/log/pix08/080711-NYSE.jpg

http://www.log24.com/log/pix08/080711-HSBClogo.jpg

The HSBC Logo Designer —

Henry Steiner

He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.

Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors’ Club.

His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995).

Yaneff.com

Related material
from the past

Wittgenstein and Fly from Fly-Bottle

Fly from Fly Bottle:

Graphic structures from Diamond Theory and from Kyocera logo

Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
  (Cambridge U. Press, 1989, p. 477) —

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

Related material
from today —

Escape from a
  cartoon graveyard:

http://www.log24.com/log/pix08/080711-BabyBlues.jpg

Saturday, June 21, 2008

Saturday June 21, 2008

Filed under: Uncategorized — m759 @ 6:00 AM
The Kyoto Prize

for lifetime achievement
in arts and philosophy
this year goes to
Charles Taylor,

Charles Margrave Taylor, professor emeritus of philosophy at McGill University

Montreal philosophy professor.

“The Kyoto Prize has been given in three domains since 1984: advanced technology, basic sciences, and the arts and philosophy. It is administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”

Kyocera Logo

“The Kyocera brand symbol is composed of a corporate mark and our corporate logotype. The mark represents the initial ‘K’ (for Kyocera) encircling a ‘C’ (for ceramics). It was introduced in October 1982 when the company name was changed from ‘Kyoto Ceramic’ to ‘Kyocera.'”

global.kyocera.com

Related material —

Wittgenstein and Fly from Fly-Bottle

Fly from Fly Bottle:

Graphic structures from Diamond Theory and from Kyocera logo

Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
  (Cambridge U. Press, 1989, p. 477) —

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

See also Talking of Michelangelo.

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: Uncategorized — m759 @ 5:01 AM
A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings.  Here is a related commentary on a different incarnation of 441.  (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the nm elements not in S.  Then show that G permutes the m elements S into

n(n -1) … (nk + 1)
____________________

m(m – 1) … (mk + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

Tuesday, March 4, 2008

Tuesday March 4, 2008

Filed under: Uncategorized — m759 @ 1:00 PM
… And for a
    Swiftly Tilting
       Shadowed Planet …

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

The pen's point:

Log24, Dec. 11, 2006

SINGER, ISAAC:
"Are Children the
Ultimate Literary Critics?"
— Top of the News 29
(Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"

An Annotated Listing
of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."

A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time,"
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937
 
A Swiftly Tilting Planet  is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

Saturday, February 16, 2008

Saturday February 16, 2008

Filed under: Uncategorized — m759 @ 9:29 AM
Bridges
Between Two Worlds


From the world of mathematics…


“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”

— Scott Carnahan at Secret Blogging Seminar, December 14, 2007

Carnahan’s remark in context:

“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GLr over function fields…. In the beginning, he drew a diagram….

If we remove all of the explanatory text, the diagram looks like this:

CheeWhye Diagram

I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.

Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”

Compare and contrast the above…

… to the world of Rudolf Kaehr:

Rudolf Kaehr on 'Diamond Structuration'

The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.

Another bridge…

Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–

From the world of mathematics

Relationship of diamond theory to other fields

“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane

… to the world of Rudolf Kaehr:

Relationship of PolyContextural Logic (PCL) to other fields

Related material:

For further details on
the “diamond theory” of
Cullinane, see

Finite Geometry of the
Square and Cube
.

For further details on
the “diamond theory” of
Kaehr, see

Rudy’s Diamond Strategies.

Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:

“Do not let me hear
Of the wisdom
of old men,
but rather of
their folly”
 
Four Quartets   

Anthony Hopkins in 'Slipstream'

Anthony Hopkins
in the film
Slipstream

Anthony Hopkins  
in the film “Proof“–

Goddamnit, open
the goddamn book!
Read me the lines!

Wednesday, July 25, 2007

Wednesday July 25, 2007

Filed under: Uncategorized — m759 @ 9:00 AM
The Comedy of
George Tabori

George Tabori

From AP “Obituaries in the News”–
Filed with The New York Times
at 11:16 p.m. ET July 24, 2007–

George Tabori

“BERLIN (AP) — Hungarian-born playwright and director George Tabori, a legend in Germany’s postwar theater world whose avant-garde works confronted anti-Semitism, died Monday [July 23, 2007]. He was 93.

Tabori, who as recently as three years ago dreamed of returning to stage to play the title role in Shakespeare’s ‘King Lear,’ died in his apartment near the theater, the Berliner Ensemble said Tuesday, noting that friends and family had accompanied him through his final days. No cause of death was given.

Born into a Jewish family in Budapest on May 24, 1914, Tabori fled in 1936 to London, where he started working for the British Broadcasting Corp., and became a British citizen. His father, and other members of his family, were killed at Auschwitz.

Tabori moved to Hollywood in the 1950s, where he worked as a scriptwriter, most notably co-writing the script for Alfred Hitchcock’s 1953 film, ‘I Confess.’

He moved to Germany in the 1970s and launched a theater career that spanned from acting to directing to writing. He used sharp wit and humor in his plays to examine the relationship between Germany and the Jews, as well as attack anti-Semitism.

Among his best-known works are ‘Mein Kampf,’ set in the Viennese hostel where Adolf Hitler lived from 1910-1913, and the ‘Goldberg Variations,’ both dark farces that poke fun at the Nazis.”

From Year of Jewish Culture:

“The year 2006 marks the 100th anniversary of the establishment of the Jewish Museum in Prague.”

From the related page Programme (October-December):

Divadlo v Dlouhé
George Tabori: GOLDBERGOVSKÉ VARIACE / THE GOLDBERG VARIATIONS, 19 October, 7 p.m. A comedy on creation and martyrdom.”

Variations on
Birth and Death

From Log24 on the date of
the Prague production of the
Tabori “Goldberg Variations,”
an illustration in honor of
Sir Thomas Browne, who
was born, and died,
on that date:

Laves tiling

The above is from
Variable Resolution 4–k Meshes:
Concepts and Applications
(pdf),
by Luiz Velho and Jonas Gomes.

See also Symmetry Framed
and The Garden of Cyrus.

Variations on
the Afterlife

 From Log24
on the date of
Tabori’s death:

Theme

(Plato, Meno)

Plato's Diamond colored

and Variations:

Diamond Theory cover, 1976

Click on “variations” above
for some material on
the “Goldberg Variations”
of Johann Sebastian Bach.

 

Monday, July 23, 2007

Monday July 23, 2007

Filed under: Uncategorized — Tags: — m759 @ 7:59 AM
 
Today’s Birthday:
Daniel Radcliffe
(“Harry Potter”)

Harry Potter and the Philosopher's Stone DVD

Theme

(Plato, Meno)

Plato's Diamond colored

and Variations:

Diamond Theory cover, 1976
Click on picture for details.

“A diamond jubilance
beyond the fire,
That gives its power
to the wild-ringed eye”

— Wallace Stevens,
“The Owl in the Sarcophagus”

Thursday, March 1, 2007

Thursday March 1, 2007

Filed under: Uncategorized — Tags: — m759 @ 6:29 AM

Senior Honors

Notes in Memory of
a Father, a Son, and a Holy Ghost

From the obituary in today's New York Times of historian Arthur M. Schlesinger Jr.–

"Mr. Schlesinger, partly through his appreciation of history, fully realized his good fortune. 'I have lived through interesting times and had the luck of knowing some interesting people,' he wrote.

A huge part of his luck was his father, who guided much of his early research, and even suggested the topic for his [Harvard] senior honors: Orestes A. Brownson, a 19th-century journalist, novelist and theologian. It was published by Little, Brown in 1938 as 'Orestes A. Brownson: A Pilgrim's Progress.'"

Douglas Martin

From The Catholic Encyclopedia:

"It is sufficient for true knowledge that it affirm as real that which is truly real."

Article on Ontologism

From The Diamond Theory of Truth:

"Was there really a cherubim waiting at the star-watching rock…?
Was he real?
What is real?

— Madeleine L'Engle, A Wind in the Door, Farrar, Straus and Giroux, 1973, conclusion of Chapter Three, "The Man in the Night"

"Oh, Euclid, I suppose."

— Madeleine L'Engle, A Wrinkle in Time, Farrar, Straus and Giroux, 1962, conclusion of Chapter Five, "The Tesseract"

Related material: Yesterday's first annual "Tell Your Story Day" at Harvard and yesterday's entry on Euclid.

Tuesday, February 27, 2007

Tuesday February 27, 2007

Filed under: Uncategorized — m759 @ 9:25 PM
Suggested by today’s 
New York Times story
on a Harvard student’s
research on pattern in
Islamic art —

and in memory of
George Sadek

From Log24 in July 2005:

Intersections

A Trinity Sunday sermon
quotes T. S. Eliot:

“… to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint.”

See also The Diamond Project.

Related material:

                                  ” … an alphabet
By which to spell out holy doom and end,
A bee for the remembering of happiness.”

— Wallace Stevens,
“The Owl in the Sarcophagus”

The image “http://www.log24.com/theory/images/HeathI47A-Illustrations.jpg” cannot be displayed, because it contains errors.

Some context for these figures:
The Diamond Theory of Truth

Monday, December 11, 2006

Monday December 11, 2006

Filed under: Uncategorized — m759 @ 7:20 AM
Geometry and Death

J. G. Ballard on “the architecture of death“:

“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”

The Guardian, March 20, 2006

Edward Hirsch on Lorca:

“For him, writing is a struggle both with geometry and death.”

— “The Duende,” American Poetry Review, July/August 1999

“Rosenblum writes with
absolute intellectual honesty,
and the effect is sheer liberation….
The disposition of the material is
a model of logic and clarity.”

Harper’s Magazine review
quoted on back cover of
Cubism and Twentieth-Century Art,
by Robert Rosenblum
(Abrams paperback, 2001)

SINGER, ISAAC:
“Are Children the Ultimate Literary Critics?”
 — Top of the News 29 (Nov. 1972): 32-36.
“Sets forth his own aims in writing for children
 and laments ‘slice of life’ and chaos in
children’s literature. Maintains that children
like good plots, logic, and clarity,
and that they have a concern for
‘so-called eternal questions.'”

An Annotated Listing of Criticism
by Linnea Hendrickson

“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”

A Swiftly Tilting Planet,
by Madeleine L’Engle

The image “http://www.log24.com/log/pix06B/061211-Swiftly2.gif” cannot be displayed, because it contains errors.

For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

For a more adult audience —

In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):

“… Philip of Austria… longing to discover the Muse and the Angel in theology, found himself imprisoned by the Duende of cold ardors in that masterwork of the Escorial, where geometry abuts with a dream and the Duende wears the mask of the Muse for the eternal chastisement of the great king.”


Perhaps. Or perhaps Philip, “the lonely
hermit of the Escorial,” is less lonely now.

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