Log24

Monday, May 9, 2022

An Old Amazon Tale

Filed under: General — Tags: — m759 @ 9:28 pm

From The DIadem of Death (May 29, 2008) —

Wonder Woman and the Secret of the Magic Tiara

Wonder Woman and the Secret of the Magic Tiara-- The End

Sunday, May 8, 2022

In Memory of a Comic-Book Artist . . .

Filed under: General — Tags: , , — m759 @ 3:28 pm

… who reportedly died on Friday, May 6, 2022.

See as well Wonder Woman in this  journal.

Friday, May 6, 2022

Interality and the Bead Game

Filed under: General — Tags: , , — m759 @ 3:00 pm

WIkipedia on the URL suffix ".io" —

"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companiesbrowser games,
and other online services."

An association with the Bead Game from a post of April 7, 2018

IMAGE- 'Solomon's Cube'

Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

The recent use by a startup company of the URL "interality.io" suggests
a fourth  reading for the 1955 list of Lévi-Strauss — in and out
i.e., inner and outer group automorphisms —  from a 2011 post
on the birthday of T. S. Eliot :

A transformation:

Inner and outer group automorphisms

Click on the picture for details.

Interality and the I Ching

Filed under: General — Tags: , , — m759 @ 12:57 am

See "Flusser and the I Ching," by Peter Zhang.

Zhang has written extensively on the concept of "interality,"
a term coined by his colleague Geling Shang.

For interality as the mathematics underlying the natural
automorphism group of the I Ching, see my own work.

Thursday, May 5, 2022

“Interality” as a Metaverse Term

Filed under: General — Tags: , , — m759 @ 11:01 pm

See also "Interality" in this  journal.

Update of 8:56 AM ET
Friday, May 6, 2022:

“You have to all have a shared language of all this stuff,
otherwise it can get pretty confusing,” Waldron said.

The Waldron quote is from . . .

"‘Doctor Strange 2’ Writer Michael Waldron Wishes
That He Didn’t Make So Many Multiverse Rules In ‘Loki’
 .

Later, at 9:29 AM ET . . .

See as well other posts now tagged Strange Change.

Friday, April 29, 2022

Code Bleu

Filed under: General — Tags: , , , — m759 @ 2:17 pm

From The New York Times  on May 5, 2011 —

"… What Paris says to me is love story, awash with painters,
shots of the Seine, Champagne. Thank God I have a
can’t-miss notion to sell you. I call it ‘Midnight in Paris.’ ”

“Romantic title,” I had to admit. “Is there a script?”

“Actually, there’s nothing on paper yet, but I can spitball
the main points,” he said, slipping on his tap shoes.

“Maybe some other time,” I said, mindful of Cubbage’s
unbroken string of theatrical Hiroshimas.

— Woody Allen

The above passage is in memory of a French film director
who, like the reporter in yesterday's post Primary Colors,
reportedly died on April 21, 2022.

See also Aitchison at Hiroshima and Easter for Aitchison.

Thursday, April 28, 2022

Primary Colors

Filed under: General — Tags: , , , — m759 @ 5:05 pm

Saturday, January 29, 2022

California Dreamin’

Filed under: General — Tags: , , , — m759 @ 2:38 pm

Bombogenesis

Filed under: General — Tags: , , — m759 @ 12:18 pm

On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: , , , , , — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.

* See a Log24 post of June 1, 2013.

Thursday, January 27, 2022

The Lexicographic Octad Generator

Filed under: General — Tags: , , , , — m759 @ 2:30 pm

A Lexicographic Basis for the Binary Golay Code:

Brouwer and Guven — "Long ago," in 
"The generating rank of the space of short vectors
in the Leech lattice mod 2," by 
Andries Brouwer & Cicek Guven,
https://www.win.tue.nl/~aeb/preprints/udim24a.pdf —

"One checks by computer" that this is a basis:

000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100

(Copied from the Brouwer-Guven paper)

_________________________________________________

Adlam at Harvard — 
"Constructing the Extended Binary Golay Code,"
by Ben Adlam, Harvard University, August 9, 2011,
https://fliphtml5.com/llqx/wppz/basic —

Adlam also asserts, citing a reference, that this same
set of twelve vectors is a basis:

000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100

(Copied from the Adlam paper)
__________________________________________________

Sources —

Background for Log24 posts on 'Embedding Change'

"One checks by computer" —

At http://magma.maths.usyd.edu.au/calc/ —

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |  
> [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],
> [0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],
> [0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1],
> [0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,1,0,1,0,1,1,0],
> [0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1],
> [0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,1,1,1,0,1,0],
> [0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,1,1,1,0,0,0],
> [0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,1,0,1],
> [0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,1,1,1,0],
> [1,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0]>;       
> Dimension(G);
12

Friday, December 25, 2020

Change Arises: Mathematical Examples

Filed under: General — Tags: , , — m759 @ 12:59 am

From old posts tagged Change Arises

From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group
Click on image for details.

For the eightfold cube
as it relates to Klein’s
simple group, see
A Reflection Group
of Order 168
.”

For an rather more
complicated theory of
Klein’s simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

The phrase “change arises” is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian

 

A related recent illustration from Quanta Magazine —

The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.

Thursday, December 24, 2020

Change Arises: A Literary Example

Filed under: General — Tags: , — m759 @ 12:12 pm

The “Change Arises” part of the title refers to the previous post.
The 1905 “geometric object” there, a 4×4 square, appeared earlier,
in 1869, in a paper by Camille Jordan. For that paper, and the
“literary example” of the title, see “Ici vient M. Jordan .”

This  post was suggested by the appearance of Jordan in today’s
memorial post for Peter M. Neumann by Peter J. Cameron.

Related remarks on Jordan and “geometrical objects” from 2016 —

These reflections are available from their author as a postprint.

Change Arises

Filed under: General — Tags: , — m759 @ 1:44 am

See posts so tagged.

“Change arises from the structure of the object.” — Arkani-Hamed

Related material from 1936 —

Related material from 1905, with the “object” a 4×4 array —

Related material from 1976, with the “object”
a 4×6 array — See Curtis.

Related material from 2018, with the “object”
a cuboctahedron — See Aitchison.

Friday, January 24, 2020

Oettinger Quote

Filed under: General — Tags: , , — m759 @ 7:39 pm

Quote Investigator on May 4, 2010* —

"QI  has traced the core of the quotation
to the work of an early researcher in
artificial intelligence, Anthony Oettinger,
who was trying to get a computer to
manipulate the English language."

See as well Oettinger in 1963.

"And that  was the state  of the  art."
— Adapted from Stephen Sondheim

* Cf.  this  journal on that date.

Lucido Dreaming

Filed under: General — Tags: , , , — m759 @ 12:51 pm

Anthony Powell's 'O, How the Wheel Becomes It!' along with Laertes' comment 'This nothing's more than matter.'

(From "Today's Sermon," Jan. 24, 2010.)

Smart Jewish Girl*…

Filed under: General — Tags: , , , — m759 @ 11:41 am

Suggests the word dreamlogic. And so

"You are getting sleepy, very sleepy "

"In this state of free-association, each new thought
resembles or overlaps or somehow connects-to
the previous thought. As our alertness continues to fall —
as we continue to grow more tired — we lose contact with
external reality.

'The sweetness/ of the gentle world you had made for him
dissolving beneath/ his drowsy eyelids, into the foretaste of
sleep — .'  (Rilke, transl. Stephen Mitchell.) Eventually we
sleep and dream."

— Edge.org, "Dream-logic, the Internet and Artificial Thought,"
by David Gelernter [7.7.10]

* Aimee Lucido

Wheel Turnin’ ’Round and ’Round

Filed under: General — Tags: , , — m759 @ 2:21 am

 

Exploring Fiction

Filed under: General — Tags: , — m759 @ 12:34 am

 

Janet Burroway's 'Writing Fiction: A Guide to Narrative Craft,' fifth edition, with I Ching coins on cover

Related material —

"Off Broadway," a post from the date
of Eric James's death.

Thursday, June 6, 2019

Strange Dots

Filed under: General — Tags: , , — m759 @ 3:43 pm

(Continued)

Wednesday, June 5, 2019

The Tigerman Kaballah

Filed under: General — Tags: , , , , — m759 @ 4:00 am

See also Life of Pi   in this  journal.

Sunday, May 26, 2019

Sunday Shul: Connecting the Dots

Filed under: General — Tags: , — m759 @ 9:00 am

"It's very easy to say, 'Well, Jeff couldn't quite connect these dots,'"
director Jeff Nichols told BuzzFeed News. "Well, I wasn't actually
looking at the dots you were looking at."

— Posted on March 21, 2016, at 1:11 p.m,
Adam B. Vary, BuzzFeed News Reporter

"Magical arrays of numbers have been the talismans of mathematicians and mystics since the time of Pythagoras in the sixth century B.C. And in the 16th century, Rabbi Isaac ben Solomon Luria devised a cosmological world view that seems to have prefigured superstring theory, at least superficially. Rabbi Luria was a sage of the Jewish cabalist movement — a school of mystics that drew inspiration from the arcane oral tradition of the Torah.

According to Rabbi Luria's cosmology, the soul and inner life of the hidden God were expressed by 10 primordial numbers [sic ], known as the sefirot. [Link added.]"

— "Things Are Stranger Than We Can Imagine,"
by Malcolm W. BrowneNew York Times Book Review ,
Sunday, March 20, 1994

Burning Bright

Filed under: General — Tags: , , — m759 @ 12:00 am

Gell-Mann's 'eightfold way' as 'a mosaic of simple triangular building blocks' — George Johnson, 1999

Compare and contrast with . . .

The Brightburn Logo:

Related material from the May 12 post

"The Collective Unconscious
in a Cartoon Graveyard
" —

"When they all finally reach their destination —
a deserted field in the Florida Panhandle…." 

" When asked about the film's similarities to the 2015 Disney movie Tomorrowland , which also posits a futuristic world that exists in an alternative dimension, Nichols sighed. 'I was a little bummed, I guess,' he said of when he first learned about the project. . . . 'Our die was cast. Sometimes this kind of collective unconscious that we're all dabbling in, sometimes you're not the first one out of the gate.' "

Saturday, May 25, 2019

Brand On Breyer

Filed under: General — Tags: , — m759 @ 1:37 pm

Or:  One Quark for Muster Mark​

"In this way the eight trigrams came into being." — Richard Wilhelm

Detail:

Midrash:

Friday, May 24, 2019

Hidden Eightfold Patterns

Filed under: General — Tags: , — m759 @ 5:23 pm

Saturday, March 3, 2018

Thanking the Academy

Filed under: General — Tags: , — m759 @ 11:05 pm

Blackboard Jungle , 1955 —

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

“Every transition from major to minor in a sonata, every transformation
of a myth or a religious cult, every classical or artistic formulation was,
I realized in that flashing moment, if seen with a truly meditative mind,
nothing but a direct route into the interior of the cosmic mystery….”

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye“

Friday, July 29, 2016

The Cumberbatch Question

Filed under: General — Tags: , — m759 @ 2:29 pm

"How do I get from here to there?"

See also Compass in this journal.

Thursday, April 14, 2016

Strange Awards

Filed under: General,Geometry — Tags: , , — m759 @ 10:38 am

From a review of a play by the late Anne Meara* —

"Meara, known primarily as an actress/comedian
(half of the team of Stiller & Meara, and mother of
Ben Stiller), is also an accomplished writer for the
stage; her After Play  was much acclaimed….
This new, more ambitious piece starts off with a sly
send-up of awards dinners as the late benefactor of
a wealthy foundation–the comically pixilated scientist
Herschel Strange (Jerry Stiller)–is seen on videotape.
This tape sets a light tone that is hilariously
heightened when John Shea, as Arthur Garden,
accepts the award given in Strange's name." 

Compare and contrast —

A circular I Ching

I of course prefer the Galois I Ching .

* See the May 25, 2015, post The Secret Life of the Public Mind.

Saturday, November 21, 2015

Brightness at Noon*

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

A recent not-too-bright book from Princeton —

Some older, brighter books from Tony Zee

Fearful Symmetry  (1986) and
Quantum Field Theory in a Nutshell  (2003).

* Continued.

Wednesday, August 5, 2015

Inking

Filed under: General — Tags: , , , , — m759 @ 9:00 am

From Doctorow's 'Jolene: A Life'

See also Go Set a Structure and Tombstone.

Tuesday, October 28, 2014

Raiders of the Lost Symbol

Filed under: General,Geometry — Tags: — 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”

Wednesday, April 2, 2014

Change Arises

Filed under: General — Tags: , — m759 @ 9:00 am

IMAGE- Search for the source of the quotation 'Change arises from the structure of the object'

For a different view of change arising, click on the tag above.

Tuesday, April 1, 2014

Kindergarten Geometry

Filed under: General,Geometry — Tags: , , , — m759 @ 11:22 pm

(Continued)

A screenshot of the new page on the eightfold cube at Froebel Decade:

IMAGE- The eightfold cube at Froebel Decade

Click screenshot to enlarge.

Saturday, November 30, 2013

Waiting for Ogdoad

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

Continued from October 30 (Devil’s Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
Classic of Change ,” the I Ching .

The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching  as the corners of a cube.

The Eightfold Cube and its Inner Structure

See also

(Click for clearer image.)

The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching  trigrams and as
seven natural ways of slicing the cube.

For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube  in a post of April 9, 2012.

Sunday, November 18, 2012

Sermon

Filed under: General,Geometry — 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, May 24, 2012

Dark, Dark, Dark

Filed under: General — Tags: , — m759 @ 12:00 pm

Damnation Morning

From her left arm hung a black handbag that closed with a drawstring and from which protruded the tip of a silvery object about which I found myself apprehensively curious.

Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question.

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


IMAGE- 'Eight-limbed asterisk' of Fritz Leiber (square version)

Except for the bangs she wore her hair pinned up. Her ears were flat, thin-edged, and nicely shaped, with the long lobes that in Chinese art mark the philosopher. Small square silver flats with rounded corners ornamented them.

Her face might have been painted by Toulouse-Lautrec or Degas. The skin was webbed with very fine lines; the eyes were darkly shadowed and there was a touch of green on the lids (Egyptian?—I asked myself); her mouth was wide, tolerant, but realistic. Yes, beyond all else, she seemed realistic.

Mary Karr

You’re not afraid to show yourself at your lowest ebb. In Lit, you stop breast-feeding because you’ve started drinking again. You describe yourself hiding in a closet with a bottle of whiskey, a bottle of Listerine, and a spit bowl.

It’s not a proud moment. The temptation in Lit was to either make myself seedy or show some glamour. But there wasn’t any. It was just dark, dark, dark for days. Ugly.

Were you surprised by how deeply people related to this dark stuff?

If I’m doing my job then I’m able to make the strange seem familiar. Bad memoirs try to make the strange stranger, to provide something for people to gawk at. I try to create an experience where no matter how bizarre something is, it seems normal. I don’t want readers to balk, I want them to be in the experience. My goal isn’t for people to go, “Oh, poor little Mary Karr,” but rather to have the reader go, “I can be an asshole too,” or just to have enthusiasm for the possibility for change.

Friday, April 27, 2012

Paradigms Lost continues…

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

This post was suggested by Paradigms Lost
(a post cited here a year ago today),
by David Weinberger's recent essay "Shift Happens,"
and by today's opening of "The Raven."

David Weinberger in The Chronicle of Higher Education April 22

"… Kuhn was trying to understand how Aristotle could be such a brilliant natural scientist except when it came to understanding motion. Aristotle's idea that stones fall and fire rises because they're trying to get to their natural places seems like a simpleton's animism.

Then it became clear to Kuhn all at once. Ever since Newton, we in the West have thought movement changes an object's position in neutral space but does not change the object itself. For Aristotle, a change in position was a change in a quality of the object, and qualitative change tended toward an asymmetric actualization of potential: an acorn becomes an oak, but an oak never becomes an acorn. Motion likewise expressed a tendency for things to actualize their essence by moving to their proper place. With that, 'another initially strange part of Aristotelian doctrine begins to fall into place,' Kuhn wrote in The Road Since Structure ."

Dr. John Raven (of Raven's Progressive Matrices)

"… these tools cannot be immediately applied within our current workplaces, educational systems, and public management systems because the operation of these systems is determined, not by personal developmental or societal needs, but by a range of latent, rarely discussed, and hard to influence sociological forces.

But this is not a cry of despair: It points to another topic which has been widely neglected by psychologists: It tells us that human behaviour is not  mainly determined by internal  properties— such as talents, attitudes, and values— but by external  social forces. Such a transformation in psychological thinking and theorising is as great as the transformation Newton introduced into physics by noting that the movement of inanimate objects is not determined by internal, 'animistic,' properties of the objects but by invisible external forces which act upon them— invisible forces that can nevertheless be mapped, measured, and harnessed to do useful work for humankind.

So this brings us to our fourth conceptualisation and measurement topic: How are these social forces to be conceptualised, mapped, measured, and harnessed in a manner analogous to the way in which Newton made it possible to harness the destructive forces of the wind and the waves to enable sailing boats to get to their destinations?"

Before Newton, boats never arrived?

Monday, January 30, 2012

Enda’s Game*

Filed under: General — Tags: , — m759 @ 7:00 pm

The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review  that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")

"Lastly, tengwesta  [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe  (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe  only in great need and urgency, and especially when lambe  is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe  will use lambe  when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe  that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe  is also often taking place; for the will to converse in lambe  is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."

* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."

   — Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar  39 (July 1998). This is copyrighted but freely available on the Web.

** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is  such a thing as a tesseract."

Saturday, November 5, 2011

Shadows

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 am

Between the idea
And the reality
Between the motion
And the act
Falls the Shadow

— T. S. Eliot, "The Hollow Men"

A passage quoted here on this date in 2005—

Douglas Hofstadter on his magnum opus:

“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions
by some central solid essence."

This refers to Hofstadter's cover image:

IMAGE- http://www.log24.com/log/pix11C/111105-GEBshadows.jpg

Also from this date in 2005:

IMAGE- www.log24.com/theory/images/GEB.jpg
 
BackgroundYesterday's link Change Logos,
                         
and Solid Symmetry.
 
Midrash:         Hearts of Darkness.

Friday, November 4, 2011

Logos

Filed under: General — Tags: , , — m759 @ 9:00 am

Continued from All Hallows Eve

The Belgian Lottery logo

http://www.log24.com/log/pix11C/111104-BelgianLotteryLogo-256w.jpg

The Belgian Lottery was a sponsor of 
last month's 25th Solvay Conference —

"The Theory of the Quantum World,"
  Brussels, October 19-22, 2011.

See also this journal in October and Change Logos

http://www.log24.com/log/pix09/090309-SqInSpace2.jpg

(Physicists will recognize the kinship
with the coat of arms of Niels Bohr.)

Friday, July 1, 2011

Symmetry Review

Filed under: General,Geometry — Tags: , , — m759 @ 5:01 am

Popular novelist Dan Brown is to speak at Chautauqua Institution on August 1.

This suggests a review of some figures discussed here in a note on Brown from February 20, 2004

IMAGE- Like motions of a pattern's parts can induce motions of the whole. Escher-'Fishes and Scales,' Cullinane-'Invariance'

Related material: Notes from Nov. 5, 1981, and from Dec. 24, 1981.

For the lower figure in context, see the diamond theorem.

Saturday, May 28, 2011

Savage Detectives

Filed under: General,Geometry — 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"

Saturday, January 22, 2011

High School Squares*

Filed under: General,Geometry — 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.

Tuesday, October 19, 2010

Savage Logic…

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

and the New York Lottery

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

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

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

by Clifford Geertz

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

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

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

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

Sunday, March 15, 2009  5:24 PM

Philosophy and Poetry:

The Origin of Change

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

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

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

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

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

Sunday, March 15, 2009  11:00 AM

Ides of March Sermon:

Angels, Demons,
"Symbology"

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

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

"Perfer et obdura: multo graviora tulisti."

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

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

This journal
on 9 March:

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

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

Related material:
The Illuminati Diamond:

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

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

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

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:07 am

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).

Notes:

  Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion  of the Self as a four-diamond figure:

http://www.log24.com/log/pix10A/100615-JungImago.gif

and Cullinane’s purely geometric four-diamond figure:

http://www.log24.com/log/pix10A/100615-FourD.gif

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Monday, April 26, 2010

Types of Ambiguity

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

From Ursula K. Le Guin’s novel
The Dispossessed: An Ambiguous Utopia
(1974)—

Chapter One

“There was a wall. It did not look important. It was built of uncut rocks roughly mortared. An adult could look right over it, and even a child could climb it. Where it crossed the roadway, instead of having a gate it degenerated into mere geometry, a line, an idea of boundary. But the idea was real. It was important. For seven generations there had been nothing in the world more important than that wall.

Like all walls it was ambiguous, two-faced. What was inside it and what was outside it depended upon which side of it you were on.”

Note—

“We note that the phrase ‘instead of having a gate it degenerated into mere geometry’ is mere fatuousness. If there is an idea here, degenerate, mere, and geometry  in concert do not fix it. They bat at it like a kitten at a piece of loose thread.”

— Samuel R. Delany, The Jewel-Hinged Jaw: Notes on the Language of Science Fiction  (Dragon Press, 1977), page 110 of revised edition, Wesleyan University Press, 2009

(For the phrase mere geometry  elsewhere, see a note of April 22. The apparently flat figures in that note’s illustration “Galois Affine Geometry” may be regarded as degenerate  views of cubes.)

Later in the Le Guin novel—

“… The Terrans had been intellectual imperialists, jealous wall builders. Even Ainsetain, the originator of the theory, had felt compelled to give warning that his physics embraced no mode but the physical and should not be taken as implying the metaphysical, the philosophical, or the ethical. Which, of course, was superficially true; and yet he had used number, the bridge between the rational and the perceived, between psyche and matter, ‘Number the Indisputable,’ as the ancient founders of the Noble Science had called it. To employ mathematics in this sense was to employ the mode that preceded and led to all other modes. Ainsetain had known that; with endearing caution he had admitted that he believed his physics did, indeed, describe reality.

Strangeness and familiarity: in every movement of the Terran’s thought Shevek caught this combination, was constantly intrigued. And sympathetic: for Ainsetain, too, had been after a unifying field theory. Having explained the force of gravity as a function of the geometry of spacetime, he had sought to extend the synthesis to include electromagnetic forces. He had not succeeded. Even during his lifetime, and for many decades after his death, the physicists of his own world had turned away from his effort and its failure, pursuing the magnificent incoherences of quantum theory with its high technological yields, at last concentrating on the technological mode so exclusively as to arrive at a dead end, a catastrophic failure of imagination. Yet their original intuition had been sound: at the point where they had been, progress had lain in the indeterminacy which old Ainsetain had refused to accept. And his refusal had been equally correct– in the long run. Only he had lacked the tools to prove it– the Saeba variables and the theories of infinite velocity and complex cause. His unified field existed, in Cetian physics, but it existed on terms which he might not have been willing to accept; for the velocity of light as a limiting factor had been essential to his great theories. Both his Theories of Relativity were as beautiful, as valid, and as useful as ever after these centuries, and yet both depended upon a hypothesis that could not be proved true and that could be and had been proved, in certain circumstances, false.

But was not a theory of which all the elements were provably true a simple tautology? In the region of the unprovable, or even the disprovable, lay the only chance for breaking out of the circle and going ahead.

In which case, did the unprovability of the hypothesis of real coexistence– the problem which Shevek had been pounding his head against desperately for these last three days. and indeed these last ten years– really matter?

He had been groping and grabbing after certainty, as if it were something he could possess. He had been demanding a security, a guarantee, which is not granted, and which, if granted, would become a prison. By simply assuming the validity of real coexistence he was left free to use the lovely geometries of relativity; and then it would be possible to go ahead. The next step was perfectly clear. The coexistence of succession could be handled by a Saeban transformation series; thus approached, successivity and presence offered no antithesis at all. The fundamental unity of the Sequency and Simultaneity points of view became plain; the concept of interval served to connect the static and the dynamic aspect of the universe. How could he have stared at reality for ten years and not seen it? There would be no trouble at all in going on. Indeed he had already gone on. He was there. He saw all that was to come in this first, seemingly casual glimpse of the method, given him by his understanding of a failure in the distant past. The wall was down. The vision was both clear and whole. What he saw was simple, simpler than anything else. It was simplicity: and contained in it all complexity, all promise. It was revelation. It was the way clear, the way home, the light.”

Related material—

Time Fold, Halloween 2005, and May and Zan.

See also The Devil and Wallace Stevens

“In a letter to Harriet Monroe, written December 23, 1926, Stevens refers to the Sapphic fragment that invokes the genius of evening: ‘Evening star that bringest back all that lightsome Dawn hath scattered afar, thou bringest the sheep, thou bringest the goat, thou bringest the child home to the mother.’ Christmas, writes Stevens, ‘is like Sappho’s evening: it brings us all home to the fold’ (Letters of Wallace Stevens, 248).”

— “The Archangel of Evening,” Chapter 5 of Wallace Stevens: The Intensest Rendezvous, by Barbara M. Fisher, The University Press of Virginia, 1990

Friday, February 19, 2010

Mimzy vs. Mimsy

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

 

Deep Play:

Mimzy vs. Mimsy

From a 2007 film, "The Last Mimzy," based on
the classic 1943 story by Lewis Padgett
  "Mimsy Were the Borogoves"–

http://www.log24.com/log/pix10/100219-LastMimzyTrailer.jpg

As the above mandala pictures show,
the film incorporates many New Age fashions.

The original story does not.

A more realistic version of the story
might replace the mandalas with
the following illustrations–

The Eightfold Cube and a related page from a 1906 edition of 'Paradise of Childhood'

Click to enlarge.

For a commentary, see "Non-Euclidean Blocks."

(Here "non-Euclidean" means simply
other than  Euclidean. It does not imply any
  violation of Euclid's parallel postulate.)

Sunday, March 29, 2009

Sunday March 29, 2009

Filed under: General,Geometry — 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

Sunday, March 15, 2009

Sunday March 15, 2009

Filed under: General,Geometry — 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: General,Geometry — 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: General,Geometry — 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.

Monday, March 2, 2009

Monday March 2, 2009

Joyce's Nightmare
continues

Today in History – March 2

Today is Monday, March 2, the 61st day of 2009. There are 304 days left in the year.

Today's Highlight in History:

On March 2, 1939, Roman Catholic Cardinal Eugenio Pacelli was elected Pope on his 63rd birthday; he took the name Pius XII.

Angels and Demons, Illuminati Diamond, pages 359-360

Log24 on June 9, 2008

From Gravity's Rainbow (Penguin Classics, 1995), page 563:

"He brings out the mandala he found.
'What's it mean?'
[….]

Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night…."

 

 

In lieu of Slothrop's mandala, here is another…

Christ and the four elements, 1495
 

 

Christ and the Four Elements

This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy
in Newton's Thought,
by B. J. T. Dobbs,
Cambridge University Press,
2002, p. 85

 

 

Related mandalas:Diamond arrangement of the four elements
and

Logo by Steven H. Cullinane for website on finite geometry

 

 

For further details,
click on any of the
three mandalas above.

Angels and Demons cross within a diamond (page 306), and Finite Geometry logo

Happy birthday to
Tom Wolfe, author of
The Painted Word.

Monday, July 21, 2008

Monday July 21, 2008


Knight Moves:

The Relativity Theory
of Kindergarten Blocks

(Continued from
January 16, 2008)

"Hmm, next paper… maybe
'An Unusually Complicated
Theory of Something.'"

Garrett Lisi at
Physics Forums, July 16

Something:

From Friedrich Froebel,
who invented kindergarten:

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

Click on image for details.

An Unusually
Complicated Theory:

From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.

For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168
."

For an even more
complicated theory of
Klein's simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:07 pm

Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler
 
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

 

Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today

 

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)


"Pauli linked this symbolism
with the concept of automorphism."

The Innermost Kernel
 (previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic
(4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible."

"There is such a thing
as a tesseract."

Madeleine L'Engle 

Friday, May 25, 2007

Friday May 25, 2007

Filed under: General,Geometry — Tags: , , , , — m759 @ 7:11 am
Dance and the Soul

From Log24 on
this date last year:

"May there be an ennui
of the first idea?
What else,
prodigious scholar,
should there be?"

— Wallace Stevens,
"Notes Toward a
Supreme Fiction"

The Associated Press,
May 25, 2007–

Thought for Today:
"I hate quotations.
 Tell me what you know."
— Ralph Waldo Emerson

[Journals, on May 3, 1849]

The First Idea:

The Line, by S. H. Cullinane

Four Elements:
 

Four Elements (Diamond)

Square Dance:

Square Dance (Diamond Theorem)

This "telling of what
I know" will of course
mean little to those
who, like Emerson,
have refused to learn
through quotations.

For those less obdurate
than Emerson —Harold Bloom
on Wallace Stevens

and Paul Valery's
   "Dance and the Soul"–

"Stevens may be playful, yet seriously so, in describing desire, at winter's end, observing not only the emergence of the blue woman of early spring, but seeing also the myosotis, whose other name is 'forget-me-not.' Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter, unable to bear what Valery's Eryximachus had called 'this cold, exact, reasonable, and moderate consideration of human life as it is.' The final form of this realization in Stevens comes in 1950, in The Course of a Particular, in the great monosyllabic line 'One feels the life of that which gives life as it is.' But even Stevens cannot bear that feeling for long. As Eryximachus goes on to say in Dance and the Soul:

A cold and perfect clarity is a poison impossible to combat. The real, in its pure state, stops the heart instantaneously….[…] To a handful of ashes is the past reduced, and the future to a tiny icicle. The soul appears to itself as an empty and measurable form. –Here, then, things as they are come together, limit one another, and are thus chained together in the most rigorous and mortal* fashion…. O Socrates, the universe cannot for one instant endure to be only what it is.

Valery's formula for reimagining the First Idea is, 'The idea introduces into what is, the leaven of what is not.' This 'murderous lucidity' can be cured only by what Valery's Socrates calls 'the intoxication due to act,' particularly Nietzschean or Dionysiac dance, for this will rescue us from the state of the Snow Man, 'the motionless and lucid observer.'" —Wallace Stevens: The Poems of Our Climate

* "la sorte… la plus mortelle":
    mortal in the sense
   "deadly, lethal"

Other quotations

(from March 28,
the birthday of
Reba McEntire):

Logical Songs

Reba McEntire, Saturday Evening Post, Mar/Apr 1995

Logical Song I
(Supertramp)

"When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they'd be singing so happily,
Joyfully, playfully watching me"

Logical Song II
(Sinatra)

"You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I'm such a happy in-
dividual….

You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots"

Tuesday, October 17, 2006

Tuesday October 17, 2006

Filed under: General — Tags: , — m759 @ 2:00 pm
To Measure
the Changes

(continued
from midnight
)

“To measure the changes
     of time and space
the smartest are nothing.”

— Shing-Tung Yau,
 The Emperor of Math
and Harvard philosopher

Illustrations —

To measure the changes:

The image “http://www.log24.com/log/pix06A/061017-Yellowbook3.jpg” cannot be displayed, because it contains errors.

The smartest are nothing:

The image “http://www.log24.com/log/pix06A/061017-Gump2A.jpg” cannot be displayed, because it contains errors.


Tuesday October 17, 2006

Filed under: General — Tags: , — m759 @ 12:00 am
To Measure
the Changes

 
(continued from
“The Legacy Codes,”
Nov. 5-6, 2003)

From this morning’s
New York Times:

The Emperor
of Math

Shing-Tung Yau
Rick Friedman for
The New York Times

The much-honored
mathematician
Shing-Tung Yau

Numbers
from the
Keystone State
on October 16:

The image “http://www.log24.com/log/pix06A/061016-PAlottery.jpg” cannot be displayed, because it contains errors.

For interpretations
of 621, see 6/21’s
Beijing String and
Go with the Flow.

For an interpretation
of 596, see Wikipedia,
596 (nuclear test):

“596 is the codename of the
People’s Republic of China’s
first nuclear weapons test,
detonated on
October 16, 1964.”

The image “http://www.log24.com/log/pix06A/061017-Fireball.jpg” cannot be displayed, because it contains errors.

Related material:

“‘In China he is a movie star,’ said Ronnie Chan, a Hong Kong real estate developer and an old friend….  And last summer Dr. Yau played the part…. He ushered Stephen Hawking into the Great Hall of the People in Tiananmen Square to kick off a meeting of some of the world’s leading physicists on string theory, and beamed as a poem he had written was performed by a music professor on the conference stage. It reads in part:
Beautiful indeed
is the source of truth.
To measure the changes
     of time and space
the smartest are nothing.”

The Emperor of Math

Friday, August 11, 2006

Friday August 11, 2006

Filed under: General — Tags: — m759 @ 11:07 pm

Under God

Adapted from August 7:

 The image “http://www.log24.com/log/pix06A/060811-Glyphs2.jpg” cannot be displayed, because it contains errors.

“Saomai, the Vietnamese name
for the planet Venus, was the
eighth major storm to hit China
during an unusually violent
typhoon season.”

AP online tonight 

Memorable Quotes 

Lieutenant Daniel Taylor:
Where the Hell is
this God of yours?
 
Forrest Gump:
[narrating]
It’s funny Lieutenant Dan
said that, ’cause right then,
God showed up.

 Wind and thunder:
the image of Increase.
Thus the superior man:
If he sees good,
he imitates it;
If he has faults,
he rids himself of them.

Hexagram 42 

For further details,
see
recent entries
(August 7-11)
and also

Symmetry and Change
In the Dreamtime
.

Update of 1:06 AM ET
from KHYI:

Mercy Now 
Written by Mary Gauthier

My father could use a little mercy now
The fruits of his labor
Fall and rot slowly on the ground
His work is almost over
It won’t be long and he won’t be around
I love my father, and he could use some mercy now

My brother could use a little mercy now
He’s a stranger to freedom
He’s shackled to his fears and doubts
The pain that he lives in is
Almost more than living will allow
I love my brother, and he could use some mercy now

My church and my country could use a little mercy now
As they sink into a poisoned pit
That’s going to take forever to climb out
They carry the weight of the faithful
Who follow them down
I love my church and country, and they could use some mercy now

Every living thing could use a little mercy now
Only the hand of grace can end the race
Towards another mushroom cloud
People in power, well
They’ll do anything to keep their crown
I love life, and life itself could use some mercy now

Yeah, we all could use a little mercy now
I know we don’t deserve it
But we need it anyhow
We hang in the balance
Dangle ‘tween hell and hallowed ground
Every single one of us could use some mercy now
Every single one of us could use some mercy now
Every single one of us could use some mercy now

Saturday, March 18, 2006

Saturday March 18, 2006

Filed under: General — Tags: — m759 @ 4:07 pm

ART WARS:
The Crimson Passion continues…

How to Grow
a Crimson Clover

Published in the Harvard Crimson
on Thursday, March 16, 2006, 6:24 PM
by Patrick R. Chesnut,
Crimson staff writer


Stephen Dedalus, James Joyce’s literary alter ego, once described the trappings of Irish culture as nets that hold a soul back from flight. By his standards, Harvard has soared.

Irish culture has been an indelible part of Boston, but the names on our red-brick buildings tell a different story: Adams, Lowell, Winthrop. It would be easy to assume that for Harvard students, Irish culture consists of little more than guzzling alcohol in Tommy Doyle’s Irish Pub or at St. Patrick’s Day Stein Club.

Recently, however, a small but lively Irish subculture, centered on Celtic music and language, has been developing at Harvard. But despite its vivacity, it remains largely unnoticed by the broader student body.

Efforts by groups like the Harvard College Celtic Club and by the producers of the upcoming Loeb mainstage of J.M. Synge’s “The Playboy of the Western World” may be just the sort of first step needed to finally make Harvard a place where Irish artistic culture lives….

REACHING OUT

“The Playboy”– which will run from April 28 through May 6– revolves around the disruption of life in a provincial Irish village when an outsider arrives with an extravagant story. All points converge at this play’s production: members of the Celtic Club coordinated and will perform the play’s music, the producers hope to draw Boston’s Irish community, and the production will present Harvard’s students with a script deeply entrenched in Irish history, but that boasts a universal appeal.

As Kelly points out, the Irish roots of “The Playboy” are clearer than in the plays of the nominally Irish, but Francophone, absurdist writer Samuel Beckett. And unlike the plays of Sean O’Casey, which are extremely rooted in Irish culture, “The Playboy” boasts a visceral appeal that will be accessible to Harvard students.

From a site linked to in yesterday’s St. Patrick’s Day sermon as the keys to the kingdom:

“In the western world, we tend to take for granted our musical scale, formed of whole tone and half tone steps. These steps are arranged in two ways: the major scale and the minor.”

From the obituary in today’s online New York Times of fashion designer Oleg Cassini, who died at 92 on St. Patrick’s Day, Friday, March 17, 2006:

“… he was always seen in the company of heiresses, debutantes, showgirls, ingenues. Between, before or after [his first] two marriages, he dated young starlets like Betty Grable and Lana Turner and actresses like Ursula Andress and Grace Kelly, to whom he was briefly engaged.

‘He was a true playboy, in the Hollywood sense,’ said Diane von Furstenberg, the fashion designer and a friend of Mr. Cassini’s. ‘Well into his 90’s, he was a flirt.'”

“How strange the change from major to minor…
      Ev’ry time we say goodbye.”
   — Cole Porter

Monday, February 27, 2006

Monday February 27, 2006

Filed under: General — Tags: , , — m759 @ 9:26 am

Sudden View

From John O'Hara's Birthday:

"We stopped at the Trocadero and there was hardly anyone there.  We had Lanson 1926.  'Drink up, sweet.  You gotta go some.  How I love music.  Frère Jacques, Cuernavaca, ach du lieber August.  All languages.  A walking Berlitz.  Berlitz sounds like you with that champagne, my sweet, or how you're gonna sound.'"

— John O'Hara, Hope of Heaven, Chapter 11, 1938

"And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance."

Acts, Chapter 2, Verse 4

"Lps. The keys to. Given! A way a lone a last a loved a long the

PARIS,
1922-1939."

— James Joyce, conclusion of Finnegans Wake

"Using illustrative material from religion, myth, and culture, he starts with the descent of the dove on Jesus and ends with the poetic ramblings of James Joyce."

Review of a biography of the Holy Spirit

Monica Potts in today's New York Times on Sybille Bedford:

"Though her works were not always widely popular, they inspired a deeply fervent following of committed admirers, starting with her first published work, A Sudden View, in 1953. Later retitled A Visit to Don Otavio, it was an account of her journey through Mexico."

… "I addressed him.  'Is Cuernavaca not below Mexico City?'
    'It is low.'
    'Then what is this?'  Another summit had sprung up above a curve.
    'At your orders, the Three Marias.'
    'What are the Three Marias?'
    'These.'
    Later, I learned from Terry that they were the three peaks by the La Cima Pass which is indeed one of the highest passes in the Republic; and still later from experience, that before running down to anywhere in this country one must first run up some six or seven thousand feet.  The descents are more alarming than the climbs.  We hurtled towards Cuernavaca down unparapeted slopes with the speed and angle, if not the precision, of a scenic railway– cacti flashed past like telegraph poles, the sun was brilliant, the air like laughing gas, below an enchanting valley, and the lack of brakes became part of a general allegro accelerando."

— Sybille Bedford, A Sudden View, Counterpoint Press, Counterpoint edition (April 2003), page 77

"How continually, how startlingly, the landscape changed!  Now the fields were full of stones: there was a row of dead trees.  An abandoned plough, silhouetted against the sky, raised its arms to heaven in mute supplication; another planet, he reflected again, a strange planet where, if you looked a little further, beyond the Tres Marias, you would find every sort of landscape at once, the Cotswolds, Windermere, New Hampshire, the meadows of the Eure-et-Loire, even the grey dunes of Cheshire, even the Sahara, a planet upon which, in the twinkling of an eye, you could change climates, and, if you cared to think so, in the crossing of a highway, three civilizations; but beautiful, there was no denying its beauty, fatal or cleansing as it happened to be, the beauty of the Earthly Paradise itself."

— Malcolm Lowry, Under the Volcano, Harper Perennial Modern Classics, 1st Perennial Classics edition (May 1, 2000), page 10

Sunday, January 9, 2005

Sunday January 9, 2005

Filed under: General — Tags: — m759 @ 3:10 pm

Light at Bologna

“Others say it is a stone that posseses mysterious powers…. often depicted as a dazzling light.  It’s a symbol representing power, a source of immense energy.  It nourishes, heals, wounds, blinds, strikes down…. Some have thought of it as the philosopher’s stone of the alchemists….”

Foucault’s Pendulum
by Umberto Eco,
Professor of Semiotics at
 Europe’s oldest university,
 the University of Bologna.

The Club Dumas

by Arturo Perez-Reverte

(Paperback, pages 346-347):

One by one, he tore the engravings from the book, until he had all nine.  He looked at them closely.  “It’s a pity you can’t follow me where I’m going.  As the fourth engraving states, fate is not the same for all.”

“Where do you believe you’re going?”

Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

“To meet someone” was his enigmatic answer. “To search for the stone that the Great Architect rejected, the philosopher’s stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso. From Faust’s black dog to the false angel of light who tried to break down Saint Anthony’s resistance.  But most of all, stupidity bores him, and he hates monotony….”

Eclogues: Eight Stories

by Guy Davenport

Johns Hopkins paperback, 1993, page 127 —

Lo Splendore della Luce a Bologna, VI:

“In 1603, at Monte Paderno, outside Bologna, an alchemist (by day a cobbler) named Vicenzo Cascariolo discovered the Philosopher’s Stone, catalyst in the transformation of base metals into gold, focus of the imagination, talisman for abstruse thought.  Silver in some lights, white in others, it glowed blue in darkness, awesome to behold.”


The Discovery of Luminescence:

The Bolognian Stone

The image “http://www.log24.com/log/pix05/050109-Bologna.jpg” cannot be displayed, because it contains errors.
Bologna, 16th Century

“For the University of Bologna hosting an International Conference on Bioluminescence and Chemiluminescence has a very special significance. Indeed, it is in our fair City that modern scientific research on these phenomena has its earliest roots….

‘After submitting the stone
to much preparation, it was not
the Pluto of Aristophanes
that resulted; instead, it was
the Luciferous Stone’ ”

From one of the best books
of the 20th century:

The Hawkline Monster

by Richard Brautigan

The image “http://www.log24.com/log/pix05/050109-Hawkline.jpg” cannot be displayed, because it contains errors.
“The Chemicals that resided in the jar were a combination of hundreds of things from all over the world.  Some of The Chemicals were ancient and very difficult to obtain.  There were a few drops of something from an Egyptian pyramid dating from the year 3000 B.C.

There were distillates from the jungles of South America and drops of things from plants that grew near the snowline in the Himalayas.

Ancient China, Rome and Greece had contributed things, too, that had found their way into the jar.  Witchcraft and modern science, the latest of discoveries, had also contributed to the contents of the jar.  There was even something that was reputed to have come all the way from Atlantis….

… they did not know that the monster was an illusion created by a mutated light in The Chemicals. a light that had the power to work its will upon mind and matter and change the very nature of reality to fit its mischievous mind.”

Friday, September 17, 2004

Friday September 17, 2004

Filed under: General — Tags: , — m759 @ 3:57 pm

3:57:09…
Time is a Weapon

In memory of rock star and NRA member Johnny Ramone, who died on Wednesday, Sept. 15:

“You’ve got to ask yourself a question.”
Clint Eastwood as Dirty Harry

“At the end, when the agent pumps Neo full of lead, the agent is using a .357 Magnum. That gun only holds 9 bullets, but the agent shoots 10 shots at Neo. I don’t know where he got that gun.”

— Jesse Baumann,
    The Matrix: The Magic Bullet 

Manufacturer:
Ta’as Israel Industries,
Ramat Hasharon, Israel

Friday, August 01, 2003:

Fearful Meditation 

Ray Price - Time

TIME, Aug. 4, 2003

Ray Price — Time

“The Max D. Barnes-penned title track, with its stark-reality lyrics, is nothing short of haunting: ‘Time is a weapon, it’s cold and it’s cruel; It knows no religion and plays by no rules; Time has no conscience when it’s all said and done; Like a beast in the jungle that devours its young.’ That’s so good, it hurts! Price’s still-amazing vocals are simply the chilling icing on the cake.”

— Lisa Berg, NashvilleCountry.com

O fearful meditation!
Where, alack,
Shall time’s best jewel
from time’s chest lie hid?

— Shakespeare, Sonnet 65

Clue: click here.  This in turn leads to my March 4 entry Fearful Symmetry, which contains the following:

“Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery….”

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye

Wednesday, September 15, 2004

Wednesday September 15, 2004

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

Shakespeare
for Rosh Hashanah

From “Walter Benjamin,
1892-1940,”
by Hannah Arendt
(Introduction to
Benjamin’s Illuminations.):

THE PEARL DIVER

Full fathom five thy father lies,
Of his bones are coral made,
Those are pearls that were his eyes.
Nothing of him that doth fade
But doth suffer a sea-change
Into something rich and strange.
THE TEMPEST, I, 2

“… we are dealing here with something which may not be unique but is certainly extremely rare: the gift of thinking poetically.

And this thinking, fed by the present, works with the ‘thought fragments’ it can wrest from the past and gather about itself.  Like a pearl diver who descends to the bottom of the sea, not to excavate the bottom and bring it to light but to pry loose the rich and the strange, the pearls and the coral in the depths, and to carry them to the surface, this thinking delves into the depths of the past– but not in order to resuscitate it the way it was and to contribute to the renewal of extinct ages. What guides this thinking is the conviction that although the living is subject to the ruin of the time, the process of decay is at the same time a process of crystallization, that in the depth of the sea, into which sinks and is dissolved what once was alive, some things ‘suffer a sea-change’ and survive in new crystallized forms and shapes that remain immune to the elements, as though they waited only for the pearl diver who one day will come down to them and bring them up into the world of the living– as ‘thought fragments,’ as something ‘rich and strange,’ and perhaps even as everlasting Urphänomene.”

For examples of everlasting Urphänomene, see Translation Plane for Rosh Hashanah and The Square Wheel; recall that on this date

“In 1935, the Nuremberg Laws deprived German Jews of their citizenship and made the swastika the official symbol of Nazi Germany.”

Today in History, the Miami Herald

(For some further reflections on square wheels, see Triumph of the Cross.)

Monday, August 30, 2004

Monday August 30, 2004

Filed under: General,Geometry — Tags: — m759 @ 12:07 pm

Q.E.D.

A Log24 entry of Aug. 17, 2004, on the
three Semitic (or “Abrahamic”) religions:

“Looney.”

From Scotsman.com News
Mon., 30 Aug., 2004
11:43 AM (UK)

Ex-Priest Sentenced
for Disrupting Marathon

By Pat Hurst, PA News, in Athens

An ex-priest who lives in Britain was given a 12-month suspended sentence today for disrupting the men’s Olympic marathon in Athens.

Cornelius Horan, 57, a former Catholic priest living in London, appeared before a Greek judge this morning, local police said.

He was sentenced and released from custody but his whereabouts are unknown.

Irishman Horan, originally from Kerry, dashed from the sidelines to attack the marathon front-runner during yesterday’s event.

He told officers he staged the disruption to “prepare for the second coming”.

A police spokesman said: “He has got mental problems. He is not very well.

“His only explanation for his behaviour was that it was for the second coming.”

Horan also disrupted last year’s Silverstone Formula One Grand Prix by dashing across the track.

Leslie Broad, of Deunant Books, which publishes Mr Horan’s books on its website, said: “We publish two of his books on biblical prophecies and he seems to be fairly convinced that the second coming is due fairly shortly.

“After the incident at Silverstone, he did say he would never do anything like that again.

“He comes across as a shy, very intelligent and compassionate man but as is often the way with people who are very intelligent, it sometimes manifests itself in very strange ways.

“I think he found prison a fairly uplifting experience. He came out feeling that he had met a lot of people he wouldn’t normally have met, people who had committed serious crimes.”

Horan’s victim yesterday, Vanderlei De Lima, from Brazil, was at the head of the race just three miles from the finish.

Horan grabbed him and bundled him into spectators at the side of the road.

After a scuffle, the runner managed to get away, but he was clearly ruffled and finished third.

The Brazilian Olympic Committee put in an official complaint to the Greeks and at one point the final medal ceremony to be staged during the closing ceremony was in doubt.

Horan was arrested and taken to the General Police Division of Attica, where he stayed overnight.

Author biography
from
Deunant Books:

Father Cornelius (“Neil”) Horan


Horan

“Neil Horan was born in 1947, in Scartaglen, County Kerry, in the Republic of Ireland. After schooling in Ireland he was ordained a Catholic Priest in Saint Mary’s Cathedral, Killarney, in 1973.

He has served all his priestly life in the Southwark Diocese, covering London south of the River Thames and Kent, his first Parish being Bexley in Kent. His interest in Bible prophecy began when he attended a lecture in 1974, given by the Apostolic Fellowship of Christ, a group which had originated with the Christadelphians. Meaning ‘Brothers in Christ’, the Christadelphians were a small Church formed in 1861 by Dr John Thomas. Father Horan says he owes a debt of gratitude to the Christadelphian tradition for the understanding of the Bible which they gave him. He regards the Bible as the greatest Book in the world and has devoted his life to making it better known, especially the Prophecies.

He is not a prophet, considering himself to be merely an interpreter, has never received a Divine message or vision, and God has never spoken to him. He feels that he is right only in so far as he interprets the Book of Books correctly.

He is still a Catholic Priest, listed in the Catholic Directory under his full name of Cornelius Horan. Cornelius, a Centurian [sic] in the Roman army, was the first Christian convert; Father Horan is proud to bear that name and hopes to meet his famous namesake soon, when Jesus comes.”

A Glorious New World
by Father Neil Horan

“Are there passages in the Bible that foretell events that were, at the time it was written, far in the future? Father Neil Horan argues eloquently, knowledgeably and persuasively in this book, first published in 1985, that this is so. It is easy to scoff at predictions of events that were, according to the book, to have taken place a few years ago but which have not happened, but to do that would be wrong. With only the most subtle changes of emphasis in interpretation, it could just as easily be argued that events in the Middle East particularly have to a large degree fulfilled the prophecies for the years since 1985.

Then there are the events yet to come. They are, according to the author and his sources, to be the most significant in the history of mankind, and are going to happen soon. With a little thought, certain current-day world figures are a disconcertingly comfortable match for some of the characters who will act out the earth-shattering dramas to come. Even the most hardened cynic will get that prickly feeling down the back of his neck as he reads this book.

Taken together with Father Horan’s later work ‘Christ Will Soon Take Power From All Governments’ (also available from Deunant Books) the two books represent one of the most remarkable and significant bodies of work seen in this field for many, many years.”

Deunant Books on Theology

Ludwig Wittgenstein,
Philosophical Investigations:

373. Grammar tells what kind of object anything is. (Theology as grammar.)

Grammar and Geometry:
The Euclidean Proposition,
by J. B. Calvert:

For more on Wittgenstein, theology, and grammar, see the Log24

entries of Jan. 14, 2004.

Related material:

God Goes Hollywood,
by Jeremiah Cullinane

Tuesday, August 10, 2004

Tuesday August 10, 2004

Filed under: General,Geometry — Tags: , , — m759 @ 3:01 pm

Battle of Gods and Giants

In checking the quotations from Dante in the previous entry, I came across the intriguing site Gigantomachia:

"A gigantomachia or primordial battle between the gods has been retold in myth, cult, art and theory for thousands of years, from the Egyptians to Heidegger. This site will present the history of the theme. But it will do so in an attempt to raise the question of the contemporary relevance of it. Does the gigantomachia take place today? Where? When? In what relation to you and me?"

Perhaps atop the Empire State Building?

(See An Affair to Remember and  Empire State Building to Honor Fay Wray.)

Perhaps in relation to what the late poet Donald Justice called "the wood within"?

Perhaps in relation to T. S. Eliot's "The Waste Land" and the Feast of the Metamorphosis?

Or perhaps not.

Perhaps at Pergamon:

Perhaps at Pergamon Press:

Invariants 

"What modern painters are trying to do,
if they only knew it, is paint invariants."

— James J. Gibson in Leonardo
(Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978)

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn.  Taken as a set, these three line diagrams describe the structure of the bottom colored figure.  After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

A more remarkable invariance — that of symmetry itself — is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2×2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry.

This sort of mathematics illustrates the invisible "form" or "idea" behind the visible two-color pattern.  Hence it exemplifies, in a way, the conflict described by Plato between those who say that "real existence belongs only to that which can be handled" and those who say that "true reality consists in certain intelligible and bodiless forms."

For further details, see a section on Plato in the Gigantomachia site.

Saturday, May 22, 2004

Saturday May 22, 2004

Filed under: General — Tags: , — m759 @ 3:00 pm

A Form

John Leonard in the June 10, 2004, New York Review of Books, on E. L. Doctorow:

"… he's got urgent things to say and seeks some form to say them in, or a form that will tease and torture secret meanings out of what he thinks he already knows, or a form, like a wishing well, down which to dream, scream, or drown."

48. The Well

The Judgment

The Well. The town may be changed,
But the well cannot be changed.
It neither decreases nor increases.
They come and go and draw from the well.
If one gets down almost to the water
And the rope does not go all the way,
Or the jug breaks, it brings misfortune.

From the Book of Ecclesiastes 12:6

or ever the silver cord be loosed, or the golden bowl be broken, or the pitcher be broken at the fountain, or the wheel broken at the cistern

From Chuck Polisher's I Ching Lexicon:

See also the following form, discussed in

Balanchine's Birthday
(1/9/03) and in

Art Theory
for Yom Kippur

(10/5/03)

Tuesday, April 6, 2004

Tuesday April 6, 2004

Filed under: General — Tags: , , , — m759 @ 10:00 pm

Ideas and Art, Part III

The first idea was not our own.  Adam
In Eden was the father of Descartes…

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"Quaedam ex his tanquam rerum imagines sunt, quibus solis proprie convenit ideae nomen: ut cùm hominem, vel Chimaeram, vel Coelum, vel Angelum, vel Deum cogito."

Descartes, Meditationes III, 5

"Of my thoughts some are, as it were, images of things, and to these alone properly belongs the name idea; as when I think [represent to my mind] a man, a chimera, the sky, an angel or God."

Descartes, Meditations III, 5

Begin, ephebe, by perceiving the idea
Of this invention, this invented world,
The inconceivable idea of the sun.

You must become an ignorant man again
And see the sun again with an ignorant eye
And see it clearly in the idea of it.

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"… Quinimo in multis saepe magnum discrimen videor deprehendisse: ut, exempli causâ, duas diversas solis ideas apud me invenio, unam tanquam a sensibus haustam, & quae maxime inter illas quas adventitias existimo est recensenda, per quam mihi valde parvus apparet, aliam verò ex rationibus Astronomiae desumptam, hoc est ex notionibus quibusdam mihi innatis elicitam, vel quocumque alio modo a me factam, per quam aliquoties major quàm terra exhibetur; utraque profecto similis eidem soli extra me existenti esse non potest, & ratio persuadet illam ei maxime esse dissimilem, quae quàm proxime ab ipso videtur emanasse."

Descartes, Meditationes III, 11

"… I have observed, in a number of instances, that there was a great difference between the object and its idea. Thus, for example, I find in my mind two wholly diverse ideas of the sun; the one, by which it appears to me extremely small draws its origin from the senses, and should be placed in the class of adventitious ideas; the other, by which it seems to be many times larger than the whole earth, is taken up on astronomical grounds, that is, elicited from certain notions born with me, or is framed by myself in some other manner. These two ideas cannot certainly both resemble the same sun; and reason teaches me that the one which seems to have immediately emanated from it is the most unlike."

Descartes, Meditations III, 11

"Et quamvis forte una idea ex aliâ nasci possit, non tamen hîc datur progressus in infinitum, sed tandem ad aliquam primam debet deveniri, cujus causa sit in star archetypi, in quo omnis realitas formaliter contineatur, quae est in ideâ tantùm objective."

Descartes, Meditationes III, 15

"And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality [or perfection] that is found objectively [or by representation] in these ideas is contained formally [and in act]."

Descartes, Meditations III, 15

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction,"

The Quest for the Fiction of the Absolute:

"Canto nine considers the movement of the poem between the particular and the general, the immanent and the transcendent: "The poem goes from the poet's gibberish to / The gibberish of the vulgate and back again. / Does it move to and fro or is it of both / At once?" The poet, the creator-figure, the shadowy god-figure, is elided, evading us, "as in a senseless element."  The poet seeks to find the transcendent in the immanent, the general in the particular, trying "by a peculiar speech to speak / The peculiar potency of the general." In playing on the senses of "peculiar" as particular and strange or uncanny, these lines play on the mystical relation of one and many, of concrete and abstract."

Brian Cronin in Foundations of Philosophy:

"The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight."

— From Ch. 2, Identifying Direct Insights

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction":

"The fourth canto returns to the theme of opposites. 'Two things of opposite natures seem to depend / On one another . . . . / This is the origin of change.'  Change resulting from a meeting of opposities is at the root of Taoism: 'Tao produced the One. / The One produced the two. / The two produced the three. / And the three produced the ten thousand things' (Tao Te Ching 42) …."

From an entry of March 7, 2004

 

From the web page

Introduction to the I Ching–
By Richard Wilhelm
:

"He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the 'great primal beginning' of all that exists, t'ai chi — in its original meaning, the 'ridgepole.' Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin,

.

This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites."

The t'ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that

"W.J.T. Mitchell calls 'iconology'
a study of the 'logos'
(the words, ideas, discourse, or 'science')
of 'icons' (images, pictures, or likenesses).
It is thus a 'rhetoric of images'
(Iconology: Image, Text, Ideology, p. 1)."

A variation on the t'ai chi symbol appears in a log24.net entry for March 5:

The Line,
by S. H. Cullinane

See too my web page Logos and Logic, which has the following:

"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi, was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies, ed. Quarto, Gallimard, 1999, p. 100

 Logos Alogos,
by S. H. Cullinane 

In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day
at the Sorbonne.
We shall return at twilight
from the lecture
Pleased that
the irrational is rational…."

This is the logoi alogoi of Simone Weil.

In "Notes toward a Supreme Fiction,"
Wallace Stevens lists three criteria
for a work of the imagination:

It Must Be Abstract

The Line,
by S.H. Cullinane 

It Must Change

 The 24,
by S. H. Cullinane

It Must Give Pleasure

Puzzle,
by S. H. Cullinane

Related material:

Logos and Logic.

 

Monday, April 5, 2004

Monday April 5, 2004

Filed under: General,Geometry — Tags: , , — m759 @ 4:03 am

Ideas and Art

 
Motto of
Plato's Academy

 

From Minimalist Fantasies,
by Roger Kimball, May 2003:

All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
—Frank Stella, 1966

Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
— Clement Greenberg, 1967

The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
— Michael Kimmelman, 2003

There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics.
— Roger Kimball, 2003

 

 

From How Not Much Is a Whole World,
by Michael Kimmelman, April 2, 2004

Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it


"William Clark,"
by Patricia Johanson, 1967

should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it.

….

The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line.

If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence….

That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one.

 

I maintain that of course
we can see ideas.

Example: the idea of
invariant structure.

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235.
    Pergamon Press Ltd., 1978

For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.

Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)…  That is to say, they are not lost in translation.  (See previous entry.)

Friday, August 1, 2003

Friday August 1, 2003

Filed under: General — Tags: , , — m759 @ 1:40 pm

Fearful Meditation

Ray Price - Time TIME, Aug. 4, 2003

Ray Price — Time

“The Max D. Barnes-penned title track, with its stark-reality lyrics, is nothing short of haunting: ‘Time is a weapon, it’s cold and it’s cruel; It knows no religion and plays by no rules; Time has no conscience when it’s all said and done; Like a beast in the jungle that devours its young.’ That’s so good, it hurts! Price’s still-amazing vocals are simply the chilling icing on the cake.”

— Lisa Berg, NashvilleCountry.com

O fearful meditation! Where, alack,
Shall time’s best jewel from time’s chest lie hid?

— Shakespeare, Sonnet 65

Clue: click here.  This in turn leads to my March 4 entry Fearful Symmetry, which contains the following:

“Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery….”

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye

Tuesday, February 11, 2003

Tuesday February 11, 2003

Filed under: General — Tags: , , — m759 @ 5:10 pm

St. John von Neumann's Song

The mathematician John von Neumann, a heavy drinker and party animal, advocated a nuclear first strike on Moscow.*  Confined to a wheelchair before his death, he was, some say, the inspiration for Kubrick's Dr. Strangelove.  He was a Jew converted to Catholicism.  His saint's day was February 8.  Here is an excerpt from a book titled Abstract Harmonic Analysis**, just one of the fields illuminated by von Neumann's brilliance:

"…von Neumann showed that an intrinsic definition can be given for the mean M(f) of an almost periodic function…. Von Neumann proved the existence and properties of M(f) by completely elementary methods…."

Should W. B. Yeats wander into the Catholic Anticommunists' section of Paradise, he might encounter, as in "Sailing to Byzantium," an unexpected set of "singing-masters" there: the Platonic archetypes of the Hollywood Argyles.

The Argyles' attire is in keeping with Yeats's desire for gold in his "artifice of eternity"… In this case, gold lamé, but hey, it's Hollywood.  The Argyles' lyrics will no doubt be somewhat more explicit in heaven.  For instance, in "Alley Oop," the line

"He's a mean motor scooter and a bad go-getter"

will in its purer heavenly version be rendered

"He's a mean M(f)er and…"

in keeping with von Neumann's artifice of eternity described above.

This theological meditation was suggested by previous entries on Yeats, music and Catholicism (see Feb. 8, von Neumann's saint's day) and by the following recent weblog entries of a Harvard senior majoring in mathematics:

"I changed my profile picture to Oedipus last night because I felt cursed by fate…."

"It's not rational for me to believe that I am cursed, that the gods are set against me.  Because I don't even believe in any gods!"

The spiritual benefits of a Harvard education are summarized by this student's new profile picture:

The image “http://log24.com/log/pix03/030211-oedipus.gif” cannot be displayed, because it contains errors.

M(f)

*Source: Von Neumann and the Development of Game Theory

**by Harvard professor Lynn H. Loomis, Van Nostrand, 1953, p. 169.

Tuesday, December 3, 2002

Tuesday December 3, 2002

Filed under: General,Geometry — Tags: , — m759 @ 1:45 pm

Symmetry, Invariance, and Objectivity

The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.

On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:

"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."

Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):

"Objectivity means invariance with respect to the group of automorphisms."

It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.

I pointed this out on June 10, 2002.

For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).

Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.

Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.

A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:

"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged."  (p. 69)

This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

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

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

Example:


For an animated version, click here.

Remarks:

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

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

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

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

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


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

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

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

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

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

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

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

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

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

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

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

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

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

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



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