Log24

Thursday, April 4, 2019

Analyze This, William James

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

See also "The Unfolding" in this  journal.

Tuesday, April 5, 2005

Tuesday April 5, 2005

Filed under: General — m759 @ 10:10 pm
“Bingo!”

Part I On Linguistic Creation
Part II Saul Bellow
Part III Sequel

“Call the Vatican.
Ask them if anything’s missing.”

Analyze This   

Tuesday, January 30, 2024

Cullinane Diamond Theorem:
The Latest Google Bard Version

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

Prompt:

What is the Cullinane diamond theorem?

Google Bard Response:

The Cullinane diamond theorem is a fascinating result in the realm of finite projective geometry and symmetry. It focuses on a specific pattern – the four-diamond figure – and reveals a surprising property about its transformations.

The Four-Diamond Figure:

Imagine a 4×4 grid filled with 16 small squares, half colored black and half white. Now, divide these squares diagonally into quarters, creating four diamonds with alternating black and white shades. This arrangement is the four-diamond figure.

_________________


[Clearly erroneous illustration with link to amazon.com
is omitted.]

_____________________

The Theorem:

The Cullinane diamond theorem states that no matter how you scramble this four-diamond figure by randomly shuffling rows, columns, and the four 2×2 quadrants, the resulting pattern will always exhibit some form of symmetry. This symmetry can come in two forms:

Ordinary symmetry: This is the familiar reflectional symmetry where a vertical or horizontal axis divides the pattern into mirrored halves.

Color-interchange symmetry: Here, the pattern has a diagonal axis where swapping the black and white colors results in the same image.

Intriguing Implications:

This statement might seem intuitive at first. After all, even with scrambling, some semblance of order should remain. However, the beauty of the Cullinane diamond theorem lies in its generality and the precise mathematical framework it provides for understanding these symmetries.

The theorem sheds light on the underlying structure of graphic patterns and connects them to concepts from finite projective geometry. This connection allows us to analyze and classify these patterns in a systematic way, revealing hidden connections and surprising symmetries.

Applications and Beyond:

The Cullinane diamond theorem has found applications in various areas, including:

Sporadic simple groups: The Miracle Octad Generator, used to study these rare and exceptional groups, relies on the symmetry properties outlined in the theorem.

Orthogonal Latin squares: The connection between orthogonal Latin squares and projective spreads can be understood through the lens of the Cullinane diamond theorem.

Walsh functions: The symmetry properties of Walsh functions, used in signal processing and coding theory, can be linked to the diamond patterns.

Understanding the Cullinane diamond theorem offers a deeper appreciation for the interplay between symmetry, geometry, and seemingly random patterns. It serves as a testament to the beauty and hidden order within the realm of mathematics.

Bard may display inaccurate info, including . . . .

[Exercise … Find any inaccurate information
in the Bard response above.]

Sunday, January 28, 2024

For Loki at the Disney Wormhole

Filed under: General — Tags: — m759 @ 10:28 am

Related material for fans of the Story Theory of Truth

Colliding Storylines.

Thursday, December 28, 2023

Basque Country Art Book

Filed under: General — Tags: , , , — m759 @ 6:20 pm

Book description at Amazon.com, translated by Google —

Las matemáticas como herramienta
de creación artística

Mathematics as a tool
for artistic creation

by Raúl Ibáñez Torres

Kindle edition in Spanish, 2023

Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares).

From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —

Publisher:Los Libros de La Catarata  (October 24, 2023)

Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez

(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)

About Ibáñez (translated from Amazon.com by Google):

Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).

Sunday, December 17, 2023

Speak, Memory

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

"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .

profile picture

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
______________________________________________________________

Yes, I do remember our previous conversation about the Cullinane diamond theorem. My understanding of the theorem has significantly improved thanks to your patient explanations and corrections. Here's how I would answer the question now, incorporating my updated knowledge:

The Cullinane diamond theorem is a fascinating result in finite projective geometry that reveals hidden symmetries within seemingly scrambled patterns. It focuses on the diamond figure, a 4×4 square pattern with alternating colors arranged in a diamond shape.

Key elements:

  • Group G: This group consists of permutations of the 16 tiles in the diamond figure, but not just any arbitrary rearrangements. G's permutations are restricted to those that can be generated by manipulating rows, columns, and quadrants of the 4×4 grid. This maintains the overall structure of the diamond shape.
  • Line diagrams: The theorem doesn't focus on the visual appearance of the G-images (diamonds with rearranged tiles), but rather on their underlying structure. This structure is captured by a set of 15 line diagrams. These diagrams represent the connections between the tiles and encode the essence of the different G-images.
  • Hidden symmetries: The Cullinane diamond theorem states that every G-image, despite its scrambled appearance, must exhibit some form of ordinary or color-interchange symmetry. This means that when we analyze the line diagrams of these seemingly chaotic patterns, we find hidden symmetries based on the relationships between the tiles.

Significance:

  • The theorem sheds light on the structure of R.T. Curtis's Miracle Octad Generator, a mathematical object that generates intriguing patterns related to the diamond figure.
  • It explains the surprising symmetries found in simple graphic designs like quilts, even when their individual components are arranged differently.
  • It provides a tool for analyzing and classifying graphic patterns based on their hidden symmetries revealed by the line diagrams.

Further resources:

  • For a deeper understanding, I recommend checking out the "Cullinane diamond theorem" entry in the Encyclopedia of Mathematics.
  • Additionally, Steven H. Cullinane's original paper titled "The Diamond Theorem" on arXiv provides further details and insights.

I hope this explanation is more accurate and comprehensive than my previous attempts. Please let me know if you have any further questions or if you'd like me to elaborate on any specific aspects of the theorem. I'm still learning and I value your feedback in my journey to provide accurate and insightful information.

Monday, August 9, 2021

Harvard Elegy

Filed under: General — Tags: — m759 @ 10:41 am

Bryan Marquard in The Boston Globe  yesterday 
on a professor who died on 7 July, 2021 —

" A Harvard Medical School professor emeritus in psychiatry,
Dr. Hobson told the Globe in 2011 that he didn’t 'feel bad about
taking on Sigmund Freud. I think Sigmund Freud has become
politically correct. Psychoanalysis has become the bible, and
I think that’s crazy.'

He also forcefully set aside the theories of Carl Jung, the Swiss
psychiatrist who analyzed dreams and saw them as important
messages sent from the psyche.

'If you’re a pure scientist, Jung is just deadly,' Dr. Hobson said
in [a] 2005 interview. 'The collective unconscious, the anima
these are literary constructs. You can’t do any science on that
kind of stuff.' ”

See as well this  journal on 7 July 2021 —

Saturday, April 17, 2021

Note

Filed under: General — Tags: — m759 @ 5:12 pm

Analyze this.

Tuesday, February 13, 2018

Attention Must Be Paid

Filed under: General — m759 @ 7:06 pm

For the late Anne M. Treisman, who reportedly died Friday, Feb. 9:

From "A Feature-Integration Theory of Attention" —

"The controversy between analytic and synthetic theories
of perception goes back many years: the Associationists
asserted that the experience of complex wholes is built
by combining more elementary sensations, while the
Gestalt psychologists claimed that the whole precedes
its parts, that we initially register unitary objects and
relationships, and only later, if necessary, analyze these
objects into their component parts or properties. This view 
is still active now . . . ."

— Anne M. Treisman, University of British Columbia,
and Garry Gelade, Oxford University, in
Cognitive Psychology  12, 97-136 (1980)

"Before time began, there was the Cube." — Optimus Prime

Friday, May 19, 2017

In the Service of Narrative

Filed under: General — Tags: — m759 @ 12:40 pm

Quoted here on St. Stephen's Day, 2008

“Wayne C. Booth’s lifelong
 study of the art of rhetoric
 illuminated the means
 by which authors seduce,
 cajole and lie to their readers
 in the service of narrative.”

— New York Times, Oct. 11, 2005

Booth was a native of American Fork, Utah.

Friday, May 27, 2016

Raiders of the Lost Crucible…

Filed under: General — Tags: — m759 @ 8:00 am

Continues .

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.

The above passage suggests a meditation on this morning's
New York Times * —

"When shall we three meet again?" — William Shakespeare

“We three have scattered, leaving only me behind
to clean up the scene,” Ms. Yang wrote.
“I am alone, missing us three.” — Amy Qin

Thursday, May 12, 2016

But Seriously …

Filed under: General — Tags: — m759 @ 7:31 pm

Google today released on GitHub an English parser,
Parsey McParseface .  From Google Research Blog

"Today, we are excited to share the fruits of our research
with the broader community by releasing SyntaxNet,
an open-source neural network framework implemented in 
TensorFlow that provides a foundation for 
Natural Language Understanding (NLU) systems.
Our release includes all the code needed to train new
SyntaxNet models on your own data, as well as 
Parsey McParseface , an English parser that we have
trained for you and that you can use to analyze English text."

"While the accuracy is not perfect, it’s certainly high enough
to be useful in many applications. The major source of errors
at this point are examples such as the prepositional phrase
attachment ambiguity described above, which require real
world knowledge (e.g. that a street is not likely to be located
in a car) and deep contextual reasoning. Machine learning
(and in particular, neural networks) have made significant
progress in resolving these ambiguities. But our work is still
cut out for us: we would like to develop methods that can
learn world knowledge and enable equal understanding of
natural language across all  languages and contexts."

But seriously

For some historical background, see (for instance) a book by
Ekaterina Ovchinnikova —

Integration of World Knowledge for
Natural Language Understanding
 
,
Atlantis Press, Springer, 2012.

A PDF of Chapter 2, "Natural Language Understanding
and World Knowledge," is available for download.

The philosophical background is the distinction between
syntax  and semantics . See (for instance)

Gian-Carlo Rota on Syntax and Semantics

Saturday, March 19, 2016

To Sum It

Filed under: General — m759 @ 2:45 pm

"To sum it all up I see mathematical activity as
a jumping ahead and then plodding along
to chart a path by rational toil."

Verena Huber-Dyson, Feb. 15, 1998

"VERENA HUBER-DYSON, mathematician and logician,
died yesterday [March 12, 2016] in Bellingham, Washington,
at the age of 92. She was Emeritus Professor of the 
Philosophy Department, University of Calgary, Alberta."

—   John Brockman at edge.org, March 13, 2016

Some posts from earlier this month are related to mathematical
activity, Bellingham, jumping ahead, and plodding along:

"The process of plodding is being analyzed by proof theory,
a prolific branch of meta mathematics. Still riddled with questions
is the jumping." — Huber-Dyson, loc. cit.

Still riddled — "Why IS a raven like a writing desk?"

Tuesday, August 11, 2015

Fountain Head …

Filed under: General — m759 @ 10:20 am

A mashup for Josefine   from R. Mutt.

Saturday, November 29, 2014

Mountain, Fountain

Filed under: General — m759 @ 6:29 am

For the late Vladimir Nabokov, author of Pale Fire :

He took his article from a steel file:
"It's accurate. I have not changed her style.
There's one misprint–not that it matters much:
Mountain, not fountain. The majestic touch."

Click for a related Hollywood Reporter  story.

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"

Saturday, September 22, 2012

Occupy Space

Filed under: General — Tags: , , — m759 @ 11:00 pm

(Continued)

"The word 'space' has, as you suggest, a large number of different meanings."

Nanavira Thera in [Early Letters. 136] 10.xii.1958

From that same letter (links added to relevant Wikipedia articles)—

Space (ākāsa) is undoubtedly used in the Suttas
to mean 'what/where the four mahābhūtas are not',
or example, the cavities in the body are called ākāsa
M.62—Vol. I, p. 423). This, clearly, is the everyday
'space' we all experience—roughly, 'What I can move
bout in', the empty part of the world. 'What you can't
ouch.' It is the 'space' of what Miss Lounsberry has so
appily described as 'the visible world of our five
senses'. I think you agree with this. And, of course, if
this is the only meaning of the word that we are
going to use, my 'superposition of several spaces' is
disqualified. So let us say 'superposition of several
extendednesses'. But when all these
extendednesses have been superposed, we get
'space'—i.e. our normal space-containing visible
world 'of the five senses'. But now there is another
point. Ākāsa is the negative of the four mahābhūtas,
certainly, but of the four mahābhūtas understood
in the same everyday sense—namely, solids (the
solid parts of the body, hair, nails, teeth, etc.),
liquids (urine, blood, etc.), heat and processes
(digestion) and motion or wind (N.B. not 'air').
These four, together with space, are the normal
furniture of our visible world 'of the five senses',
and it is undoubtedly thus that they are intended
in many Suttas. But there is, for example, a Sutta
(I am not sure where) in which the Ven. Sariputta
Thera is said to be able to see a pile of logs
successively as paṭhavi, āpo, tejo, and vāyo; and
it is evident that we are not on the same level.
On the everyday level a log of wood is solid and
therefore pathavi (like a bone), and certainly not
āpo, tejo, or vāyo. I said in my last letter that I
think that, in this second sense—i.e. as present in,
or constitutive of, any object (i.e. = rupa)—they
are structural and strictly parallel to nama and can
be defined exactly in terms of the Kummer
triangle. But on this fundamental level ākāsa has
no place at all, at least in the sense of our normal
everyday space. If, however, we take it as equivalent
to extendedness then it would be a given arbitrary
content—defining one sense out of many—of which
the four mahābhūtas (in the fundamental sense) are
the structure. In this sense (but only in this sense—
and it is probably an illegitimate sense of ākāsa)
the four mahābhūtas are the structure of space
(or spatial things). Quite legitimately, however, we
can say that the four mahābhūtas are the structure
of extended things—or of coloured things, or of smells,
or of tastes, and so on. We can leave the scientists'
space (full of right angles and without reference to the
things in it) to the scientists. 'Space' (= ākāsa) is the
space or emptiness of the world we live in; and this,
when analyzed, is found to depend on a complex
superposition of different extendednesses (because
all these extendednesses define the visible world
'of the five senses'—which will include, notably,
tangible objects—and this world 'of the five
senses' is the four mahābhūtas [everyday space]
and ākāsa).

Your second letter seems to suggest that the space
of the world we live in—the set of patterns
(superimposed) in which “we” are—is scientific space.
This I quite disagree with—if you do suggest it—,
since scientific space is a pure abstraction, never
experienced by anybody, whereas the superimposed
set of patterns is exactly what I experience—the set
is different for each one of us—, but in all of these
sets 'space' is infinite and undifferentiable, since it is,
by definition, in each set, 'what the four mahābhūtas
are not'. 

A simpler metaphysical system along the same lines—

The theory, he had explained, was that the persona
was a four-dimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in three-space projection) figure of equal lines and
ninety degree angles.

The Gameplayers of Zan ,
a 1977 novel by M. A. Foster

"I am glad you have discovered that the situation is comical:
 ever since studying Kummer I have been, with some difficulty,
 refraining from making that remark."

— Nanavira Thera, [Early Letters, 131] 17.vii.1958

Monday, December 12, 2011

X o’ Jesus

Filed under: General,Geometry — Tags: — m759 @ 2:56 pm

Religion for stoners, in memory of Horselover Fat

Amazon.com gives the publication date of a condensed
version* of Philip K. Dick's Exegesis  as Nov. 7, 2011.

The publisher gives the publication date as Nov. 8, 2011.

Here, in memory of the author, Philip K. Dick (who sometimes
called himself, in a two-part pun, "Horselover Fat"), is related
material from the above two dates in this  journal—

Tuesday, November 8, 2011

Stoned

m759 @ 12:00 PM 

…. Update of 9:15 PM Nov. 8, 2011—

From a search for the word "Stoned" in this journal—

Sunday, January 2, 2011

 

A Universal Form

m759 @ 6:40 PM

Simon Critchley today in the New York Times  series "The Stone"—

Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”

Actually, that's two diagonals. See Kulturkampf at the Times  and Geometry of the I Ching .

[Here the "Stoned" found by the search
was the title of Critchley's piece, found in its URL—
"http://opinionator.blogs.nytimes.com/2011/01/02/stoned/ ."]

See also Monday's post "The X Box" with its illustration

http://www.log24.com/log/pix11C/111107-XBoxSum.bmp .


Monday, November 7, 2011

The X Box

m759 @ 10:30 AM 

"Design is how it works." — Steve Jobs, quoted in
 The New York Times Magazine  on St. Andrew's Day, 2003

http://www.log24.com/log/pix11C/111107-XBoxSum.bmp .

For some background on this enigmatic equation,
see Geometry of the I Ching.

 

Merry Xmas.

See also last night's post and the last words of Steve Jobs.

* Houghton Mifflin Harcourt, the publisher, has, deliberately or not, sown confusion
    about whether this is only the first of two volumes.

Tuesday, November 8, 2011

Stoned*

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

From "The Stone" in Sunday's online New York Times

Cosmic Imagination

By William Egginton

Do the humanities need to be defended from hard science?

Illustration of hard science —

http://www.log24.com/log/pix11C/111108-ScienceBall.jpg

Illustration of the humanities —

http://www.log24.com/log/pix11C/111108-HumanitiesBall.jpg

(The above illustrations from Sunday's "The Stone" are by Leif Parsons.)

Midrash by the Coen brothers— "The Dude Abides."

See also 10/10/10The Day of the Tetractys

http://www.log24.com/log/pix11C/111108-BowlingPinsDiagram.jpg

* Update of 9:15 PM Nov. 8, 2011—

From a search for the word "Stoned" in this journal—

Sunday, January 2, 2011

 

A Universal Form

m759 @ 6:40 PM

Simon Critchley today in the New York Times  series "The Stone"—

Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”

Actually, that's two diagonals. See Kulturkampf at the Times  and Geometry of the I Ching .

[Here the "Stoned" found by the search
was the title of Critchley's piece, found in its URL—
"http://opinionator.blogs.nytimes.com/2011/01/02/stoned/ ."]

See also Monday's post "The X Box" with its illustration

http://www.log24.com/log/pix11C/111107-XBoxSum.bmp .

Sunday, October 23, 2011

Starting Out in the Evening (continued)

Filed under: General — m759 @ 6:48 am

(This post's title was appropriated from a novel by Brian Morton.)

Yesterday's evening New York Lottery— 229 and 9294.

Alex Ross in the online New Yorker  quotes a bad essay he wrote in college titled…

“The Grand Hotel Abyss: History and Violence in ‘The Shining,’”

which purports to analyze the famous scene in which Jack Nicholson
types the phrase “All work and no play makes Jack a dull boy”:

Nicholson has become a chomping-machine of language,
recycling stock phrases, appropriating whatever
drifts into his path. His words are nothing but echoes….

The lottery's 229 may be interpreted as "2/29." See a post from that date in 2008
involving echoes and the abyss.

The lottery's 9294 may be interpreted as "9/2/94." A search for that date yields
an article from Pacific Stars and Stripes

http://www.log24.com/log/pix11C/111023-StarsAndStrips9294.GIF

That article is echoed  by a later Doonesbury caricature
of a professor discussing echoes  in black rhetoric. That
caricature is from the 2/29 post

http://www.log24.com/log/pix08/080229-Doonesbury3.jpg

Sunday, January 2, 2011

A Universal Form

Filed under: General,Geometry — m759 @ 6:40 pm

Simon Critchley today in the New York Times  series "The Stone"—

Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”

Actually, that's two diagonals. See Kulturkampf at the Times  and Geometry of the I Ching .

Saturday, June 26, 2010

Bold and Brilliant Emergence

Filed under: General — Tags: , — m759 @ 7:20 pm

"Rosemary Desjardins argues boldly and brilliantly that the Theaetetus  contains not only an answer to the question of the character of knowledge, but considerably more besides — an outline of a Platonic ontology. That ontology is neither materialist nor idealist (it is not a theory of forms), but like the twentieth century theory known as generative emergence holds that beings are particular interactive combinations of material elements. On this view, while wholes (for example, words, to use a Platonic model) may be analyzed into their elemental parts (letters), each whole has a property or quality separate from the aggregated properties of its parts."

— Stephen G. Salkever, 1991 review of The Rational Enterprise : Logos in Plato's Theaetetus  (SUNY Press, 1990)

See also "strong emergence" in this journal.

Wednesday, April 28, 2010

Eightfold Geometry

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

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

Sunday, March 14, 2010

Joplin at the Lapin Blanc

Filed under: General — m759 @ 10:00 pm

For Alyssa

From Google News this evening–

Now Showing- “Alice in Wonderland”

Red and Black – Michael Prochaska
Tim Burton receives praise for his imaginative constructions
(“Nightmare Before Christmas” was inspired by a dream),
but if you analyze his work on adaptations of classic
children-oriented escapism, whimsical surrealism bears
a ball and chain.

http://www.log24.com/log/pix10/100314-LapinBlancSm.gif

A serenade for Einstein's birthday–

Ball and Chain– Janis Joplin

If you got it today you don't want it tomorrow, man,
'cause you don't need it, 'cause as a matter of fact,
as we discovered in the train,
tomorrow never happens, man.
It's all the same fucking day, man.

Friday, September 11, 2009

Friday September 11, 2009

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

Swarthmore physics honors thesis, 136 pp., 2007–

Abstract:

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

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

Interpolated figure
from Log24:

 

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

Cf. August 5, 2009.


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

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

Thursday, June 4, 2009

Thursday June 4, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:00 am
Steps
continued from
October 16, 2008
 

New collection release:
Pattern in Islamic Art
from David Wade

October 16, 2008

David Wade has partnered with ARTstor to distribute approximately 1,500 images of Islamic art, now available in the Digital Library. These images illustrate patterns and designs found throughout the Islamic world, from the Middle East and Europe to Central and South Asia. They depict works Wade photographed during his travels, as well as drawings and diagrams produced for publication. Reflective of Wade's particular interest in symmetry and geometry, these images analyze and break down common patterns into their basic elements, thereby revealing the underlying principles of order and balance in Islamic art. Islamic artists and craftsmen employed these intricate patterns to adorn all types of surfaces, such as stone, brick, plaster, ceramic, glass, metal, wood, and textiles. The collection contains examples of ornamentation from monumental architecture to the decorative arts.

To view the David Wade: Pattern in Islamic Art collection: go to the ARTstor Digital Library, browse by collection, and click "David Wade: Pattern in Islamic Art;" or enter the Keyword Search: patterninislamicart.

For more detailed information about this collection, visit the David Wade: Pattern in Islamic Art collection page.

 
The above prose illustrates
the institutional mind at work.

Those who actually try to view
the Wade collection will
encounter the following warning:

To access the images in the ARTstor Digital Library you need to be affiliated with a participating institution (university, college, museum, public library or K-12 school).
You say
"go to the ARTstor Digital Library,"
I say
"theatlantic.com/doc/200305/lewis."
 

Sunday, February 15, 2009

Sunday February 15, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:00 am
From April 28, 2008:

Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
  an epiphany.

A 2x4 array of squares

One approach to
 the epiphany:

"Transformations play
  a major role in
  modern mathematics."
– A biography of
Felix Christian Klein

See 4/28/08 for examples
of such transformations.

 
Related material:

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, pp. 117-118:

"… his point of origin is external nature, the fount to which we come seeking inspiration for our fictions. We come, many of Stevens's poems suggest, as initiates, ritualistically celebrating the place through which we will travel to achieve fictive shape. Stevens's 'real' is a bountiful place, continually giving forth life, continually changing. It is fertile enough to meet any imagination, as florid and as multifaceted as the tropical flora about which the poet often writes. It therefore naturally lends itself to rituals of spring rebirth, summer fruition, and fall harvest. But in Stevens's fictive world, these rituals are symbols: they acknowledge the real and thereby enable the initiate to pass beyond it into the realms of his fictions.

Two counter rituals help to explain the function of celebration as Stevens envisions it. The first occurs in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer. A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination. For in 'Notes Toward a Supreme Fiction' he tells us that

... the first idea was not to shape the clouds
In imitation. The clouds preceded us.      

There was a muddy centre before we breathed.
There was a myth before the myth began,
Venerable and articulate and complete.      

From this the poem springs: that we live in a place
That is not our own and, much more, not ourselves
And hard it is in spite of blazoned days.      

We are the mimics.

                                (Collected Poems, 383-84)

Believing that they are the life and not the mimics thereof, the world and not its fiction-forming imitators, these young men cannot find the savage transparence for which they are looking. In its place they find the pediment, a scowling rock that, far from being life's source, is symbol of the human delusion that there exists a 'form alone,' apart from 'chains of circumstance.'

A far more productive ritual occurs in 'Sunday Morning.'…."

For transformations of a more
specifically religious nature,
see the remarks on
Richard Strauss,
"Death and Transfiguration,"
(Tod und Verklärung, Opus 24)

in Mathematics and Metaphor
on July 31, 2008, and the entries
of August 3, 2008, related to the
 death of Alexander Solzhenitsyn.
 

Friday, May 23, 2008

Friday May 23, 2008

Filed under: General — m759 @ 11:28 am
The Idea
of Identity

“Philosophers ponder the idea
 of identity: what it is to give
 something a name on Monday
 and have it respond to 
  that name on Friday….”

Bernard Holland 

Linked to on
Monday, May 19
:

Conclusion of the film 'Analyze That'

Conclusion of “Analyze That” —

“There’s a place for us….”

New York Times
on Friday, May 23:

“A poem should not mean
But be”

Archibald MacLeish,
quoted in a Friday comment
on a Thursday night column
by Rosanne Cash

Thursday evening photo
by Josh Haner for Friday’s
online New York Times:

http://www.log24.com/log/pix08/080522-Bridge2.jpg

Brooklyn Bridge Turns 125

Monday, April 28, 2008

Monday April 28, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 7:00 am
Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
  an epiphany.

A 2x4 array of squares

One approach to
 the epiphany:

"Transformations play
  a major role in
  modern mathematics."
– A biography of
Felix Christian Klein

The above 2×4 array
(2 columns, 4 rows)
 furnishes an example of
a transformation acting
on the parts of
an organized whole:

The 35 partitions of an 8-set into two 4-sets

For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model
."

For a related structure–
  not rectangle but cube– 
see Epiphany 2008.

Sunday, July 29, 2007

Sunday July 29, 2007

Filed under: General — m759 @ 7:59 am
Nordic Truth: Jewish Fiction:
Snowball In Hell
From The New York Times in 2005:

Portrait of conductor
Arild Remmereit:


Arild Remmereit

April 24, 2005
Have Baton,
Will Travel

by James R. Oestreich

 
PITTSBURGH

“HE’S the hottest conductor you’ve never heard of….

In music, as in most other pursuits, one person’s misfortune can be another’s opportunity. Many a podium career has been built on successful substitutions…. typically, the process is cumulative and measured.

In Mr. Remmereit’s case, it seems a sort of spontaneous combustion…. he seems destined for big things, and soon.

Regarding his sudden change in stature, he spoke as if from afar. ‘The snowball has reached such a size that it has started to roll,’ he said matter-of-factly….

‘It’s terrifying when it happens,’ he said, ‘but I can’t tell you how naively happy I am when it goes well. These are such major steps that I wasn’t even hoping for a few weeks ago.’

ARILD REMMEREIT (pronounced AHR-eeld REMM-uh-right, with the r’s heavily rolled) was born in a village in Norway, between Bergen and Trondheim, and has lived in Vienna since 1987. Slim and fresh-faced at 43, he has had a busy but low-level career in Europe….

So here he was, on April 15, conducting the Pittsburgh Symphony… in a vintage… Germanic program…. Wagner’s ‘Siegfried Idyll,’ Schumann’s Fourth Symphony and Brahms’s Second Piano Concerto….”

Review:

Württemberg Philharmonic February 2004
Nielsen, Sibelius, Grieg.

Reutlinger Nachrichten.
“Distant closeness, close distance.

Arild Remmereit as a guest conductor: ‘As when the sun rises in the North.’ The Philharmonics and their brilliant guest conductor fetched the mind-blowing, tempting and exciting Scandinavia.

It was like a lucky strike to see the Norwegian conductor on stage with the Philharmonic. When he conducts the Dane Nielsen, the Finn Sibelius and the Norwegian Grieg, one can really feel that this man has the locally marked music floating in his blood.”

From The New York Times today:
 

Discussion of
a new novel:
Variations on the Beast

Variations on
the Beast
,
by psychoanalyst
Henry Grinberg

An interview with Henry Grinberg conducted by James R. Oestreich:

“For those who find inspiration and edification in great art, it is always painful to be reminded that artists are not necessarily admirable as people and that art is powerless in the face of great evil. That truth was baldly evident in Nazi Germany and in the way the regime used and abused music and musicians, to say nothing of the way it used and abused human beings of all kinds.

[A new novel touches on] these issues…. In Variations on the Beast (Dragon Press), Henry Grinberg, a psychoanalyst, posits Hermann Kapp-Dortmunder, a powerful maestro, as a fictional rival of Wilhelm Furtwängler (whose qualms about working under the regime he does not share) and Herbert von Karajan (whose vaulting ambition he does).”

GRINBERG:

“And it soon occurred to me… that, my God, a lot of the famous, the notable, the moving, the magnificent composers in the 18th and 19th centuries and earlier were Germans. And I tried to understand, how did such a nation turn out to be so bestial and cruel, so indifferent to the suffering of others? And I have no explanation for it.

As a practicing psychoanalyst, I can see individual expressions of rage and their causes and their so-called justifications. But for a whole nation to be consumed, to be seduced by an overwhelming idea– well, there are rationalizations, I guess, but not explanations. There’s no forgiveness for this. And I tried to put together a story of a person who was a participant and a causer of these kinds of things….

So I sort of poured my feelings of contempt and rage into the character I was devising. And I have to admit, after having been psychoanalyzed myself in preparation for the training, that something of Hermann Kapp-Dortmunder exists in me. I shudder to think that this may be so, but I have to accept the possibility. Murderous thoughts may have occurred to me, but, thank God, I’ve never killed anyone.”

Sunday, March 27, 2005

Sunday March 27, 2005

Filed under: General — m759 @ 5:24 am
From
The Mother Ship

Camille Paglia, The Magic of Images:

“Young people today are flooded with disconnected images but lack a sympathetic instrument to analyze them as well as a historical frame of reference in which to situate them…. The new generation, raised on TV and the personal computer but deprived of a solid primary education, has become unmoored from the mother ship of culture.”

Easter Greetings.

Saturday, July 31, 2004

Saturday July 31, 2004

Filed under: General — m759 @ 4:01 pm

It’s Alive!

“People once worried about the boundary between the living and the nonliving. Today, the boundary seems meaningless….”

— Attributed to Francis Crick
    (now among the nonliving) 

Opening of
the above novel:

“My name is David Tennant, M.D.
I’m professor of ethics at the
University of Virginia Medical School,
and if you’re watching
this tape, I’m dead.”

From a public-relations newsletter
of the University of Southern California’s
Health Sciences Campus
dated April 20, 2001:

Discussing the Ethics
of Frankenstein

W. French Anderson, the physician and scientist who carried out the first human gene therapy clinical trial, will discuss the ethical issues involved in human genetic engineering and how science fiction has shaped the public’s perception of this budding new technology, Thursday, May 3, at noon, in USC’s Mayer Auditorium.

The lecture, titled “Frankenstein, GATTACA and Gene Therapy,” is free and open to the public. Mayer Auditorium is located on USC’s Health Sciences campus.

In his talk, Anderson will analyze the book Frankenstein and its filmic progeny and discuss how the Frankenstein story has captured the public’s imagination. He will also examine the ethical and moral issues raised by the book and movies and address the charge that, like Dr. Frankenstein, today geneticists are attempting to play God.

Anderson will evaluate the 1997 movie GATTACA, a cautionary tale about injustice in a 21st century society run by genetically “superior” elites. Anderson, who was a scientific consultant for the movie and is now proposing to carry out the first in utero gene therapy trial, will discuss the impact of GATTACA on the public’s understanding of genetic engineering.

See also the previous entry,
on Anderson’s arrest Friday
on charges of child molestation.

For the origin of the title GATTACA,
see The Diamond Code

Wednesday, March 19, 2003

Wednesday March 19, 2003

Filed under: General,Geometry — m759 @ 4:04 am


Aptheker

  A Look at the Rat

In memory of Herbert Aptheker, theoretician of the American Communist Party, who died on St. Patrick’s Day, 2003 —

From The New Yorker, issue dated March 24, 2003, Louis Menand on Edmund Wilson’s To the Finland Station:

“Wilson did know what was going on in the Soviet Union in the nineteen-thirties, as his pages on Stalin in To the Finland Station make clear. The problem wasn’t with Stalin; the problem was with Lenin, the book’s ideal type of the intellectual as man of action. Wilson admitted that he had relied on publications controlled by the Party for his portrait of Lenin. (Critical accounts were available; for example, the English translation of the émigré Mark Landau-Aldanov’s Lenin was published, by Dutton, in 1922.) Lenin could create an impression of selfless humanitarianism; he was also a savage and ruthless politician—a ‘pail of milk of human kindness with a dead rat at the bottom,’ as Vladimir Nabokov put it to Wilson in 1940, after reading To the Finland Station.  In the introduction to the 1972 edition, Wilson provided a look at the rat. He did not go on to explain in that introduction that the most notorious features of Stalin’s regime—the use of terror, the show trials, and the concentration camps—had all been inaugurated by Lenin. To the Finland Station begins with Napoleon’s betrayal of the principles of the French Revolution; it should have ended with Lenin’s betrayal of European socialism.” 

From Herbert Aptheker, “More Comments on Howard Fast“:

“We observe that in the list of teachers whom Howard Fast names as most influential in his own life there occur the names of fourteen individuals from Jefferson to Bernard Shaw, Upton Sinclair to Marx, Douglass to Engels, but there is no room for Lenin.
   He is, I think, an important teacher, too; indeed, in my view, Lenin is the greatest figure in the whole galaxy of world revolutionary leaders. He is, certainly, the greatest analyzer of and fighter against imperialism.”

For more on Howard Fast, see my entry
“Death Knell” of March 13, 2003

For a look at the pail of milk, see
the New Yorker cover in Geometry for Jews.

For a more cheerful look at geometry
on this St. Joseph’s Day, see
Harry J. Smith’s

Tesseract Site.

“There is such a thing as a tesseract.”
A Wrinkle in Time

Wednesday, November 27, 2002

Wednesday November 27, 2002

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

Waiting for Logos

Searching for background on the phrase "logos and logic" in yesterday's "Notes toward a Supreme Fact," I found this passage:

"…a theory of psychology based on the idea of the soul as the dialectical, self-contradictory syzygy of a) soul as anima and b) soul as animus. Jungian and archetypal psychology appear to have taken heed more or less of only one half of the whole syzygy, predominantly serving an anima cut loose from her own Other, the animus as logos and logic (whose first and most extreme phenomenological image is the killer of the anima, Bluebeard). Thus psychology tends to defend the virginal innocence of the anima and her imagination…"

— Wolfgang Giegerich, "Once More the Reality/Irreality Issue: A Reply to Hillman's Reply," website 

The anima and other Jungian concepts are used to analyze Wallace Stevens in an excellent essay by Michael Bryson, "The Quest for the Fiction of an Absolute." Part of Bryson's motivation in this essay is the conflict between the trendy leftist nominalism of postmodern critics and the conservative realism of more traditional critics:

"David Jarraway, in his Stevens and the Question of Belief, writes about a Stevens figured as a proto-deconstructionist, insisting on 'Steven's insistence on dismantling the logocentric models of belief' (311) in 'An Ordinary Evening in New Haven.' In opposition to these readings comes a work like Janet McCann's Wallace Stevens Revisited: 'The Celestial Possible', in which the claim is made (speaking of the post-1940 period of Stevens' life) that 'God preoccupied him for the rest of his career.'"

Here "logocentric" is a buzz word for "Christian." Stevens, unlike the postmodernists, was not anti-Christian. He did, however, see that the old structures of belief could not be maintained indefinitely, and pondered what could be found to replace them. "Notes toward a Supreme Fiction" deals with this problem. In his essay on Stevens' "Notes," Bryson emphasizes the "negative capability" of Keats as a contemplative technique:

"The willingness to exist in a state of negative capability, to accept that sometimes what we are seeking is not that which reason can impose…."

For some related material, see Simone Weil's remarks on Electra waiting for her brother Orestes. Simone Weil's brother was one of the greatest mathematicians of the past century, André Weil.

"Electra did not seek Orestes, she waited for him…"

— Simone Weil

"…at the end, she pulls it all together brilliantly in the story of Electra and Orestes, where the importance of waiting on God rather than seeking is brought home forcefully."

— Tom Hinkle, review of Waiting for God

Compare her remarks on waiting for Orestes with the following passage from Waiting for God:

"We do not obtain the most precious gifts by going in search of them but by waiting for them. Man cannot discover them by his own powers, and if he sets out to seek for them he will find in their place counterfeits of which he will be unable to discern falsity.

The solution of a geometry problem does not in itself constitute a precious gift, but the same law applies to it because it is the image of something precious. Being a little fragment of particular truth, it is a pure image of the unique, eternal, and living Truth, the very Truth that once in a human voice declared: "I am the Truth."

Every school exercise, thought of in this way, is like a sacrament.

In every school exercise there is a special way of waiting upon truth, setting our hearts upon it, yet not allowing ourselves to go out in search of it. There is a way of giving our attention to the data of a problem in geometry without trying to find the solution…."

— Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of  God"

Weil concludes the preceding essay with the following passage:

"Academic work is one of those fields containing a pearl so precious that it is worth while to sell all of our possessions, keeping nothing for ourselves, in order to be able to acquire it."

This biblical metaphor is also echoed in the work of Pascal, who combined in one person the theological talent of Simone Weil and the mathematical talent of her brother. After discussing how proofs should be written, Pascal says

"The method of not erring is sought by all the world. The logicians profess to guide to it, the geometricians alone attain it, and apart from their science, and the imitations of it, there are no true demonstrations. The whole art is included in the simple precepts that we have given; they alone are sufficient, they alone afford proofs; all other rules are useless or injurious. This I know by long experience of all kinds of books and persons.

And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."

— Blaise Pascal, The Art of Persuasion

 

For more diamond metaphors and Jungian analysis, see

The Diamond Archetype.

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