Log24

Thursday, May 7, 2020

Kant as Diamond Cutter

Filed under: General — Tags: , — m759 @ 4:26 AM

“He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter.”

— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.

Kant’s  “category theory” —

“In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….

The categories must be ‘schematized’ because their non-empirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rule-governed way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori  in origin, and yet they are sensible since they are constructed in concreto . ”

— Shabel, Lisa, “Kant’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy  (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kant-mathematics/>.

See also The Diamond Theorem and Octad.us.

Friday, March 23, 2018

From the Personal to the Platonic

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

On the Oslo artist Josefine Lyche —

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

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

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic

The Diamond Cube.

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

Friday, January 5, 2018

Subway Art for Plato’s Ghost

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

Suggested by the previous post

See also the post Plato's Ghost of March 3, 2010.

Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: General,Geometry — Tags: — m759 @ 12:35 PM

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .

DOI

Monday, October 30, 2017

Plato and His Modern Rivals

Filed under: General,Geometry — m759 @ 1:00 PM

The previous post's Lewis Carroll cover,
modified to illustrate Plato's diamond

Book cover modified to illustrate 'Plato and His Modern Rivals'

See also "To Forge
a Head
" (Oct. 27).

Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM
 

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by  V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Saturday, October 8, 2016

Unity of Opposites: Plato and Beyond

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

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

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

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

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

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

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

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

Saturday, September 6, 2014

Plato Thanks the Academy

Filed under: General — m759 @ 9:00 PM

(Continued)

Plato at Stanford:
Lacan and the Matheme of Fantasy

“… [in] the matheme of fantasy ($ ◊ ),
the diamond-shaped “lozenge” (poinçon )
can be read as a condensation of four symbols:
one, (the logical symbol for conjunction [“and”]);
two, (the logical symbol for disjunction [“or”]);
three, > (the mathematical symbol for “greater than”); and,
four, < (the mathematical symbol for “less than”). As per
Lacan’s matheme, the subject’s desires are scripted and
orchestrated by an unconscious fundamental fantasy
in which the desiring subject ($) is positioned in relation to
its corresponding object-cause of desire ( ).”

plato.stanford.edu, Stanford Encyclopedia of Philosophy

Plato.stanford.edu on Lacan, and Halle Berry in 'Frankie and Alice'

The Stanford author: 

The author is a professor in Albuquerque.
For other perspectives, see that city in this journal.

For the film  authors, see IMDb.

Friday, August 1, 2014

The Diamond-Theorem Correlation

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

Click image for a larger, clearer version.

IMAGE- The symplectic correlation underlying Rosenhain and Göpel tetrads

Tuesday, December 3, 2013

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A new website illustrates its URL.
See DiamondSpace.net.

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

Monday, February 11, 2013

The Penrose Diamond

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

IMAGE- The Penrose Diamond

Related material:

(Click to enlarge.)

See also remarks on Penrose linked to in Sacerdotal Jargon.

(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)

Thursday, May 31, 2012

Black Diamond

Filed under: General — Tags: — m759 @ 12:26 PM

IMAGE- Four-elements-diamond test problem in the style of Raven's Progressive Matrices (answer: the black diamond)

“To say more is to say less.”
― Harlan Ellison, as quoted at goodreads.com

Saying less—

Saturday, March 17, 2012

The Purloined Diamond

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

(Continued)

The diamond from the Chi-rho page
of the Book of Kells —

The diamond at the center of Euclid's
Proposition I, according to James Joyce
(i.e., the Diamond in the Mandorla) —

Geometry lesson: the vesica piscis in Finnegans Wake

The Diamond in the Football

Football-mandorla

“He pointed at the football
  on his desk. ‘There it is.’”
         – Glory Road
   

Wednesday, December 21, 2011

The Purloined Diamond

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

Stephen Rachman on "The Purloined Letter"

"Poe’s tale established the modern paradigm (which, as it happens, Dashiell Hammett and John Huston followed) of the hermetically sealed fiction of cross and double-cross in which spirited antagonists pursue a prized artifact of dubious or uncertain value."

For one such artifact, the diamond rhombus formed by two equilateral triangles, see Osserman in this journal.

Some background on the artifact is given by John T. Irwin's essay "Mysteries We Reread…" reprinted in Detecting Texts: The Metaphysical Detective Story from Poe to Postmodernism .

Related material—

Mathematics vulgarizer Robert Osserman died on St. Andrew's Day, 2011.

A Rhetorical Question

Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer

http://www.log24.com/log/pix11C/111130-SunshineCleaning.jpg

Above: Amy Adams in "Sunshine Cleaning"

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

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

Some background

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

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

From yesterday's Sunday morning New York Times

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

Drew Westen, professor at Emory University

From May 22, 2009

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

The above ad is by
  Diane Robertson Design—

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

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

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

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

Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest  thing there is
(Tractatus Logico-Philosophicus  No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach  sein. Sie ist vor  aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh.  No. 5.5563.)

See also

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

* Updates of 9:29 PM ET July 10, 2011—

A  mnemonic  from a course titled "Galois Connections and Modal Logics"—

"Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and diamond (diamond) for possibly. Then, for example, diamondP  can be read as
'it is possibly the case that P .'"

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

"The diamond symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’"

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

Monday, December 27, 2010

Church Diamond

Filed under: General,Geometry — Tags: — m759 @ 3:09 PM

IMAGE- The diamond property

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

From “NYU Lambda Seminar, Week 2” —

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

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

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

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

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

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

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

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

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

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

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

Link Two — Divine Symmetry

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

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

George Steiner, Grammars of Creation

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

Link Three – Spanning the Arc —

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

Part B — “Span” in category theory at nLab —

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

Also from nLab — Completing Spans to Diamonds

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

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

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

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

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

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

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

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

Thursday, July 1, 2010

Plato’s Code

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

John Allen Paulos yesterday at Twitter

"Plato's code cracked? http://bit.ly/ad6k1S
Fascinating if not a hoax or hype."

The story that Paulos linked to is about a British
academic who claims to have found some
symbolism hidden in Plato's writings by
splitting each into 12 parts and correlating
the 12 parts with semitones of a musical scale.

I prefer a different approach to Plato that is
related to the following hoax and hype—

HOAX:

From Dan Brown's novel Angels & Demons  (2000)

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

HYPE:

Image-- From 'Alchemy,' by Holmyard, the diamond of Aristotle's 4 elements and 4 qualities

This  four-elements diamond summarizes the classical
four elements and four qualities neatly, but some scholars
might call the figure "hype" since it deals with an academically
disreputable subject, alchemy, and since its origin is unclear.

For the four elements' role in some literature more respectable
than Dan Brown's, see Poetry's Bones.

Although an author like Brown might spin the remarks
below into a narrative—  The Plato Code — they are
neither  hoax nor hype.

NOT  HOAX:

Image-- From the Diamond in Plato's Meno to Modern Finite Geometry

NOT  HYPE:

http://www.log24.com/log/pix10A/100626-CrossOnSocratesSm.gif

For related non-hoax, non-hype remarks, see
The Rational Enterprise: Logos in Plato's Theaetetus,
by Rosemary Desjardins.

Those who prefer  hoax and hype in their philosophy may consult
the writings of, say, Barbara Johnson, Rosalind Krauss, and—
in yesterday's NY Times's  "The Stone" columnNancy Bauer.

Image-- The Philosophers' Stone according to The New York Times

— The New York Times

Saturday, June 26, 2010

Plato’s Logos

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

“The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was ‘The Idea of the Beautiful in Plato’s Dialogues’…. My investigation traces the historical destiny of the same concept….”

* See Cassirer’s Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.

— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924

On a figure from Plato’s Meno

IMAGE- Plato's diamond and finite geometry

The above figures illustrate Husserl’s phrase  “eidetic variation”
a phrase based on Plato’s use of eidos, a word
closely related to the word “idea” in Panofsky’s title.

For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his “The Concept of Group and the Theory of Perception.”

Sketch of some further remarks—

http://www.log24.com/log/pix10A/100626-Theories.jpg

The Waterfield question in the sketch above
is from his edition of Plato’s Theaetetus
(Penguin Classics, 1987).

The “design theory” referred to in the sketch
is that of graphic  design, which includes the design
of commercial logos. The Greek  word logos
has more to do with mathematics and theology.

“If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity.”

— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato’s Theaetetus
, SUNY Press, 1990

Related material—

(Click to enlarge.)

http://www.log24.com/log/pix10A/100626-CrossOnSocratesSm.gif

Tuesday, June 8, 2010

From Plato to Finite Geometry

Filed under: General,Geometry — m759 @ 12:30 PM

A supplement to yesterday's post on variation of an eidos

Image-- Plato's diamond and a modern version from finite geometry

Enlarge.

Monday, September 14, 2020

Socrates in the Marketplace

Filed under: General — Tags: — m759 @ 7:39 AM

Plato's diamond in Jowett's version of the Meno dialogue

Diamond Matrix slide template at presentationgo.com

“The 2×2 matrix is commonly used in business strategy
as a representational tool to show conflicting concepts and
for decision making. This four-quadrant matrix diagram
is perfect to be used for business or marketing matrices
like BCG, SWOT, Ansoff, risk assessment…

Additionally, it will also be suitable to illustrate 4 ideas or
concepts.” [Link on “illustrate” added.]

See also a Log24 search for “Resplendent.”

Saturday, August 22, 2020

An Object Lesson

Filed under: General — Tags: — m759 @ 9:15 AM

Posts tagged Plato’s Video continue.

IMAGE- A Jesuit on words and shadows

Related literary remarks from this  journal on Oct. 1, 2016

— A Heart for the Gods of Mexico , Conrad Aiken, 1939

Related imagery this morning from the Gulf of Mexico —

Meanwhile, also on Oct. 1, 2016, related imagery from Star Wars Rebels —

Click here to enlarge the holocrons.

Sunday, August 2, 2020

The Sword and the Stone

Filed under: General — Tags: , — m759 @ 12:42 PM

A post of May 26, 2005, displays, if not the sword,
a place  for it —

Drama of the Diagonal

“The beautiful in mathematics resides in contradiction.
Incommensurability, logoi alogoi, was the first splendor
in mathematics.” — Simone Weil, Oeuvres Choisies,
éd. Quarto
, Gallimard, 1999, p. 100

Logos Alogos  by S. H. Cullinane

“To a mathematician, mathematical entities have their own existence,
they habitate spaces created by their intention.  They do things,
things happen to them, they relate to one another.  We can imagine
on their behalf all sorts of stories, providing they don’t contradict
what we know of them.  The drama of the diagonal, of the square…

— Dennis Guedj, abstract of “The Drama of Mathematics,” a talk
to be given this July at the Mykonos conference on mathematics and
narrative. For the drama of the diagonal of the square, see

Wednesday, July 1, 2020

The Long Strange Road

Filed under: General — Tags: — m759 @ 12:55 PM

“The road is long
With many a winding turn”

Neil Diamond

Magic Child

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

Wednesday, April 15, 2020

Oslo Prophet (after Varignon)

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

See also Invariance, a Log24 post from yesterday morning —

Note the resemblance to Plato’s Diamond.

“Causal Invariance” According to Wolfram

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

Stephen Wolfram yesterday —

“Causal invariance may at first seem like a rather obscure property.
But in the context of our models, we will see in what follows that
it may in fact be the key to a remarkable range of fundamental features
of physics, including relativistic invariance, general covariance, and
local gauge invariance, as well as the possibility of objective reality in
quantum mechanics.”

From . . .

Tuesday, April 14, 2020

Confluence, or:

Filed under: General — Tags: — m759 @ 7:21 PM

Church Diamond   Continued

The above article leads to remarks by Stephen Wolfram published today :

See also “Invariance” as the title of the previous post  here.

Invariance

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

Note the resemblance to Plato’s Diamond.

Click the Pritchard passage above for an interactive version.

Sunday, February 2, 2020

Tetrads for McLuhan, or “Blame It on Video”

Filed under: General — Tags: — m759 @ 11:22 PM

"I like to put people on myself by skipping logical steps
in the conversation until they're dizzy." — Jemima Brown
in The Eiger Sanction

Related posts — See "McLuhan Tetrad" in this journal.

Related theology — See  "The Meaning of Perichoresis."
Background — The New Yorker , "On Religion:
Richard Rohr Reorders the Universe," by Eliza Griswold
on February 2, 2020, and a different reordering in posts
tagged Eightfold Metaphysics.

Game of Shadows

Filed under: General — Tags: — m759 @ 2:27 PM

A search in this journal for "Game of Shadows" yields

IMAGE- A Jesuit on words and shadows

“Krauss, Portman; Portman, Krauss.”

Filed under: General — Tags: , , — m759 @ 12:58 AM

Prominent in the oeuvre  of art theorist Rosalind Krauss, the Klein group
is a four-element group named for Felix Christian Klein.

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

It is commonly known as the four-group.
Mathematicians sometimes call this group
"V," for its German name, Vierergruppe .

For those who prefer narrative to mathematics

Thursday, January 23, 2020

The Demarcation of Nothing

Filed under: General — Tags: , , , , — m759 @ 3:50 PM

" nothing could be demarcated as 'hors d'oeuvre'…"

Geoffrey Hartman in his Haskins Lecture for 2000
(quoted here on Columbus Day, 2004).

See also May Day 2016 and Gap Dance.

Saturday, January 18, 2020

Interplay

Filed under: General — Tags: — m759 @ 1:40 PM

"This interplay of necessity and contingency
produces our anxious— and highly pleasurable—
speculation about the future path of the story."

— Michel Chaouli in "How Interactive Can Fiction Be?"
(Critical Inquiry  31, Spring 2005, page 613.)

See also . . .

Nietzsche, 'law in becoming' and 'play in necessity'

Continuing previous Modal Diamond Box posts:

Nietzsche on Heraclitus— 'play in necessity' and 'law in becoming'— illustrated.

Tuesday, October 15, 2019

A White Stone for Bloom

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

Excerpt from a long poem by Eliza Griswold 
in a recent New Yorker —

Monday, June 10, 2019

Revelations of a Transcendental Subject

Filed under: General — Tags: — m759 @ 10:45 PM

(Namely, Plato's ghost)

Background: "Transcendental subject" is Kant's term for, more or less, the self.

Rosalind Krauss in 1978 —

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions 
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 

More from Krauss —

A book by an author with somewhat wider "cultural experience" —

See also "Plato's Diamond" in this journal.

Monday, May 13, 2019

“The Eyes of Orson Welles”…

Filed under: General — m759 @ 7:49 PM

is a TCM special at 8 PM ET this evening.

A snow-globe phrase from April 28 —

Bauble, Babel . . . Bubble —

Images including Plato's diamond on a tombstone

The "bubble" cited above —

For more metaphysical accounting, see
The Church of Synchronology.

Friday, January 11, 2019

Permutations at Oslo

Filed under: General — Tags: , , — m759 @ 8:45 PM

Webpage at Oslo of Josefine Lyche, 'Plato's Diamond'

See also yesterday’s  Archimedes at Hiroshima  and the
above 24 graphic permutations on  All Souls’ Day 2010.

For some backstory, see Narrative Line (November 10, 2014).

Sunday, October 28, 2018

Commonwealth Tales, or “Lost in Physics”

Filed under: General — m759 @ 11:00 PM

From Ulysses , by James Joyce —

John Eglinton, frowning, said, waxing wroth:

—Upon my word it makes my blood boil to hear anyone compare Aristotle with Plato.

—Which of the two, Stephen asked, would have banished me from his commonwealth?

Compare and contrast:

Plato's diamond in Jowett's version of the Meno dialogue

Fans of Plato might enjoy tales of Narnia, but fans of
James Joyce and Edgar Allan Poe might prefer
a tale by Michael Chabon from April 2001 about a
"doleful little corner of western Pennsylvania."

Monday, June 11, 2018

Glitter

Filed under: General,Geometry — Tags: — m759 @ 8:32 PM

A Scientific American  headline today —

Glittering Diamond Dust in Space
Might Solve a 20-Year-Old Mystery

Related art —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Background:  "Diamond Dust" + Glitter in this journal.

Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Monday, May 14, 2018

Logos at Harvard

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

In 2013, Harvard University Press changed its logo to an abstract "H."

Harvard University Press Logo, Before and After

Both logos now accompany a Harvard video first published in 2012,
"The World of Mathematical Reality." 

In the video, author Paul Lockhart discusses Varignon's theorem
without naming Varignon (1654-1722) . . .

Paul Lockhart on geometry

A related view of "mathematical reality" —

Note the resemblance to Plato's Diamond.

Saturday, April 14, 2018

Immanentizing the Transcendence

Filed under: General — m759 @ 10:15 AM

The title refers to the previous two posts.

Related literature —

Plato's Ghost: The Modernist Transformation of Mathematics
(Princeton University Press, 2008)  and . . .

Plato's diamond-in-a-matrix:

Plato's diamond in Jowett's version of the Meno dialogue

Friday, April 6, 2018

A Service

Filed under: General — m759 @ 11:36 AM

From a Boston Globe obituary for Andrew Lewis, an Oscar-nominated
screenwriter who reportedly died at 92 on Feb. 28, 2018 —

"A service has been held for Mr. Lewis . . . ."

—  Bryan Marquard, Globe staff, April 5, 2018

From this  journal on the reported date of his death —

The Globe reports that Lewis's father was Clarence Irving Lewis,
a professor of philosophy at Harvard University.

Fact check:  See page 246 of C. I. Lewis: The Last Great Pragmatist ,
by Murray G. Murphey (SUNY Press, 2005).

Figure (a) above is not unrelated to philosophy. See Plato 's Meno  dialogue.
See also a different diamond — a symbol devised by C. I. Lewis for use in
modal logic — in the post Wittgenstein's Diamond (July 10, 2011).

Wednesday, February 28, 2018

Mathematics and Narrative

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

(Continued)

Excerpts from a post of May 25, 2005 —

Above is an example I like of mathematics….

Here is an example I like of narrative:

Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn  certain  that
it  was  the  sort of experience that her mother would not have
approved of on a first date.
     "Is this all part of what we have to do to go to  Asgard?"
she said. "Or are you just fooling around?"
     "We will go to Asgard...now," he said.
     At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect  was as if he had twisted the entire world through a
billionth part of a billionth  part  of  a  degree.  Everything
shifted,  was  for  a  moment  minutely  out of focus, and then
snapped back again as a suddenly different world.

— Douglas Adams, The Long Dark Tea-Time of the Soul

Image from a different  different world —

Hat-tip to a related Feb. 26 weblog post
at the American Mathematical Society.

Sunday, December 10, 2017

Geometry

Google search result for Plato + Statesman + interlacing + interweaving

See also Symplectic in this journal.

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of  linear  (or line ) complex
in the finite projective space PG(3,2) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

Friday, December 8, 2017

Logos (Continued)

Filed under: General — Tags: — m759 @ 3:00 PM

Nietzsche, 'law in becoming' and 'play in necessity'

"Denn die Welt braucht ewig die Wahrheit,
also braucht sie ewig Heraklit:
obschon er ihrer nicht bedarf.
Was geht ihn sein Ruhm an?
Der Ruhm bei »immer fortfließenden Sterblichen!«,
wie er höhnisch ausruft.
Sein Ruhm geht die Menschen etwas an, nicht ihn,
die Unsterblichkeit der Menschheit braucht ihn,
nicht er die Unsterblichkeit des Menschen Heraklit.
Das, was er schaute, die Lehre vom Gesetz im Werden
und vom
Spiel in der Notwendigkeit 
, muß von jetzt
ab ewig geschaut werden: er hat von diesem größten
Schauspiel den Vorhang aufgezogen."

Logos for Philosophers
(Suggested by Modal Logic) —

Nietzsche, 'law in becoming' and 'play in necessity'

Saturday, October 28, 2017

Lowell Brown at Vanity Fair

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:18 PM

A sequel to the post  CP  is for Consolation Prize  (Sept. 3, 2016)

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry

Monday, July 17, 2017

Athens Meets Jerusalem . . .

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

At the Googleplex .

For those whose only interest in higher mathematics
is as a path to the occult

Plato's Diamond and the Hebrew letter Aleph —

          

and some related (if only graphically) mathematics —

Click the above image for some related purely mathematical  remarks.

Monday, June 19, 2017

Final Club

Filed under: General — m759 @ 11:20 AM

Today’s New York Times  on a character in a 1978 film —

“Cluelessly upbeat and charmingly idiotic.”

Related material from a post Saturday —

Plato's Formula: A Hollywood version of Plato's diamond from the Meno dialogue

Director with Oscar

Coda —

See as well this  journal on the above date — Sept. 24, 2015.

Thursday, June 15, 2017

Early Personal Computer

Filed under: General — m759 @ 10:01 AM

(The title is from yesterday morning's Graphical Interfaces.)

Plato's diamond in Jowett's version of the Meno dialogue

Wednesday, June 14, 2017

Graphical Interfaces

Filed under: General — Tags: — m759 @ 10:52 AM

Thacker reportedly died on Monday, June 12, 2017.

This journal on that date —

Images including Plato's diamond on a tombstone

Thacker retired from Microsoft in February.

Monday, May 15, 2017

Appropriation at MoMA

Filed under: General,Geometry — m759 @ 1:14 PM

For example, Plato's diamond as an object to be transformed —

Plato's diamond in Jowett's version of the Meno dialogue

Versions of the transformed object —

See also The 4×4 Relativity Problem in this journal.

Friday, February 17, 2017

Kostant Is Dead

Filed under: General,Geometry — m759 @ 1:10 PM

"Bertram Kostant, professor emeritus of mathematics at MIT,
died at the Hebrew Senior Rehabilitation Center in Roslindale,
Massachusetts, on Thursday, Feb. 2, at the age of 88."

MIT News, story dated Feb. 16, 2017

See also a search for Kostant in this journal.

Regarding the discussions of symmetries and "facets" found in
that search —

Kostant:

A word about E(8). In my opinion, and shared by others,
E(8) is the most magnificent ‘object’ in all of mathematics.
It is like a diamond with thousands of facets. Each facet
offering a different view of its unbelievable intricate internal
structure.”

Cullinane:

In the Steiner system S(5, 8, 24) each octad might be
regarded as a "facet," with the order of the system's
automorphism group, the Mathieu group M24 , obtained
by multiplying the number of such facets, 759, by the
order of the octad stabilizer group, 322,560. 

Analogously

Platonic solids' symmetry groups   

Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: , — m759 @ 12:31 PM

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices

Related material from a post of Aug. 6, 2014

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

(Here "five point sets" should be "five-point sets.")

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate 
abuse of language .  For the real definition of 
“symplectic structure,” see (for instance) 
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry 
, Vol 2.) To establish that the above
figure is indeed symplectic , see the post 
Zero System of July 31, 2014.

Tuesday, September 27, 2016

Chomsky and Lévi-Strauss in China

Filed under: General,Geometry — Tags: — m759 @ 7:31 AM

Or:  Philosophy for Jews

From a New Yorker  weblog post dated Dec. 6, 2012 —

"Happy Birthday, Noam Chomsky" by Gary Marcus—

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

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

Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).

Consider that not-particularly-deep structure from the Meno dialogue
in the light of the following…

The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the Lévi-Strauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji  denotes the fundamental concept in
Chinese philosophy that such a going-beyond is both useful
and possible.)

The matrix at left below represents the feminine yin  principle
and the diamond at right represents the masculine yang .

      From a post of Sept. 22,
  "Binary Opposition Illustrated" —

A symbol of the unity of yin and yang —

Related material:

A much more sophisticated approach to the "deep structure" of the
Meno diagram —

The larger cases —

The diamond theorem

Sunday, June 19, 2016

Making Gatsby Great Again

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

Image-- From the Diamond in Plato's Meno to Modern Finite Geometry

See also the previous post.

Saturday, June 4, 2016

Icons

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

From this morning's news, a  cultural icon —

From November 18, 2015, four  icons —

— the three favicons above, and the following:

Jack in the Box, by Natasha Wescoat

Sunday, May 15, 2016

One Ring

Filed under: General,Geometry — Tags: , — m759 @ 5:06 PM

(Continued from May 11 and May 15.)

Poem by Eleanor Wilner from 'Reversing the Spell' speaks of diamonds and 'glitter.' (Pbk. publ. Nov. 1, 1997)

Friday, April 29, 2016

Blackboard Jungle…

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

Continues .

An older and wiser James Spader —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Tuesday, April 26, 2016

Interacting

Filed under: General,Geometry — m759 @ 8:31 PM

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

Saturday, January 23, 2016

Hard

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

"Hard Science Fiction in the era of short attention spans,
crowd-sourcing, and rapid obsolescence"

— May 26, 2012, Dragon Press Bookstore symposium

Related material:  Posts now tagged Black Diamond.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Friday, January 22, 2016

Story

Filed under: General — Tags: — m759 @ 10:21 AM

The New Yorker , April 12, 2004 —

Wednesday, November 18, 2015

Tuesday, November 17, 2015

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Friday, October 23, 2015

Retro or Not?

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

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

See also Diamond Theory Roulette —

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

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

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

Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: General,Geometry — Tags: — m759 @ 2:01 AM

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

The diamond-theorem correlation

 

Saturday, July 4, 2015

Context

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

Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks 
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .

See also the diamond-theorem correlation.  

Monday, June 15, 2015

Slow Art

Filed under: General — Tags: — m759 @ 2:03 PM

Slowness is sometimes in the eye of the beholder.

See this journal on Slow Art Day 2015.

Related material: Epistemic States in this journal.

Saturday, June 13, 2015

Egg Tales

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

"And not all the king's men nor his horses
 Will resurrect his corpus."

Finnegans Wake

See as well Andy Weir's "The Egg" and Working Backward.

Thursday, May 7, 2015

Paradigm for Pedagogues

Filed under: General,Geometry — Tags: , — m759 @ 7:14 PM

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Saturday, April 11, 2015

The Starbird Manifesto

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

"But what was supposed to be the source of a compound's
authority? Why, the same as that of all new religious movements:
direct access to the godhead, which in this case was Creativity."

— Tom Wolfe, From Bauhaus to Our House

"Creativity is not a matter of magical inspiration."

— Burger and Starbird, The 5 Elements of Effective Thinking  (2012) 

Video published on Oct 19, 2012

"In this fifth of five videos, mathematics professor
Michael Starbird talks about the fifth element
in his new book, The 5 Elements of Effective Thinking ,
co-authored with Williams College professor
Edward B. Burger." 

For more on the Starbird manifesto, see Princeton University Press.

An excerpt —

See also a post for Abel's Birthday, 2011 —  
Midnight in Oslo — and a four-elements image from
the Jan. 26, 2010, post Symbology —

Logo for 'Elements of Finite Geometry'.

Wednesday, February 25, 2015

Words and Images

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

The words:  "symplectic polarity"—

The images:

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

Picturing the Smallest Projective 3-Space

Quilt Block Designs

Saturday, February 21, 2015

High and Low Concepts

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European. 
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

Sunday, November 9, 2014

Sermon

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

The Ideas

“We tell ourselves stories in order to live….
We interpret what we see, select the most workable
of multiple choices. We live entirely, especially if we
are writers, by the imposition of a narrative line upon
disparate images, by the ‘ideas’  with which we have
learned to freeze the shifting phantasmagoria
which is our actual experience.”
— Joan Didion

See Didion and the I Ching  and posts tagged Plato in China .

Tuesday, September 16, 2014

Where the Joints Are

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

An image related to the recent posts Sense and Sensibility:

A quote from yesterday's post The Eight:

A possible source for the above phrase about phenomena "carved at their joints":

See also the carving at the joints of Plato's diamond from the Meno :

Image-- Plato's diamond and a modern version from finite geometry

Related material: Phaedrus on Kant as a diamond cutter
in Zen and the Art of Motorcycle Maintenance .

Wednesday, August 27, 2014

Altar

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

"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"

— Macaulay, quoted in the April 2013 film "Oblivion"

"Leave a space." — Tom Stoppard, "Jumpers"

Related material: The August 16, 2014, sudden death in Scotland
of an architect of the above Cardross seminary, and a Log24 post,
Plato's Logos, from the date of the above photo: June 26, 2010.

See also…

IMAGE- T. Lux Feininger on 'Gestaltung'

Here “eidolon” should instead be “eidos .”

An example of eidos — Plato's diamond (from the Meno ) —

http://www.log24.com/log/pix10A/100607-PlatoDiamond.gif

Sunday, August 24, 2014

Symplectic Structure…

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the
3-subsets.

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Wednesday, August 13, 2014

Symplectic Structure continued

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

Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —

Related material in this journal (click image for posts) —

Monday, August 11, 2014

Syntactic/Symplectic

(Continued from August 9, 2014.)

Syntactic:

Symplectic:

"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive 
. They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."

– Susanne K. LangerPhilosophy in a New Key

For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).

This is a symplectic  correlation,* constructed using the following
visual structure:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.

Wednesday, August 6, 2014

Symplectic Structure*

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

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens  54, 59-79 (1992):

"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: , — m759 @ 6:11 PM

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Thursday, July 17, 2014

Paradigm Shift:

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

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Wednesday, July 9, 2014

Fashion Statements

Filed under: General — Tags: — m759 @ 3:28 AM

From Monday in this journal —

Related news this morning —

Anne Hollander, Scholar of Style, Dies at 83
By William Yardley in The New York Times ,
10:26 PM ET July 8, 2014

Anne Hollander, a historian who helped elevate
the study of art and dress by revealing the often striking
relationships between the two, died on Sunday at her home
in Manhattan. She was 83.

The cause was cancer, said her husband, the philosopher
Thomas Nagel.
. . . .
She received a degree in art history from Barnard College
in 1952. The next year she married the poet John Hollander.
Their marriage ended in divorce.

Related material from this journal last year —

“Be serious, because
The stone may have contempt
For too-familiar hands”

Adrienne Rich in “The Diamond Cutters” (1955)

Tuesday, July 8, 2014

Toward Freedom

Filed under: General — m759 @ 12:00 PM

A search for “Dark Fields of the Republic,”
an F. Scott Fitzgerald phrase mentioned in
the previous post, yields a book by that title.

“When does a life bend toward freedom?
grasp its direction?”
— Adrienne Rich on page 275 of
Later Poems Selected and New: 1971-2012

The book’s author, Adrienne Rich, died at 82 on
March 27, 2012. See that date in this journal.

See also the following:

The Diamond Cutters
by Adrienne Rich (1955)

However legendary,
The stone is still a stone,
though it had once resisted
The weight of Africa,
The hammer-blows of time
That wear to bits of rubble
The mountain and the pebble–
But not this coldest one.

Now, you intelligence
So late dredged up from dark
Upon whose smoky walls
Bison took fumbling form
Or flint was edged on flint–
Now, careful arriviste,
Delineate at will
Incisions in the ice.

Be serious, because
The stone may have contempt
For too-familiar hands,
And because all you do
Loses or gains by this:
Respect the adversary,
Meet it with tools refined,
And thereby set your price.

Be hard of heart, because
the stone must leave your hand.
Although you liberate
Pure and expensive fires
Fit to enamor Shebas,
Keep your desire apart.
Love only what you do,
And not what you have done.

Be proud, when you have set
The final spoke of flame
In that prismatic wheel,
And nothing’s left this day
Except to see the sun
Shine on the false and the true,
And know that Africa
will yield you more to do.

Monday, July 7, 2014

“‘Consider,’ said I…”

Filed under: General — m759 @ 8:28 PM

Roger Cooke in The History of Mathematics: A Brief Course
(2nd ed., Wiley-Interscience, 2005)—

“Like all numbers, the number four is bound to occur
in many contexts.”

— Ch. 1: “The Origin and Prehistory of Mathematics,”
Part 3, “Symbols,” footnote 1, page 11.

As is the number 382:

Click the above image for some related material.

Commentary:

“Once the students are taken in by the story, it will be
the instructor’s job to elaborate on the historical
calculations and proofs.”

— Gary S. Stoudt, Professor of Mathematics,
Indiana U. of Pennsylvania, review of Cooke’s book
at the Mathematical Association of America

Sunday, June 8, 2014

Vide

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

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Tuesday, March 11, 2014

Dark Fields of the Republic

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

This post was suggested by today's previous post, Depth,
by Plato's Diamond, and by Rebecca Newberger Goldstein's
recent fanciful fiction about Plato.

Plato, Republic , Book II, Paul Shorey translation at Perseus

“Consider,” [382a] said I; “would a god wish to deceive, or lie, by presenting in either word or action what is only appearance?” “I don’t know,” said he. “Don’t you know,” said I, “that the veritable lie, if the expression is permissible, is a thing that all gods and men abhor?” “What do you mean?” he said. “This,” said I, “that falsehood in the most vital part of themselves, and about their most vital concerns, is something that no one willingly accepts, but it is there above all that everyone fears it.” “I don’t understand yet either.” “That is because you suspect me of some grand meaning,” [382b] I said; “but what I mean is, that deception in the soul about realities, to have been deceived and to be blindly ignorant and to have and hold the falsehood there, is what all men would least of all accept, and it is in that case that they loathe it most of all.” “Quite so,” he said.

Related material —

A meditation from the Feast of St. Francis, 2012 —

A post from Sept. 30, 2012, the reported date of  death
for British children's author Helen Nicoll —

The New Criterion  on the death of Hilton Kramer —

This uncredited translation of Plato is, Google Books tells us,
by "Francis MacDonald Cornfield."  The name is an error,
but the error is illuminating —

Signs Movie Stills: Mel Gibson, Joaquin Phoenix, Patricia Kalember, M. Night Shyamalan

Sunday, March 2, 2014

Sermon

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

Raiders of the Lost  (Continued)

"Socrates: They say that the soul of man is immortal…."

From August 16, 2012

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not ,  at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

A more correct version, from hermes-press.com —

Socrates: He only guesses that because the square is double, the line is double.Meno: True.

 

Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line?

[Boy] Yes.

Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)?

[Boy] Certainly.

Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet?

[Boy] Yes.

Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet?

[Boy] Yes.

Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN)

[Boy] True.

Socrates: And is not that four times four?

[Boy] Certainly.

Socrates: And four times is not double?

[Boy] No, indeed.

Socrates: But how much?

[Boy] Four times as much.

Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much.

[Boy] True.

Socrates: Four times four are sixteen— are they not?

[Boy] Yes.

As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect  version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:

A 4×4 square version, by Coxeter in 1950, of  a tesseract

This square version we may call the Galois  tesseract.

Friday, February 21, 2014

Night’s Hymn of the Rock

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

One way of interpreting the symbol  IMAGE- Modal Diamond in a square 
at the end of yesterday's post is via
the phrase "necessary possibility."

See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .

The Tablet  post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—

IMAGE- Tom Wolfe in 'The Painted Word' on conceptual art

On that  day (July 23) another weblog had
a post titled

Wallace Stevens: Night's Hymn of the Rock.

Some related narrative —

IMAGE- The 2001 film 'The Discovery of Heaven'

I prefer the following narrative —

Part I:  Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"

Part II:  Mystery Box III: Inside, Outside (2014)

Thursday, February 20, 2014

Relativity Blues

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

(Continued)

A review of this date in 2005 —

Modal Theology

“We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes)).”

— Keith Allen Korcz

And what do we  
   symbolize by   The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

Wednesday, December 25, 2013

Rotating the Facets

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

Previous post

“… her mind rotated the facts….”

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

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

“I’ve heard of affairs that are strictly Platonic

Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

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

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

See also Diamond Theory in 1937.

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

Wednesday, December 18, 2013

Bing Bang Theory

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

Microsoft in 2009 on its new search engine name—

"We like Bing because it sounds off in our heads
when we think about that moment of discovery
and decision making— when you resolve those
important tasks."

A search on Bing today —

IMAGE- Top search result on Bing for 'diamond space' on Dec. 18, 2013

A colorful tale —

IMAGE- The Diamond 16 Puzzle, with commentary

"Bing bang, I saw the whole gang
Bobby Darin, 1958

Thursday, December 12, 2013

Outsider Art

Filed under: General,Geometry — Tags: — m759 @ 4:10 PM

"… Galois was a mathematical outsider…."

— Tony Mann, "head of the department of mathematical sciences,
University of Greenwich, and president, British Society for the
History of Mathematics," in a May 6, 2010, review of Duel at Dawn
in Times Higher Education.

Related art: 

(Click for a larger image.)

IMAGE- Google search for 'Diamond Space' + Galois

For a less outside  version of the central image
above, see Kunstkritikk  on Oct. 15, 2013.

Wednesday, September 11, 2013

Language Game

Filed under: General — Tags: , — m759 @ 9:29 AM

In which Plato continues to thank the Academy.

From the Academy, a lead balloon for 9/11 —
continued from March First, 2002.

A search today for the name Eisenman
(see previous post) yields the following :

"We need a cameo from Plato, a safecracker,
a wrinkle or two to be ironed out, some ice,
some diamonds, and, above all, laughter
for this irony of ironies."

Jeffrey Kipnis, "Twisting the Separatrix,"
Assemblage  No. 14, April 1991, MIT Press

Monday, September 9, 2013

Viking Book

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

For the late Billy Wilder, director of Ace in the Hole  (1951)

IMAGE- Book by Halvor Bodin on the art of Josefine Lyche and others. See halvorbodin.com.

Click image for a larger version.

See, too, this morning's quarter-to-three post, and The Vikings  (1958)—

The art by Josefine Lyche in the Bodin book shown 
above is, as the artist notes, based on my own work.

Monday, August 12, 2013

Form

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

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

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

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

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

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

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

Sunday, July 28, 2013

Sermon

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

(Simplicity continued)

"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by  the
demonstration or the metaphor.

What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already-
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."

— Jan Zwicky, Wisdom & Metaphor , page 36 (left)

Zwicky illustrates this with Plato's diamond figure
​from the Meno  on the facing page— her page 36 (right).

A more sophisticated geometrical figure—

Galois-geometry key to
Desargues' theorem:

   D   E   F
 S'  P Q R
 S  P' Q' R'
 O  P1 Q1 R1

For an explanation, see 
Classical Geometry in Light of Galois Geometry.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

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

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

Compare to an image of Vril muse Maria Orsitsch.

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

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

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

(See also the original catalog page.)

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

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

Saturday, June 29, 2013

Thursday, June 27, 2013

Tuesday, May 28, 2013

Codes

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

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

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

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

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

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

IMAGE- The 35 square patterns within the Curtis MOG

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

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

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

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

Update of May 29, 2013:

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

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

Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

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

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

IMAGE- The Penrose diamond and the Klein quadric

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

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

IMAGE- Stroppel on the Klein quadric, 2008

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

Related material:

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

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

Those who prefer story to structure may consult 

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

Transgressing the Boundary

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

The title refers not to the 1996 Sokal hoax (which has
Boundaries , plural, in the title), but to the boundary
discussed in Monday's Penrose diamond post

"Science is a differential equation.
Religion is a boundary condition."

Alan Turing in the epigraph to the
first chapter of a book by Terence Tao

From the Tao book, page 170—

"Typically the transformed solution extends to the
boundary of the Penrose diamond and beyond…."

Transgressing the boundary between science
and religion is the topic of a 1991 paper available
at JSTOR for $29.

For the Pope on Ash Wednesday:

"Think you might have access 
to this content via your library?" —JSTOR

See also Durkheim at Harvard.

Friday, February 1, 2013

Get Quotes

Filed under: General — Tags: , — m759 @ 4:01 PM

For Tony Kushner fans:

For logic fans:

IMAGE- NY Times market quotes, American Express Gold Card ad, Kevin Spacey in 'House of Cards' ad

John Searle on Derrida:

On necessity, possibility, and 'necessary possibility'

In the box-diamond notation, the axiom Searle quotes is

.

"The euclidean property guarantees the truth of this." — Wikipedia

Linking to Euclid

Clicking on "euclidean" above yields another Wikipedia article

"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."

Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.

Thursday, January 31, 2013

Scholarship in 1961…

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

Before Derrida's writings on Plato and on inscription

A remark by the late William Harris:

"Scholarship has many dark ages, and they do not all fall
in the safe confines of remote antiquity."

For more about Harris, see the previous post.

Discussing an approach to solving a geometrical problem 
from section 86e of the Meno , Harris wrote that

"… this is a very important element of method and purpose,
one which must be taken with great seriousness and respect.
In fact it is as good an example of the master describing for us
his method as Plato ever gives us. Tricked by the appearance
of brevity and unwilling to follow through Plato's thought on
the road to Euclid, we have garbled or passed over a unique
piece of philosophical information."

Harris, though not a geometer, was an admirable man.
His remark on the Meno  method is itself worthy of respect.

In memory of Harris, Plato, and pre-Derrida scholarship, here
are some pages from 1961 on the problem Harris discussed.

A pair of figures from the 1961 pages indicates how one view of the
section 86e problem (at right below) resembles the better-known 
demonstration earlier in the Meno  of how to construct
a square of area 2 —

Wednesday, January 30, 2013

Abstract Possibility

Filed under: General — m759 @ 1:01 PM

Today's NY Times  "Stone Links" to philosophy include
a link to a review of a collection of Hilary Putnam's papers.

Related material, from Putnam's "What is Mathematical
Truth?
" (Historia Mathematica  2 (1975): 529-543)—

"In this paper I argue that mathematics should be interpreted realistically – that is, that mathematics makes assertions that are objectively true or false, independently of the human mind, and that something answers to such mathematical notions as ‘set’ and ‘function’. This is not to say that reality is somehow bifurcated – that there is one reality of material things, and then, over and above it, a second reality of ‘mathematical things’. A set of objects, for example, depends for its existence on those objects: if they are destroyed, then there is no longer such a set. (Of course, we may say that the set exists ‘tenselessly’, but we may also say the objects exist ‘tenselessly’: this is just to say that in pure mathematics we can sometimes ignore the important difference between ‘exists now’ and ‘did exist, exists now, or will exist’.) Not only are the ‘objects’ of pure mathematics conditional upon material objects; they are, in a sense, merely abstract possibilities. Studying how mathematical objects behave might better be described as studying what structures are abstractly possible and what structures are not abstractly possible."

See also Wittgenstein's Diamond and Plato's Diamond.

Tuesday, January 22, 2013

Raven Light

Filed under: General — Tags: , — m759 @ 11:40 AM

"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of things. …

…matrices are considered the gold standard of fluid-intelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."

— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012

See also "Raven Steals the Light" in this  journal.

Related material:

Plan 9 from MIT and, perhaps exemplifying crystallized  rather than fluid  intelligence, Black Diamond.

Wednesday, January 16, 2013

Treasure Hunt

Filed under: General,Geometry — Tags: — m759 @ 3:17 PM

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

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

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

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

IMAGE- Plato's Diamond

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

Saturday, December 8, 2012

Defining the Contest…

Filed under: General,Geometry — Tags: , , , , — m759 @ 5:48 AM

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

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

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

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

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

The Meno Embedding

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

For related truths about geometry, see the diamond theorem.

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

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

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

The sorts of patterns discussed in the 2012 paper —

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

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

* See Gombrich-Douat in this journal.

Thursday, November 1, 2012

For All Saints’ Day

Filed under: General — Tags: , — m759 @ 5:31 AM

Conclusion of "The Storyteller," a story 
by Cynthia Zarin about author Madeleine L'Engle—

The New Yorker , April 12, 2004 —

Note the black diamond at the story's end.

Monday, September 17, 2012

The Count

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

… I saw a shadow
sliding around the ropes
to get at me. The referee
moved it back, and then
went over and picked up the count.
"One!" The fog was clearing.

I rose to a knee,
and at "nine" to my feet.

— Louis Simpson, "The Appointment"

Simpson reportedly died on Holy Cross Day.

That day in this journal—

IMAGE- Log24 posts 'Please Mister Please' and 'Plan 9'

Friday, August 24, 2012

Formal Pattern

Filed under: General,Geometry — m759 @ 4:28 PM

(Continued from In Memoriam (Aug. 22), Chapman's Homer (Aug. 23),
and this morning's Colorful Tale)

An informative, but undated, critique of the late Marvin W. Meyer
by April D. DeConick at the website of the Society of Biblical Literature
appeared in more popular form in an earlier New York Times
op-ed piece, "Gospel Truth," dated Dec. 1, 2007.

A check, in accord with Jungian synchronicity, of this  journal
on that date yields a quotation from Plato's Phaedrus  —

"The soul or animate being has the care of the inanimate."

Related verses from T. S. Eliot's Four Quartets

"The detail of the pattern is movement."

"So we moved, and they, in a formal pattern."

Some background from pure mathematics (what the late
William P. Thurston called "the theory of formal patterns")—

The Animated Diamond Theorem.

Thursday, August 16, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

(Continued from August 13. See also Coxeter Graveyard.)

Coxeter exhuming Geometry

Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.

A similar 1950 figure by Coxeter illustrating a tesseract

For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.

Saturday, May 26, 2012

Talk Amongst Yourselves

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

Hard Science Fiction weekend at Dragon Press Bookstore

Saturday May 26:
11am-noon Playing with the net up:
Hard Science Fiction in the era of
short attention spans, crowd-sourcing,
and rapid obsolescence
( Greg Benford, James Cambias, Kathryn Cramer)
….
3pm-4:30 Technological optimism and pessimism;
utopia and dystopia; happy endings & sad endings:
what do these oppositions have to do with one another?
Are they all the same thing? How are they different
from one another? Group discussion.

My own interests in this area include…

(Click image for some context)

IMAGE- 'The Stars My Destination' (with cover slightly changed)

    The above was adapted from a 1996 cover

IMAGE- PyrE on the 1996 Vintage Books cover of 'The Stars My Destination'

 Vintage Books, July 1996. Cover: Evan Gaffney.

For the significance of the flames, 
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.

Monday, May 21, 2012

Child’s Play (continued*)

Filed under: General — Tags: — m759 @ 7:59 PM

You and I …

we are just like a couple of tots…

Sinatra

JOSEFINE LYCHE

Born 1973 in Bergen. Lives and works in Oslo.

Education

2000 – 2004 National Academy of Fine Arts, Oslo
1998 – 2000 Strykejernet Art School, Oslo, NO
1995 – 1998 Philosophy, University of Bergen

University of Bergen—

 It might therefore seem that the idea of digital and analogical systems as rival fundaments to human experience is a new suggestion and, like digital technology, very modern. In fact, however, the idea is as old as philosophy itself (and may be much older). In his Sophist, Plato sets out the following ‘battle’ over the question of ‘true reality’:

What we shall see is something like a battle of gods and giants going on between them over their quarrel about reality [γιγαντομαχία περì της ουσίας] ….One party is trying to drag everything down to earth out of heaven and the unseen, literally grasping rocks and trees in their hands, for they lay hold upon every stock and stone and strenuously affirm that real existence belongs only to that which can be handled and offers resistance to the touch. They define reality as the same thing as body, and as soon as one of the opposite party asserts that anything without a body is real, they are utterly contemptuous and will not listen to another word. (…) Their adversaries are very wary in defending their position somewhere in the heights of the unseen, maintaining with all their force that true reality [την αληθινήν ουσίαν] consists in certain intelligible and bodiless forms. In the clash of argument they shatter and pulverize those bodies which their opponents wield, and what those others allege to be true reality they call, not real being, but a sort of moving process of becoming. On this issue an interminable battle is always going on between the two camps [εν μέσω δε περι ταυτα απλετος αμφοτέρων μάχη τις (…) αει συνέστηκεν]. (…) It seems that only one course is open to the philosopher who values knowledge and truth above all else. He must refuse to accept from the champions of the forms the doctrine that all reality is changeless [and exclusively immaterial], and he must turn a deaf ear to the other party who represent reality as everywhere changing [and as only material]. Like a child begging for 'both', he must declare that reality or the sum of things is both at once [το όν τε και το παν συναμφότερα] (Sophist 246a-249d).

The gods and the giants in Plato’s battle present two varieties of the analog position. Each believes that ‘true reality’ is singular, that "real existence belongs only to" one side or other of competing possibilities. For them, difference and complexity are secondary and, as secondary, deficient in respect to truth, reality and being (την αληθινήν ουσίαν, το όν τε και το παν). Difference and complexity are therefore matters of "interminable battle" whose intended end for each is, and must be (given their shared analogical logic), only to eradicate the other. The philosophical child, by contrast, holds to ‘both’ and therefore represents the digital position where the differentiated two yet belong originally together. Here difference, complexity and systematicity are primary and exemplary.

It is an unfailing mark of the greatest thinkers of the tradition, like Plato, that they recognize the digital possibility and therefore recognize the principal difference of it from analog possibilities.

— Cameron McEwen, "The Digital Wittgenstein,"
    The Wittgenstein Archives at the University of Bergen

* See that phrase in this journal.

Sunday, April 1, 2012

The Palpatine Dimension

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

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

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

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

See also uses of the word triality  in mathematics. For instance…

A discussion of triality by Edward Witten

Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6  thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron

There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality  link from above
and Cameron's Klein Correspondence and Triality.)

Cameron applies his  triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Friday, January 20, 2012

Poetry and Thought*

Filed under: General — m759 @ 10:30 AM

* Title courtesy of George Steiner.
   For the "thought" part, see Plato's diamond
   in last night's Mathematical Imagery.

Thursday, October 20, 2011

The Thing Itself

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

Suggested by an Oct. 18 piece in the Book Bench section
of the online New Yorker  magazine—

http://www.log24.com/log/pix11C/111020-Derrida.GIF

http://www.log24.com/log/pix11C/111020-Topia122.GIF

Related material suggested by the "Shouts and Murmurs" piece
in The New Yorker , issue dated Oct. 24, 2011—

"a series of e-mails from a preschool teacher planning to celebrate
the Day of the Dead instead of Halloween…"

A search for Coxeter + Graveyard in this journal yields…

Coxeter exhuming Geometry

Here the tombstone says "GEOMETRY… 600 BC — 1900 AD… R.I.P."

A related search for Plato + Tombstone yields an image from July 6, 2007…

The image “http://www.log24.com/log/pix06A/061019-Tombstones.jpg” cannot be displayed, because it contains errors.

Here Plato's poems to Aster suggested
the "Star and Diamond" tombstone.

The eight-rayed star is an ancient symbol of Venus
and the diamond is from Plato's Meno .

The star and diamond are combined in a figure from
12 AM on September 6th, 2011—

The Diamond Star

http://www.log24.com/log/pix11B/110905-StellaOctangulaView.jpg

See Configurations and Squares.

That webpage explains how Coxeter
united the diamond and the star.

Those who prefer narrative to mathematics may consult
a definition of the Spanish word lucero  from March 28, 2003.

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