Log24

Sunday, May 1, 2016

Sunday Appetizer from 1984

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

Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —

“What would an organic Christian Sabbath look like today?”

The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —  

Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich
unter http://​www.​encyclopediaofma​th.​org/​index.​php/​
Cullinane_​diamond_​theorem
und http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."

The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the 
1988 first edition of Sphere Packings, Lattices and Groups
by J. H. Conway and N. J. A. Sloane —

A passage from the 1984 document —

Sunday, March 6, 2016

Raven Writing

Filed under: General — Tags: — m759 @ 9:45 pm

Continued.

IMAGE- Raven's Progressive Matrices item with symbols from Cullinane's box-style I Ching

Friday, December 11, 2015

Street View

Filed under: General — m759 @ 2:00 pm

Continued from Once Upon a Matrix  (November 27, 2015).

Click image below to enlarge.

Icon Parking, W. 54th St.

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

— American adaptation of Shakespeare's Tempest , 1956

Midrash —

"Remember me to Herald Square."

Thursday, August 27, 2015

Tears in the Rain

Filed under: General — Tags: , , — m759 @ 12:21 pm

For a Norwegian historian

Game Over

The film "The Matrix," illustrated
Coordinates for generating the Miracle Octad Generator

Friday, July 24, 2015

Field of Manifestation

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

This post was suggested by a book title in
the previous post: "Pieces of the Action."

A group action  is a mathematical concept.

Related meditation:

"The number 9, that is to say, relates traditionally
to the Great Goddess of Many Names (Devi, 
Inanna, Ishtar, Astarte, Artemis, Venus, etc.)
as matrix of the cosmic process, whether in the
macrocosm or in a microcosmic field of manifestation."

— Joseph Campbell,
     The Inner Reaches of Outer Space

Examples:

Click to enlarge —

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Monday, July 13, 2015

Weaving World

Filed under: General — Tags: — m759 @ 5:45 pm

(The title was suggested by the novel Weaveworld .)

Recent public selfie by Oslo artist Josefine Lyche —

Lyche's shirt honors the late Kurt Cobain.

Not-so-recent image of Hugo Weaving as
Agent Smith in "The Matrix" —

"Smells like teen spirit."

See also Weaving in the new film "Strangerland."

Saturday, June 13, 2015

Plan 9 Continues…

Filed under: General — Tags: — m759 @ 11:32 am

A version of the song in the previous post that I prefer:

A related meditation —

IMAGE- Joseph Campbell, 'The Inner Reaches of Outer Space,' meditation on the number nine, the Goddess, and the Angelus

For a more abstract version of the
"matrix of the cosmic process,"
see "3×3 Grid" in this journal.

Saturday, April 25, 2015

Ageometretos Medeis Eisito

Filed under: General — m759 @ 10:30 am

Tuesday, December 30, 2014

An Education

Filed under: General — m759 @ 1:01 am

AP Today in History 
Thought for the Day:

“I respect faith, but doubt is what
gives you an education.”
Wilson Mizner
     American playwright (1876-1933)*

From this journal on the (wide) release date
of "X-Men: First Class" —

A minimalist 3×3 matrix favicon—

http://www.log24.com/log/pix11A/110518-3x3FaviconURL.jpg

This may, if one likes, be viewed as the "nothing"
present at the Creation.  See Jim Holt on physics.

* A source —

Wednesday, December 17, 2014

Launched from Cuber

Filed under: General — Tags: , — m759 @ 5:55 pm

Continued from Nobel Note (Jan. 29, 2014).

IMAGE- 'Launched from Cuber' scene in 'X-Men: First Class'

From Tradition in Action , "The Missal Crisis of '62,"
remarks on the revision of the Catholic missal in that year—

"Neither can the claim that none of these changes
is heretical in content be used as an argument
in favor of its use, for neither is the employment of
hula girls, fireworks, and mariachis strictly speaking
heretical in itself, but they belong to that class of novel
and profane things that do not belong in the Mass."

— Fr. Patrick Perez, posted Sept. 11, 2007 

See also this  journal on November 22, 2014

Say Bingo to my little friend

    … and on Bruce Springsteen's birthday this year —

Tuesday, September 23, 2014

Matrix

Filed under: Uncategorized — m759 @ 11:00 AM 

From AP’s Today in History:

Happy birthday.

“It all adds up.” — Saul Bellow

The Matrix:

 

 

 

 

 

 

Thursday, December 4, 2014

The Pony Argument

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

The title was suggested by this morning's post "Follow This."

From a previous incarnation of my home website, m759.com —
The second of the site's three pages mentions authors
Alfred Bester and Zenna Henderson :

"Bester and Henderson are particularly good at
fictional accounts of telepathy. The noted Harvard
philosopher W. V. Quine doubts such a thing exists,
but I prefer the 'There must be a pony' argument."

Related material: The date Nov. 27, 2014, in a web search today 


 in The Washington Post 

and in this  journal

Thursday, November 27, 2014

In the Details

Filed under: General,Geometry — m759 @ 1:01 pm

For the late novelist P. D. James, author of, among other things,
The Children of Men .

Michael Caine in the film of that book —

Detail —

Raven's Progressive Matrices example —

 A quote for tellers of tales —

“There have long been rumors of a mythical Ninth Element
that grants ultimate power to the Wizard who masters it.
The Order of Magick says there is no such thing. But….”

— Website of Magicka: The Ninth Element Novel

Tuesday, November 18, 2014

The Abacus Conundrum…

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

Continues.

http://www.log24.com/log/pix10B/101206-AbacusConundrum.jpg

Prequel from 1961 (click image for context):

Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—

"Ting-a-ling." — Kurt Vonnegut.

Our Most Important Product

Filed under: General — Tags: — m759 @ 12:45 am

Hexagram 35:
Progress

"Then came a 'Robot Psychologist,' known as a Psychological Matrix Rotator,
developed for the Department of Defense. It is still used to literally 'see' that
the right man gets the right Army job."

Ronald Reagan, 1961 GE Sales Meeting

"Always with a little humor." — Yen Lo

In memory of Dr. Irving Peress,
who reportedly died on Thursday,
November 13, 2014.

Monday, September 22, 2014

A Holy Cross for Brooklyn

Filed under: General — m759 @ 12:00 pm

Saturday night’s 9  PM post:

“Once again, Harvard defeats Holy Cross.”

That post quoted Joseph Campbell describing a

“… matrix of the cosmic process, whether in the macrocosm
or in a microcosmic field of manifestation.”

Related material —

Macrocosm:

Microcosmic field of manifestation:

Wednesday, August 13, 2014

Stranger than Dreams*

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

Illustration from a discussion of a symplectic structure 
in a 4×4 array quoted here on January 17, 2014 —

See symplectic structure in this journal.

* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.

Thursday, August 7, 2014

The Omega Mystery

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

See a post,  The Omega Matrix, from the date of her death.

Related material:

"When Death tells a story, you really have to listen."
— Cover of The Book Thief

A scene from the film of the above book —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.

Tuesday, August 5, 2014

The Omega Story

Filed under: General — Tags: — m759 @ 1:00 am
 

"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 also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).

Monday, August 4, 2014

A Wrinkle in Space

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

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

An approach via the Omega Matrix:

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

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

Tuesday, June 3, 2014

Galois Matrices

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

The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra  (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
Symmetry invariance in a diamond ring.”

See also related historical remarks by Weyl and by Atiyah.

Thursday, May 29, 2014

Lost in Translation

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

IMAGE- Original French of text from 'The Shining of May 29'

Translation by Barbara Johnson:

"The minimum number of rows— lines or columns—
that contain all the zeros in a matrix is equal to
the maximum number of zeros
located in any individual line or column ."

In the original:

"situés sur des lignes ou des colonnes distinctes "

Update of 11:30 PM ET May 29, 2014:

Derrida in 1972 was quoting Philippe Sollers, Nombres
(Paris: Éditions du Seuil , 1968).  Sollers in turn was
perhaps quoting A. Kaufmann, Méthodes et Modèles
de la Recherche Opérationnelle , Paris, Dunod , 1964,
L'Économie d'Entreprise 10 , vol. 2, page 305:

"Le nombre minimal de rangées
(lignes et/ou colonnes) contenant
tous les zéros d'une matrice, est égal
au nombre maximal de zéros
situés 
sur des lignes et des colonnes distinctes."

Wednesday, May 7, 2014

Meanwhile, in São Paulo…

Filed under: General — m759 @ 9:38 am

This post was suggested by last night’s posts on conceptual art
(in São Paulo) and on a quarter-to-three story.

From this journal on May 31, 2012:

Matrix Problem:
IMAGE- Raven's Progressive Matrices problem based on triangular half- and quarter-diamonds
Click image for some related material.

Meanwhile…

A game released on the above date:

IMAGE- 'Max Payne 3,' a PC game set in São Paulo

Friday, January 17, 2014

The 4×4 Relativity Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

Wednesday, December 25, 2013

A Midnight Clear

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

Click image for a meditation.

Sunday, December 22, 2013

The Bronfman Catechism

Filed under: General — m759 @ 10:21 am

Meanwhile

Log24 on Sunday, October 5, 2008

Theologian James Edwin Loder:

“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.”

Related material:

Terence McKenna:

“Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey ‘haunts time like a ghost.’”

Anonymous author:

“‘Knight’s move thinking’ is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia.”

Related journalism

IMAGE- Scene from a blackboard jungle

"What's the 'S'  stand for?" — Amy Adams

Wednesday, December 11, 2013

The Well-Tempered Monolith

Filed under: General — m759 @ 12:00 pm

IMAGE- The Cullinane frequency matrix

"My God, it's full of numbers!"

Tuesday, December 10, 2013

Blue Note

Filed under: General — m759 @ 10:25 pm

For the late Jim Hall

Backstory:   Icon, 1:44 PM ET today.

Update of 11 PM ET Dec. 10, 2013 — 

For all  the notes, see Da Capo  (11 AM today)
and the Cullinane frequency matrix (12×12).

IMAGE- Matrix used to illustrate the well-tempered scale. The integer frequency-ratio values are only approximate in such a scale.

Matrix used to illustrate the well-tempered
scale. The integer frequency-ratio values
are only approximate in such a scale.

Friday, September 6, 2013

Space

Filed under: General,Geometry — Tags: , — m759 @ 9:15 pm

"A vast space that travels down to the bedrock
upon which the towers were built, the museum
winds its way deeper and deeper underground,
taking visitors on a journey to the very bottom."

— The Associated Press in
this evening's Washington Post

This suggests a review of a different sort of
bedrock:—

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

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.

Tuesday, July 9, 2013

Vril Chick

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

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'

Monday, April 15, 2013

M Theory

Filed under: General,Geometry — m759 @ 9:25 am

This morning's previous post pictured the cover
of a book titled "The Mystery of the Quantum World."

That title, together with Peter Woit's post on Hawking
yesterday, suggests a review of the phrase "-theory."

See remarks on that topic in the October 1998
Notices of the American Mathematical Society :

"The richer theory, which has as limiting cases
the five string theories studied in the last generation,
has come to be called -theory, where M  stands for
magic, mystery, or matrix, according to taste."

— "Magic, Mystery, and Matrix," by Edward Witten

See also, in this  journal, a post mentioning Hawking's
vulgarized M -theory book The Grand Design , as well
as a post of January 9, 2012, titled "M Theory."

Of Witten's three alternative meanings for M , I prefer "matrix."

Tuesday, March 26, 2013

Nothing Revisited

Filed under: General — m759 @ 11:00 am

"… the primal ground of this 'one' is 'nothing.'
The elucidation of the generative matrix of the
myth-work is thus completed…." — Jadran Mimica

Related material— The Nothing That Is.

Wednesday, March 6, 2013

The Red Pill

Filed under: General — m759 @ 1:20 am

On the late Hugo Chávez:

"A couple of years later, I asked him why,
so late in the day, he had decided to
adopt socialism. He acknowledged that he
had come to it late, long after most of
the world had abandoned it, but said that it
had clicked for him after he had read
Victor Hugo’s epic novel Les Misérables .
That, and listening to Fidel."

The New Yorker 's  Jon Lee Anderson

"As you can see, we've had our eye on you
for some time now, Mr. Anderson."

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.

Monday, December 24, 2012

Eternal Recreation

Memories, Dreams, Reflections
by C. G. Jung

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

From pages 195-196:

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

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

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

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

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

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

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

Part I: Mothers

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

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

Part II: Matrices

        

Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Sunday, December 23, 2012

In a Nutshell…

Filed under: General — m759 @ 1:00 am

The Kernel of the Concept of the Object

according to the New York Lottery yesterday—

From 4/27

From 11/24

IMAGE- Agent Smith from 'The Matrix,' 1999

A page numbered 176

A page numbered 187

Thursday, December 13, 2012

Vibrations

Filed under: General,Geometry — m759 @ 11:18 am

 Or:  A Funny Thing Happened
     on the Way to the Embedding

This journal on the morning of Saturday, Dec. 8, 2012:

Plato's Diamond embedded in The Matrix

Marilyn Monroe and her music coach in 1954,
from last night's online New York Times :

" 'We were very close to making love; I don’t remember
the stage we were at, but I would say half-dressed,'
Mr. Schaefer recalled. He added: 'And all of a sudden
for some reason, Marilyn got these vibrations, and
we went over to the window….' "  more »

"Mr. Schaefer died on Saturday at 87 at his home in
Fort Lauderdale, Fla. ….

He [had] coached Monroe through 'Diamonds Are
a Girl’s Best Friend,' her signature number in the
1953 movie 'Gentlemen Prefer Blondes' (he arranged
the music as well)…."

Perhaps on Saturday she returned the favor.

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

Plato's Diamond embedded in The Matrix

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, December 6, 2012

The Embedding

Filed under: General — m759 @ 6:29 pm

Part I

Embedding the Stone (March 23, 2012) —

The Meno Embedding

Plato's Diamond embedded in The Matrix

Part II

ReverbNation.com — Lawrence Class —

Sunday, December 2, 2012

Knight’s Labyrinth

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

A magic— indeed, diabolic— square:

IMAGE- 5x5 magic- in fact, diabolic- square

For the construction, see a book
by W. W. Rouse Ball, founding president
of a Cambridge University magic society.

For some related religious remarks,
see Raiders of the Lost Matrix.

Saturday, November 24, 2012

Deep Agents

Filed under: General — m759 @ 6:00 am

New today
John Markoff in this morning's New York Times

"…what is new in recent months is the growing speed and accuracy
of deep-learning programs, often called artificial neural networks
or just 'neural nets' for their resemblance to the neural connections
in the brain."

Not so new
From the early days of Mike Lynch's Autonomy Inc.

"Autonomy 's intelligent agents use neural networking
to search for patterns of information, rather than
specific words or phrases, thereby distinguishing relevant
from irrelevant information." (Business Wire , October 8, 1996)

And from 1999

IMAGE- Agent Smith from 'The Matrix,' 1999

Tuesday, October 30, 2012

Black October

Filed under: General — m759 @ 9:00 am

From AntiChristmas 2010

Image-- Rosalind Krauss and The Ninefold Square

Art theorist Rosalind Krauss and The Ninefold Square

Krauss is a co-founder of the art journal October .
For some backgound, see the overlapping searches
Krauss October and Matrix Architect.

Thursday, October 25, 2012

Bunker Bingo*

Filed under: General — Tags: — m759 @ 11:09 am

Mathematics and Narrative continues…

IMAGE- In 'Downfall,' Hitler says that 'Steiner's assault will bring it under control.'

Steiner's version of "classical functional analysis"—

"Mein Führer Steiner"

IMAGE- Classical functional analysis according to Moscow State University

* See the story by Kilgore Trout. See also On Linguistic Creation,
   The Matrix of Abraham, and The Thoreau Foundation.

Monday, October 1, 2012

Foundation

Filed under: General,Geometry — m759 @ 3:09 pm

IMAGE- 'Square Acquires a New York Design Firm' (NY Times, noon on October 1st, 2012)

» more

This, together with this morning's post involving the squares 16 and 9,
suggests a review of Conceit at Harvard (October 25, 2006), which
contains the following figure involving the squares 16, 9, and 25—

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Thursday, September 6, 2012

Glory Road (continued)

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

"In ancient Greece, 9 was the number of the Muses,
patron goddesses of the arts. They were the daughters
of Mnemosyne ('memory'), the source
of imagination, which in turn is the carrier of archetypal,
elementary ideas to artistic realization in the field
of space-time. The number 9, that is to say, relates
traditionally to the Great Goddess of Many Names
(Devi, Inanna, Ishtar, Astarte, Artemis, Venus, etc.),
as matrix of the cosmic process, whether in the
macrocosm or in a microcosmic field of manifestation."

— Joseph Campbell in The Inner Reaches of Outer Space ,
      first published in 1986

From Robert A. Heinlein’s Glory Road  (1963):

Her face turned thoughtful. “Would you like to call me ‘Ettarre’?”

“Is that one of your names?”

“It is much like one of them, allowing for different spelling and accent.  Or it could be ‘Esther’ just as closely.  Or ‘Aster.’  Or even ‘Estrellita.’ “

” ‘Aster,’ ” I repeated. “Star. Lucky Star!”

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:

Plato's Diamond embedded in The Matrix

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, March 23, 2012

Embedding the Stone

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

"Imbedding the God character in a holy book's very detailed narrative
and building an entire culture around this narrative
seems by itself to confer a kind of existence on Him."

John Allen Paulos in the philosophy column "The Stone,"
     New York Times  online, Oct. 24, 2010

A related post from Log24 later that year—

Sunday, November 28, 2010

The Embedding

 — m759 @ 6:00 AM

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

Plato's Diamond embedded in The Matrix

See also Plato's Code and
 Plato Thanks the Academy.

 

 

"Next come the crown of thorns and Jesus' agonized crawl across the stage,
bearing the weight of his own crucifix. And at last, after making
yet another entrance, Mr. Nolan strikes the pose immortalized
in centuries of art, clad in a demure loincloth, arms held out to his sides,
one leg artfully bent in front of the other, head hanging down
in tortured exhaustion. Gently spotlighted, he rises from the stage
as if by magic, while a giant cross, pulsing with hot gold lights,
descends from above to meet him. Mr. Lloyd Webber's churning guitar rock
hits a climactic note, and the audience erupts in excited applause."

— Charles Isherwood, review of "Jesus Christ Superstar" in today's  New York Times

Other remarks on embedding —

Part I

Review of a new book on linguistics, embedding, and a South American tribe—

"Imagine a linguist from Mars lands on Earth to survey the planet's languages…."
Chronicle of Higher Education , March 20, 2012

Part II

The Embedding , by Ian Watson (Review of a 1973 novel from Shakespeare's birthday, 2006)

Saturday, October 22, 2011

The Meter is Running

Filed under: General — m759 @ 10:30 am

Suggested by this morning's previous post (Araby) as well as
by Thursday's posts Jack and Jill and The Thing Itself
an item from Google News—

http://www.log24.com/log/pix11C/111022-SaudiShares.JPG

Note the beauty of the headline's meter.

A midrash for Bloomberg—

"Let us return to the insertions." —André Topia, "The Matrix and the Echo."

Monday, October 10, 2011

The Aleph

Filed under: General — m759 @ 12:00 am

COLLEGE OF THE DESERT
Minutes — Organization Meeting
11:00 a.m., Saturday, July 1, 1961—

15. Preparation of College Seal:

By unanimous consent preparation of a College
Seal to contain the following features was
authorized: A likeness of the Library building
set in a matrix of date palms, backed by
a mountain skyline and rising sun; before
the Library an open book, the Greek symbol
Alpha on one page and Omega on the other;
the Latin Lux et Veritas, College of the
Desert, and 1958 to be imprinted within or
around the periphery of the seal.

From the website http://geofhagopian.net/ of
Geoff Hagopian, Professor of Mathematics,
College of the Desert—

http://www.log24.com/log/pix11C/111010-CollegeOfTheDesert-Seal.gif

Note that this version of the seal contains
an Aleph  and Omega instead of Alpha and Omega.

From another Hagopian website, another seal.

Sunday, September 11, 2011

First Lady

Filed under: General — m759 @ 7:59 am

Betty Skelton, "the First Lady of Firsts," died on the last day of August.

http://www.log24.com/log/pix11B/110911-NYTobitsSm.jpg

From this  journal on August thirty-first—

"The Tesseract was the jewel of Odin's treasure room."

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

Hugo Weaving also played Agent Smith
in The Matrix Trilogy .

For Cynthia Zarin, biographer of Madeleine L'Engle

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

Saturday, September 3, 2011

The Galois Tesseract (continued)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Wednesday, August 31, 2011

Odin’s Day

Filed under: General — m759 @ 12:00 pm

Today is Wednesday.

O.E. Wodnesdæg  "Woden's day," a Gmc. loan-translation of L. dies Mercurii  "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin  . — Online Etymology Dictionary

http://www.log24.com/log/pix11B/110831-HopkinsAsOdin.jpg

Above: Anthony Hopkins as Odin in the 2011 film "Thor"

Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—

"The Tesseract* was the jewel of Odin's treasure room."

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

Weaving also played Agent Smith in The Matrix Trilogy.

The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.

"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."

— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com

For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—

http://www.log24.com/log/pix11B/110831-BrightStar.jpg

Today's New York Times  has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—

"He treated it like a puzzle."

Sometimes that's the best you can do.

* See also tesseract  in this journal.

** For a different arrangement of 13 things, see the cube's 13 axes in this journal.

Friday, June 3, 2011

MIT Day

Filed under: General — m759 @ 7:59 am

Today is Commencement Day at MIT.

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

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

To measure the changes:

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

The smartest are nothing:

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

Well, perhaps not quite nothing.

The above pictures were posted here on the day the following book was published—

http://www.log24.com/log/pix11A/110603-NineLives.jpg

The lives of the nine Jews in the above book amount to more than Yau's "nothing."

Note, however, that claims by Jews (see Jill Abramson yesterday
that their secular publications constitute a substitute for religion
and contain only "absolute truth" should be viewed with at least one
raised eyebrow.

Abramson's remark yesterday that her promotion to New York Times  executive editor
was like "ascending to Valhalla" had a religious flavor worthy of yesterday's
Feast of the Ascension.

In related news from yesterday's Times

IMAGE- 'Fearless Ascent, as a God or a Jet' -NYT

See also a symbol related to Apollo, to nine, and to "nothing"

A minimalist 3×3 matrix favicon—

http://www.log24.com/log/pix11A/110518-3x3FaviconURL.jpg

This may, if one likes, be viewed as the "nothing"
present at the Creation.  See Jim Holt on physics.

Wednesday, May 18, 2011

Minimalist Icon

Filed under: General,Geometry — Tags: — m759 @ 6:48 am

The source of the mysterious generic
3×3 favicon with one green cell

http://www.log24.com/log/pix11A/110518-GenericFavicon.jpg

— has been identified.

For minimalists, here is a purer 3×3 matrix favicon—

http://www.log24.com/log/pix11A/110518-3x3FaviconURL.jpg

This may, if one likes, be viewed as the "nothing"
present at the Creation.  See Jim Holt on physics.

See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.

Sunday, April 10, 2011

Bedeviled

Filed under: General — Tags: — m759 @ 10:30 pm

From tonight's online New York Times

John McCracken, Sculptor of Geometric Forms, Dies at 76

McCracken died in Manhattan on Friday, April 8.

From Christopher Knight in tonight's online LA Times

… the works embody perceptual and philosophical conundrums. The colored planks stand on the floor like sculptures….

McCracken was bedeviled by Stanley Kubrick's famously obscure science-fiction epic, "2001: A Space Odyssey," with its iconic image of an ancient monolith floating in outer space. The 1968 blockbuster was released two years after the artist made his first plank.

"At the time, some people thought I had designed the monolith or that it had been derived from my work," he told art critic Frances Colpitt of the coincidence in a 1998 interview.

Two photos of McCracken's 1967 Black Plank  seem relevant—

November 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCrackenPlank1967400w.jpg

December 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCracken-NatGallery-NothingToSeeHere-400w.jpg

Material that an artist might view as related, if only synchronistically—

Two posts in this journal on the dates the photos were taken—
The Embedding on November 28 and Dry Bones on December 28.

The photos are of an exhibition titled "There is nothing to see here" at the
National Gallery of Art, October 30, 2010-April 24, 2011 —

Click to enlarge.

http://www.log24.com/log/pix11/110410-NothingToSee-400w.jpg

For related nihilism from the National Gallery, see "Pictures of Nothing" in this journal.

Some less nihilistic illustrations—

The Meno  Embedding

Plato's Diamond embedded in The Matrix

A photo by one of the artists whose work is displayed above beside McCracken's—

http://www.log24.com/log/pix11/110410-Sugimoto-AndoChurch.jpg

"Accentuate the Positive."
 — Clint Eastwood

Sunday, March 13, 2011

The Counter

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

"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010

Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—

Cover Illustration Number 1 (1976):

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

Cover Illustration Number 2 (1991):

http://www.log24.com/log/pix11/110313-CombinatorialMatrixTheorySm.jpg

   The Shrikhande Graph

http://www.log24.com/log/pix11/110313-BrualdiRyser153.jpg

______________________________________________________________________________

This post was prompted by two remarks…

1.  In a different weblog, also on August 26, 2010—

    The Accidental Mathematician— "The Girl Who Played with Fermat's Theorem."

"The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire….

Salander is fascinated by a theorem on perfect numbers—
one can verify it for as many numbers as one wishes, and it never fails!—
and then advances through 'Archimedes, Newton, Martin Gardner,*
and a dozen other classical mathematicians,' all the way to Fermat’s last theorem."

2.  "The fact that the pattern retains its symmetry when you permute the rows and columns
     is very well known to combinatorial theorists who work with matrices."
     [My italics; note resemblance to the Brualdi-Ryser title above.]

     –Martin Gardner in 1976 on the diamond theorem

* Compare Eric Temple Bell (as quoted at the MacTutor history of mathematics site)—

    "Archimedes, Newton, and Gauss, these three, are in a class by themselves
     among the great mathematicians, and it is not for ordinary mortals
     to attempt to range them in order of merit."

     This is from the chapter on Gauss in Men of Mathematics .

Saturday, January 8, 2011

True Grid (continued)

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

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

Sunday December 10, 2006

 

 m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
– Thomas Pynchon   

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Monday, November 29, 2010

On Time

Filed under: General — m759 @ 5:45 am

and under budget —

The Tiffany Case

http://www.log24.com/log/pix10B/101129-TheTiffanyCase.jpg

See also Sunday morning’s pictures

Plato's Diamond embedded in The Matrix

The sound of singing grows louder.

Sunday, November 28, 2010

The Embedding

Filed under: General,Geometry — m759 @ 6:00 am

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

Plato's Diamond embedded in The Matrix

See also Plato's Code and
 Plato Thanks the Academy.

Saturday, October 30, 2010

Keanu vs. the Devil

Filed under: General — Tags: — m759 @ 4:04 am

(Continued from Little Buddha  (1994), The Matrix  (1999), and Constantine  (2005))

This post was suggested by yesterday's post on Habermas and by his 1962 book The Structural Transformation of the Public Sphere  (English translation, 1989).

The "public sphere" of Habermas has come to pass; it is, of course, the World-Wide Web.

For October 30, the day leading up to Devil's Night, a more private sphere—though in a public setting— seems appropriate…

http://www.log24.com/log/pix10B/101030-CentralParkSphere.jpg

The Day the Earth Stood Still  (2008)

A Keanu Reeves scene related to this image—

"The low point of the movie’s persuasiveness is the single scene with Professor Barnhardt (John Cleese) — in the original an Einstein-like scientist who impresses Klaatu with his highly evolved thinking, here a caricature of professorial enlightenment. Helen decides to bring Klaatu to Professor Barnhardt when Klaatu professes his disappointment with earth’s leaders. 'Those aren’t our leaders!' she protests earnestly. 'Let me take you to one of our leaders!'"

A perhaps more persuasive scene, from today's New York Times

Prize in Hand, He Keeps His Eye on Teaching

http://www.log24.com/log/pix10B/101030-VargasLlosa.jpg

Nobel winner Mario Vargas Llosa teaches
a seminar on Borges at Princeton

(Photo by James Leynse for The New York Times )

Thursday, October 14, 2010

Diamond Theory and Magic Squares

Filed under: General,Geometry — Tags: , — m759 @ 6:19 pm

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

— e. e. cummings, 1944

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

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

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

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

— e. e. cummings

Happy birthday, e. e.

Monday, September 27, 2010

The Social Network…

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

… In the Age of Citation

1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).

— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology,  Vol. 23, pp. 127–146 (1993)

From this paper's CiteSeer page

Citations

766  Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994
100 The act of creation – Koestler – 1964
 75 Visual Thinking – Arnheim – 1969

Visual Image of the Problem—

From a Google search today:

http://www.log24.com/log/pix10B/100927-GardnerGaloisSearch.jpg

Related material—

http://www.log24.com/log/pix10B/100927-GoogleBirthdayCake.jpg

"It is better to light one candle…"

"… the early favorite for best picture at the Oscars" — Roger Moore

Saturday, August 7, 2010

Rift Designs

Filed under: General — m759 @ 8:28 pm

From the current index to obituaries at Telegraph.co.uk—

Teufel is also featured in today's New York Times

"Mr. Teufel became a semicelebrity, helped in no small part by his last name, which means 'devil' in German."

From Group Analysis ,  June 1993, vol. 26 no. 2, 203-212—

The Problem of Good and Evil

by Ronald Sandison, Ledbury, Herefordshire HR8 2EY, UK

In my contribution to the Group Analysis Special Section: "Aspects of Religion in Group Analysis" (Sandison, 1993) I hinted that any consideration of a spiritual dimension to the group involves us in a discussion on whether we are dealing with good or evil spirits. But if we say that God is in the group, why is not the Devil there also? Can good and evil coexist in the same group matrix? Is the recognition of evil "nothing but" the ability to distinguish between good and bad? If not, then what is evil? Is it no more than the absence of good?

These and other questions were worked on at a joint Institute of Group Analysis and Group-Analytic Society (London) Workshop entitled "The Problem of Good and Evil." We considered the likelihood that good and evil coexist in all of us, as well as in the whole of the natural world, not only on earth, but in the cosmos and in God himself What we actually do with good and evil is to split them apart, thereby shelving the problem but at the same time creating irreconcilable opposites. This article examines this splitting and how we can work with it psychoanalytically.

This suggests a biblical remark—

"Now there was a day… when the sons of God
came to present themselves before the Lord,
and Satan came also among them."

Job 1:6, quoted by Chesterton in The Man Who Was Thursday

Sandison died on June 18. See the Thursday, August 5, Log24 post "The Matrix."

Teufel died on July 6. See the Log24 posts for that day.

The title of this  post, "rift designs," refers to a recurring theme in the July 6 posts. It is taken from Heidegger.

From a recent New Yorker  review of Absence of Mind  by Marilynne Robinson—

"Robinson is eloquent in her defense of the mind’s prerogatives, but her call for a renewed metaphysics might be better served by rereading Heidegger than by dusting off the Psalms."

Following this advice, we find—

"Propriation gathers the rift-design of the saying and unfolds it  in such a way that it becomes the well-joined structure of a manifold showing."

p. 415 of Heidegger's Basic Writings , edited by David Farrell Krell, HarperCollins paperback, 1993

"Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des vielfältigen Zeigens." 

— Heidegger, Weg zur Sprache

Monday, April 12, 2010

Weaveworld

Filed under: General — m759 @ 10:31 am

"Fact and fiction weave in and out of novels
 like a shell game." –R. B. Kitaj
 as quoted by Marco Livingstone

'The Bourne Ultimatum,' starring Matt Damon, cannot be displayed, because it contains errors.

Related material: Prime Designer,
Shell Game, and Evil Daemon.

See also remarks by root@matrix.net.

Wednesday, April 7, 2010

Veritas

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

Some historians consider today's date, April 7, to be the date of the Crucifixion in the Roman calendar (a solar calendar, as opposed to the Jewish lunar scheme).

Since the ninefold square has been called both a symbol of Apollo and the matrix of a cross, it will serve as an icon for today–

The 3x3 square

The image “http://www.log24.com/log/pix05B/051202-Cross.gif” cannot be displayed, because it contains errors.
Adapted from
Ad Reinhardt

Sunday, January 17, 2010

Welcome to the Ape Stuff

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

NY Times obituary of Knox Burger,
book editor and agent, who died at 87 on January 4

"As a magazine editor in the 1950s, Knox Burger published Kurt Vonnegut’s first short story….

During Mr. Burger’s tenure at Collier’s, a short story by Vonnegut, whom he had known slightly when both were at Cornell and who was then working in public relations for General Electric, crossed his desk. He asked for changes, which Vonnegut made, and the story, 'Report on the Barnhouse Effect,' appeared in the magazine in February 1950. It was the first published work of fiction for Vonnegut, who recounted the episode decades later….

At least half a dozen authors… honored Mr. Burger by dedicating books to him. Vonnegut, who died in 2007, did, too. His dedication of Welcome to the Monkey House, a 1968 collection of short stories that included 'Report on the Barnhouse Effect,' read:

'To Knox Burger. Ten days older than I am. He has been a very good father to me.'"

A Jesuit at the
Gerard Manley Hopkins Archive

"Bisociation": The Act of Creation

"Koestler’s concept of ‘bisociation’… enters into the very ‘act of creation.’ In every such act, writes Koestler, the creator ‘bisociates,’ that is, combines, two ‘matrices’– two diverse patterns of knowing or perceiving– in a new way. As each matrix carries its own images, concepts, values, and ‘codes,’ the creative person brings together– ‘bisociates’– two diverse matrices not normally connected."

– Joseph J. Feeney, S.J.

Robert Stone in A Flag for Sunrise
(Knopf hardcover, 1981)–

"The eye you see him with is the same eye with which he sees you."

– Father Egan on page 333

Pablo on page 425–

"'…You know, he told me– that old man told me– the eye you look at it with, well, that's the eye it sees you with. That's what he told me.'

Holliwell was moved to recall an experiment he had once read about; he had clipped the report of it for his class. An experimenter endeavoring to observe chimpanzee behavior had fashioned a spy hole in the door of the animals' chamber through which he might watch them unobserved. Putting his eye to it, he had seen nothing more than what he finally identified as the eye of a chimpanzee on the other side of the door. Ape stuff."

More ape stuff from a Jesuit–

"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond,
                Is immortal diamond."

— Gerard Manley Hopkins,
"That Nature is a Heraclitean Fire
and of the comfort of the Resurrection
"

More ape stuff from myself–

http://www.log24.com/log/pix10/100117-TradingPlaces.jpg

Problem: Perform this transformation
by combining the sorts of permutations allowed
in the diamond puzzle. A solution: click here.

Wednesday, December 30, 2009

Fearful Symmetry

Filed under: General,Geometry — m759 @ 4:12 pm

http://www.log24.com/log/pix09A/091230-Koestlerr-NYRB19641217.gif

Arthur Koestler by David Levine,
New York Review of Books,
December 17, 1964

A Jesuit at the
Gerard Manley Hopkins Archive
:

‘Bisociation’: The Act of Creation

Koestler’s concept of ‘bisociation’… enters into the very ‘act of creation.’ In every such act, writes Koestler, the creator ‘bisociates,’ that is, combines, two ‘matrices’– two diverse patterns of knowing or perceiving– in a new way. As each matrix carries its own images, concepts, values, and ‘codes,’ the creative person brings together– ‘bisociates’– two diverse matrices not normally connected.

— Joseph J. Feeney, S.J.

See also December 9, 2009:

The theme of the January 2010 issue of the
Notices of the American Mathematical Society
is “Mathematics and the Arts.”

Related material:

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

Saturday, November 28, 2009

Annals of Intelligence:

Filed under: General — m759 @ 1:06 am

A Cross for the
Goat Men

Cross made from 9 squares of a 25-square matrix

Related material:

Krauss, Malevich, and
Linguistic Creation

Tuesday, November 3, 2009

Summa Mythologica

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

Book review by Jadran Mimica in Oceania, Vol. 74, 2003:

"In his classic essay of 1955 'The Structural Study of Myth' Levi-Strauss came up with a universal formula of mythopeic dynamics

[fx(a) : fy(b) :: fx(b) : fa-1(y)]

that he called canonical 'for it can represent any mythic transformation'. This formulation received its consummation in the four massive Mythologiques volumes, the last of which crystallises the fundamental dialectics of mythopoeic thought: that there is 'one myth only' and the primal ground of this 'one' is 'nothing'. The elucidation of the generative matrix of the myth-work is thus completed as is the self-totalisation of both the thinker and his object."

So there.

At least one mathematician has claimed that the Levi-Strauss formula makes sense. (Jack Morava, arXiv pdf, 2003.)

I prefer the earlier (1943) remarks of Hermann Hesse on transformations of myth:

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

Wednesday, August 19, 2009

Wednesday August 19, 2009

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

Group Actions, 1984-2009

From a 1984 book review:

"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

— Jonathan L. Alperin,
   review of books on group theory,
   Bulletin (New Series) of the American
   Mathematical Society
10 (1984) 121, doi:
   10.1090/S0273-0979-1984-15210-8
 

A more specific example:


Actions of GL(2,3) on a 3x3 coordinate-array

The same example
at Wolfram.com:

Ed Pegg Jr.'s program at Wolfram.com to display a large number of actions of small linear groups over finite fields

Caption from Wolfram.com:
 
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."

Citation data from Wolfram.com:

"GL(2,p) and GL(3,3) Acting on Points"
 from The Wolfram Demonstrations Project,
 http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
 Contributed by: Ed Pegg Jr"

As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:

"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."

Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
 

Tuesday, February 10, 2009

Tuesday February 10, 2009

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

“Most crime dramas
have a gimmick.”

March 2009 Notices of the
American Mathematical Society

Feb. 10,
2009
PA NY
Midday 106 407
Evening 829 216


Brams… uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….”

Revelation Game payoff matrix

(The number-pairs here reflect
relative values of the situations
the author assigns to SB and to P.)

Related Material

on theology and drama —
the two Log24 entries on
Stephen King’s birthday, 2008.

Monday, January 5, 2009

Monday January 5, 2009

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

A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

 

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”

 

(Received June 15 1974)

Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Click for details.

Sunday, October 5, 2008

Sunday October 5, 2008

Filed under: General — Tags: — m759 @ 4:23 pm
Nash Equilibrium or:
To Make a Short Story Long

Last night's entry presented a
short story summarized by
four lottery numbers.

Today's mid-day lotteries
and associated material:

Pennsylvania, 201– i.e., 2/01:
Kindergarten Theology

Theologian James Edwin Loder:

"In a game of chess, the knight's move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a 'knight's move' in intelligence."

New York, 229– i.e., 2/29:
I Have a Dreamtime

"One must join forces with friends of like mind"

Related material:

Terence McKenna:

"Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey 'haunts time like a ghost.'"

Anonymous author:

"'Knight's move thinking' is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia."

Star Wars:
 
John Nash, as portrayed by Russell Crowe

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

Tom O'Bedlam's Song

For more on the sleep of Apollo,
see the front page of today's
New York Times Book Review.

Garrison Keillor's piece there,
"Dying of the Light," is
about the fear of death felt
by an agnostic British twit.

For relevant remarks by
a British non-twit, see
William Dunbar–

Timor Mortis conturbat me.

Saturday, August 16, 2008

Saturday August 16, 2008

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

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Finite Structure
 on a Book Cover:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.


Walsh Series
states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

Sunday, July 6, 2008

Sunday July 6, 2008

Filed under: General — m759 @ 9:00 pm

"Hancock" Powers to the Top
of July Fourth Box Office

This evening's online
  New York Times

New York Lottery
Sunday, July 6,
2008:
Mid-day 307
Evening  921

Log24  3/07:

Symbols:


Three 3x3 symbols of a language game:  the field, the game, checkmate

Log24  9/21:

"The consolations of form,
the clean crystalline work"
— Iris Murdoch, 
"Against Dryness"

Will Smith
on Chess

Will Smith with chessboard

Will Smith

The Independent, 9 July 2004:

"A devoted father, Smith passes on his philosophy of life to his children through chess, among other things.

'My father taught me how to play chess at seven and introduced beautiful concepts that I try to pass on to my kids. The elements and concepts of life are so perfectly illustrated on a chess board. The ability to accurately assess your position is the key to chess, which I also think is the key to life.'

He pauses, searching for an example. 'Everything you do in your life is a move. You wake up in the morning, you strap on a gun, and you walk out on the street– that's a move. You've made a move and the universe is going to respond with its move.

'Whatever move you're going to make in your life to be successful, you have to accurately access the next couple of moves– like what's going to happen if you do this? Because once you've made your move, you can't take it back. The universe is going to respond.'

Smith has just finished reading The Alchemist, by the Brazilian writer Paulo Coelho: 'It says the entire world is contained in one grain of sand, and you can learn everything you need to learn about the entire universe from that one grain of sand. That is the kind of concept I'm teaching my kids.'"

Related material:

"Philosophers' Stone"
and other entries
of June 25, 2008

Monday, June 16, 2008

Monday June 16, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 pm
Bloomsday for Nash:
The Revelation Game

(American Mathematical Society Feb. 2008
review of Steven Brams’s Superior Beings:
If They Exist, How Would We Know?)

(pdf, 15 megabytes)

"Brams does not attempt to prove or disprove God. He uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….

In the 'Revelation Game,' for example, the Person (P) has two options:
1) P can believe in SB's existence
2) P can not believe in SB's existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself

Each player also has a primary and secondary goal. For the Person, the primary goal is to have his belief (or non-belief) confirmed by evidence (or lack thereof). The secondary goal is to 'prefer to believe in SB’s existence.' For the Supreme Being, the primary goal is to have P believe in His existence, while the secondary goal is to not reveal Himself. These goals allow us to rank all the outcomes for each player from best (4) to worst (1). We end up with a matrix as follows (the first number in the parentheses represents the SB's ranking for that box; the second number represents P's ranking):

Revelation Game payoff matrix

The question we must answer is: what is the Nash equilibrium in this case?"

Analogously:

Lotteries on
Bloomsday,
June 16,
2008
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
418

 

The Exorcist

No belief,
no revelation

064

 

4x4x4 cube summarizing geometry of the I Ching

Revelation
without belief

Evening
(Belief)
709

Human Conflict Number Five album by The 10,000 Maniacs

 

Belief without
revelation

198

 

(A Cheap
Epiphany)

Black disc from end of Ch. 17 of Ulysses

Belief and
revelation

The holy image

Black disc from end of Ch. 17 of Ulysses

denoting belief and revelation
may be interpreted as
a black hole or as a
symbol by James Joyce:

When?

Going to dark bed there was a square round Sinbad the Sailor roc's auk's egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.

Where?

Black disc from end of Ch. 17 in Ulysses

Ulysses, conclusion of Chapter 17

Friday, June 6, 2008

Friday June 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 2:45 am
The Dance of Chance

"Harvard seniors have
every right to demand a
    Harvard-calibre speaker."

— Adam Goldenberg in
The Harvard Crimson

"Look down now, Cotton Mather"

— Wallace Stevens,
Harvard College
Class of 1901

For Thursday, June 5, 2008,
commencement day for Harvard's
Class of 2008, here are the
Pennsylvania Lottery numbers:

Mid-day 025
Evening 761

Thanks to the late
Harvard professor
Willard Van Orman Quine,
the mid-day number 025
suggests the name
"Isaac Newton."

(For the logic of this suggestion,
see On Linguistic Creation
and Raiders of the Lost Matrix.)

Thanks to Google search, the
  name of Newton, combined with
  Thursday's evening number 761,
suggests the following essay:

Science 10 August 2007:
Vol. 317. no. 5839, pp. 761-762

PHILOSOPHY OF SCIENCE:
The Cha-Cha-Cha Theory
of Scientific Discovery

Daniel E. Koshland Jr.*

* D. E. Koshland Jr. passed away on 23 July 2007. He was a professor of biochemistry and molecular and cell biology at the University of California, Berkeley, since 1965. He served as Science's editor-in-chief from 1985 to 1995.
 


What can a non-scientist add?

Perhaps the Log24 entries for
the date of Koshland's death:

The Philosopher's Stone
and The Rock.

Or perhaps the following
observations:

On the figure of 25 parts
discussed in
"On Linguistic Creation"–

5x5 ultra super magic square

"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah.
"

— Clifford Pickover  

"At the still point,
there the dance is.
"

— T. S. Eliot,
Harvard College
Class of 1910

Thursday, March 6, 2008

Thursday March 6, 2008

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

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

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

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

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

See also Log24, May 21, 2007.

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

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

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

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

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

Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube

and
Jewel in the Crown

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

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

They may also consult
the poet Rubén Darío:

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


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

Wednesday, October 3, 2007

Wednesday October 3, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:09 pm
Janitor Monitor

 

The image “http://www.log24.com/log/pix07A/070803-Trees.jpg” cannot be displayed, because it contains errors.

Will Hunting may be
interested in the following
vacant editorships at
The Open Directory:

Graph Theory
and
Combinatorics.

Related material:

The Long Hello and
On the Holy Trinity

"Hey, Carrie-Anne, what's
your game now….?
"

The image “http://www.log24.com/log/pix07A/071003-Magdalene.GIF” cannot be displayed, because it contains errors.

Picture sources:
azstarnet.com,
vibrationdata.com.

Personally, I prefer
Carol Ann:

From Criticism,  Fall, 2001,
by Carol Ann Johnston

"Drawing upon Platonic thought, Augustine argues that ideas are actually God's objective pattern and as such exist in God's mind. These ideas appear in the mirror of the soul. (35)."

(35.) In Augustine, De Trinitate, trans., Stephen McKenna (Washington, D.C.: Catholic University Press, 1970). See A. B. Acton, "Idealism," in The Encyclopedia of Philosophy, ed., Paul Edwards. Vol. 4 (New York: Macmillan, 1967): 110-118; Robert McRae, "`Idea' as a Philosophical Term in the Seventeenth Century," JHI 26 (1965): 175-190, and Erwin Panofsky, Idea: A Concept in Art History, trans., Joseph J. S. Peake (Columbia, S.C.: University of South Carolina Press, 1968) for explications of this term.

 

See also
Art Wars: Geometry as Conceptual Art
and Ideas and Art: Notes on Iconology.

 

For more on Augustine and geometry,
see Today's Sinner (Aug. 28, 2006).

 

 

Monday, October 1, 2007

Monday October 1, 2007

Filed under: General,Geometry — Tags: , , — m759 @ 7:20 am
Bright as Magnesium

"Definitive"

— The New York Times,  
Sept. 30, 2007, on
Blade Runner:
The Final Cut

Institute for Advanced Study, Princeton, N.J.

"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."

Hal Crowther

"His final lecture was an eloquent, prophetic flight of free association….

Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"


Related material: 
tears in the rain–

Game Over
(Nov. 5, 2003):
 

The film "The Matrix," illustrated

Coordinates for generating the Miracle Octad Generator

Tuesday, August 21, 2007

Tuesday August 21, 2007

Filed under: General — Tags: — m759 @ 3:29 pm
Shell Game

The Bourne Ultimatum, starring Matt Damon” cannot be displayed, because it contains errors.

Part I:

Overview of Unix
at pangea.stanford.edu

Last revision August 2, 2004

“The Unix operating environment is organized into three layers. The innermost level of Unix is the kernel. This is the actual operating system, a single large program that always resides in memory. Sections of the code in this program are executed on behalf of users to do needed tasks, like access files or terminals. Strictly speaking, the kernel is Unix.

The next level of the Unix environment is composed of programs, commands, and utilities. In Unix, the basic commands like copying or removing files are implemented not as part of the kernel, but as individual programs, no different really from any program you could write. What we think of as the commands and utilities of Unix are simply a set of programs that have become standardized and distributed. There are hundreds of these, plus many additional utilities in the public domain that can be installed.

The final level of the Unix environment, which stands like an umbrella over the others, is the shell. The shell processes your terminal input and starts up the programs that you request. It also allows you to manipulate the environment in which those programs will execute in a way that is transparent to the program. The program can be written to handle standard cases, and then made to handle unusual cases simply by manipulating its environment, without having to have a special version of the program.” (My italics.)

Part II:

Programs

From my paper journal
on the date
“Good Will Hunting”
was released:

Friday, December 5, 1997

To: The executive editor, The New York Times

Re: The Front Page/His Girl Friday

Match the speaker with the speech–

The Speech
“The son of a
bitch stole my…”
  The Speaker Frame of Reference
 1. rosebud A. J. Paul Getty The front page, N.Y. Times, Monday, 12/1/97
 2. clock B. Joel Silver Page 126, The New Yorker, 3/21/94
 3. act C. Blanche DuBois The Elysian Fields
 4. waltz D. Bob Geldof People Weekly 12/8/97
 5. temple E. St. Michael Heaven’s Gate
 6. watch F. Susanna Moore In the Cut (pbk., Dec. ’96) p. 261
 7. line G. Joseph Lelyveld Page A21, The New York Times, 12/1/97
 8. chair H. Kylie Minogue Page 69, People Weekly, 12/8/97
 9. religion I. Carol Gilligan The Garden of Good and Evil
10. wife J. John Travolta “Michael,” the movie
11. harp K. Shylock Page 40, N.Y. Review of Books, 12/4/97
12. Oscar L. Stephen King The Shining (pbk., 1997), pp. 316, 317

Postscript of June 5, 2003:

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

Vladimir Nabokov

Part III:

The Bourne Shell

“The binary program of the Bourne shell or a compatible program is located at /bin/sh on most Unix systems, and is still the default shell for the root superuser on many current Unix implementations.” –Wikipedia

Afterword:

See also
the recent comments
of root@matrix.net in
Peter Woit’s weblog.

“Hey, Carrie-Anne,
what’s your game now….”

— The Hollies, 1967   

Sunday, August 12, 2007

Sunday August 12, 2007

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

The Geometry of Qubits

In the context of quantum information theory, the following structure seems to be of interest–

"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

A 16-element affine space and a corresponding 16-element matrix ring

This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field.  (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.)  (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.)  Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring.  This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)

Saniga concludes:
 

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

  • not only offers a principally new geometrically-underlined insight into their intrinsic nature,
  • but also gives their applications a wholly new perspective
  • and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
It would seem that my own
study of pure mathematics
for instance, of the following
"diamond ring"–
 
The image “http://www.log24.com/theory/images/FourD.gif” cannot be displayed, because it contains errors.
 
is not without relevance to
the physics of quantum theory.

Sunday, June 3, 2007

Sunday June 3, 2007

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

Haunting Time

"Macquarrie remains one of the most
important commentators [on] …
Heidegger's work. His co-translation
of Being and Time into English is
considered the canonical version."
Wikipedia

The Rev. John Macquarrie, Scottish Theologian, Dies at 87

The Rev. Macquarrie died on
May 28.  The Log24 entry
for that date contains the
following illustration:

A 4x4x4 cube

The part of the illustration
relevant to the death of
Macquarrie is the color.
From my reply to

a comment on the
May 28 entry:

"I checked out [Terence] McKenna and found this site on the aging druggie. I didn't like the hippie scene in the sixties and I don't like it now. Booze was always my drug of choice. Still, checking further, I found that McKenna's afterword to Dick's In Pursuit of Valis was well written."

From McKenna's afterword:

"Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey 'haunts time like a ghost.'"

I can find no source for
any remarks of Whitehead
on the color "dove grey"
(or "gray") but Whitehead
did say that

"A colour is eternal.  It haunts time like a spirit.  It comes and it goes.  But where it comes it is the same colour.  It neither survives nor does it live.  It appears when it is wanted." —Science and the Modern World, 1925

The poetic remark of
McKenna on the color
"dove grey" may be
taken, in a schizophrenic
(or, similarly, a Christian) way,
 as a reference to the Holy Spirit.

My own remarks on the hippie
scene seem appropriate as a
response to media celebration
of today's 40th anniversary of
the beginning of the 1967
"summer of love."

Friday, May 4, 2007

Friday May 4, 2007

Filed under: General — Tags: , , , — m759 @ 5:01 pm

May '68 Revisited

"At his final Paris campaign rally… Mr. Sarkozy declared himself the candidate of the 'silent majority,' tired of a 'moral crisis in France not seen since the time of Joan of Arc.'

'I want to turn the page on May 1968,' he said of the student protests cum social revolution that rocked France almost four decades ago.

'The heirs of May '68 have imposed the idea that everything has the same worth, that there is no difference between good and evil, no difference between the true and the false, between the beautiful and the ugly and that the victim counts for less than the delinquent.'

Denouncing the eradication of 'values and hierarchy,' Mr. Sarkozy accused the Left of being the true heirs and perpetuators of the ideology of 1968."

— Emma-Kate Symons, Paris, May 1, 2007, in The Australian

Related material:

From the translator's introduction to Dissemination, by Jacques Derrida, translated by Barbara Johnson, University of Chicago Press, 1981, page xxxi —

"Both Numbers and 'Dissemination' are attempts to enact rather than simply state the theoretical upheavals produced in the course of a radical reevaluation of the nature and function of writing undertaken by Derrida, Sollers, Roland Barthes, Julia Kristeva and other contributors to the journal Tel Quel in the late 1960s. Ideological and political as well as literary and critical, the Tel Quel program attempted to push to their utmost limits the theoretical revolutions wrought by Marx, Freud, Nietzsche, Mallarme, Levi-Strauss, Saussure, and Heidegger."

This is the same Barbara Johnson who has served as the Frederic Wertham Professor of Law and Psychiatry in Society at Harvard.

Johnson has attacked "the very essence of Logic"–

"… the logic of binary opposition, the principle of non-contradiction, often thought of as the very essence of Logic as such….

Now, my understanding of what is most radical in deconstruction is precisely that it questions this basic logic of binary opposition….

Instead of a simple 'either/or' structure, deconstruction attempts to elaborate a discourse that says neither 'either/or', nor 'both/and' nor even 'neither/nor', while at the same time not totally abandoning these logics either."

— "Nothing Fails Like Success," SCE Reports 8, 1980

Such contempt for logic has resulted, for instance, in the following passage, quoted approvingly on page 342 of Johnson's  translation of Dissemination, from Philippe Sollers's Nombres (1966):

"The minimum number of rows– lines or columns– that contain all the zeros in a matrix is equal to the maximum number of zeros located in any individual line or column."

For a correction of Sollers's  Johnson's damned nonsense, click here.

Update of May 29, 2014:

The error, as noted above, was not Sollers's, but Johnson's.
See also the post of May 29, 2014 titled 'Lost in Translation.'

Thursday, March 8, 2007

Thursday March 8, 2007

Filed under: General,Geometry — m759 @ 9:00 am
Dia de la
Mujer Trabajadora

The image “http://www.log24.com/log/pix07/070308-Aldecoa.jpg” cannot be displayed, because it contains errors.

“Yo es que nací un 8 de marzo,
Día de la Mujer Trabajadora,
y no he hecho más que
trabajar toda mi vida.”

Josefina Aldecoa

For background on Aldecoa,
see a paper (pdf) by
Sara Brenneis:

“Josefina Aldecoa intertwines
history, collective memory
and individual testimony in her
historical memory trilogy…”

HISTORICAL MEMORY–

History:

The Triangle Shirtwaist Factory Fire in New York City on March 25, 1911, was the largest industrial disaster in the history of the city of New York, causing the death of 146 garment workers who either died in the fire or jumped to their deaths.

Propaganda, March 1977:

“On March 8, 1908, after the death of 128 women trapped in a fire at the Triangle Shirtwaist Factory in New York City, 15,000 women workers from the garment and textile industry marched echoing the demands of their sisters 50 years earlier…”

Propaganda, March 2006:

“First of all, on March 8th, 1857, a large number of factory workers in the United States took to the streets to demand their economic and political rights. The owners called the police who arrived immediately and opened fire, engaging in blind repression… Later on, in 1908, the same date of March 8th was once again a memorable date of struggle. On this day, capitalist bosses in Chicago set fire to a textile factory where over a thousand women worked. A very large number was terribly burnt. 120 died!”

Propaganda disguised as news, March 2007:

From today’s top story in 24 HoursTM, a commuter daily in Vancouver published by Sun Media Corporation:

Fight still on for equality

By Robyn Stubbs and Carly Krug

“International Women’s Day commemorates a march by female garment workers protesting low wages, 12-hour workdays and bad working conditions in New York City on March 8, 1857.

Then in 1908, after 128 women were trapped and killed in a fire at a New York City garment and textile factory, 15,000 women workers again took their protests to the street.”

Related historical fiction:

A version of the
I Ching’s Hexagram 19:

The image “http://www.log24.com/log/pix05B/051202-Hex19.jpg” cannot be displayed, because it contains errors.

Log24 12/3/05:

The image “http://www.log24.com/log/pix05B/051202-Axe.jpg” cannot be displayed, because it contains errors.

Katherine Neville, The Eight

    “What does this have to do with why we’re here?”
    “I saw it in a chess book Mordecai showed me.  The most ancient chess service ever discovered was found at the palace of King Minos on Crete– the place where the famous Labyrinth was built, named after this sacred axe.  The chess service dates to 2000 B.C.  It was made of gold and silver and jewels…. And in the center was carved a labrys.”
… “But I thought chess wasn’t even invented until six or seven hundred A.D.,” I added.  “They always say it came from Persia or India.  How could this Minoan chess service be so old?”
    “Mordecai’s written a lot himself on the history of chess,” said Lily…. “He thinks that chess set in Crete was designed by the same guy who built the Labyrinth– the sculptor Daedalus….”
    Now things were beginning to click into place….
    “Why was this axe carved on the chessboard?” I asked Lily, knowing the answer in my heart before she spoke.  “What did Mordecai say was the connection?”….
    “That’s what it’s all about,” she said quietly.  “To kill the King.”
 
     The sacred axe was used to kill the King.  The ritual had been the same since the beginning of time. The game of chess was merely a reenactment.  Why hadn’t I recognized it before?

Perhaps at the center of
Aldecoa’s labyrinth lurk the
  capitalist bosses from Chicago
who, some say, set fire
to a textile factory
on this date in 1908.

For a Freudian perspective
on the above passage,
see yesterday’s entry
In the Labyrinth of Time,
with its link to
John Irwin‘s essay

The False Artaxerxes:
Borges and the
Dream of Chess
.”

The image “http://www.log24.com/log/pix07/070307-Symbols.gif” cannot be displayed, because it contains errors.

Symbols
S. H. Cullinane
March 7, 2007

Today, by the way, is the
feast of a chess saint.

Friday, December 15, 2006

Friday December 15, 2006

Filed under: General — m759 @ 2:02 am

 Putting the
X
in Xmas

“It’s all in Plato, all in Plato;
bless me, what do they
teach them at these schools?”

C. S. Lewis

Apparently they teach them nihilism, empty rhetoric, and despair, as reflected in Borges, Baudrillard, and Benjamin, according to the art review below from today’s New York Times.  Let us hope that the late Peter Boyle, who died on Tuesday, Dec. 12, has moved beyond these now– singing “Heaven, I’m in Heaven,” rather than “Puttin’ on the Ritz.”

Ritz and Heaven

Black, White, and
Read All Over

by Randy Kennedy
in The New York Times
Friday, Dec. 15, 2006

“In one of Jorge Luis Borges’s best-known short stories, ‘Pierre Menard, Author of the Quixote,’ a 20th-century French writer sets out to compose a verbatim copy of Cervantes’s 17th-century masterpiece simply because he thinks he can, originality perhaps not being all it’s cracked up to be.

He manages two chapters word for word, a spontaneous duplicate that Borges’s narrator finds to be ‘infinitely richer’ than the original because it contains all manner of new meanings and inflections, wrenched as it is from its proper time and context….”

[An artist’s version of a newspaper is]….

“a drawing of a copy of a version of what happened, holding a mirror up to nature with a refraction or two in between.  In a way that mixes Borges with a dollop of Jean Baudrillard and a heavy helping of Walter Benjamin, the work also upends ideas….”

The Work:

Pennsylvania Lottery
December 2006
Daily Number (Day):

Borges,
Menard’s Quixote, and
The Harvard Crimson
Mon., Dec. 11:
133
Baudrillard
(via a white Matrix)
Sun., Dec. 10:
569
Benjamin and
a black view of life in
“The Garden of Allah”
Sat., Dec. 9:
602

Click on numbers
for commentary.

Borges and Benjamin are
  referenced directly in the
  commentary. For Baudrillard,
  see Richard Hanley on
  Baudrillard and The Matrix:

“There is nothing new under the sun. With the death of the real, or rather with its (re)surrection, hyperreality both emerges and is already always reproducing itself.”  –Jean Baudrillard

Related material:

To Be,”

The Transcendent Signified,”

and…

Postmodern Religion


.

Sunday, December 10, 2006

Sunday December 10, 2006

Filed under: General,Geometry — Tags: , — m759 @ 6:00 am
The Matrix:

Time and Chance
on the 90th Birthday
of Kirk Douglas,
star of
The Garden of Allah

The Lottery 12/9/06 Mid-day Evening
New York 036

See

The Quest
for the 36

331

See 3/31

“square crystal” and “the symbolism could not have been more perfect.”

Pennsylvania 602

See 6/02

Walter Benjamin
on
“Adamic language.”

111

See 1/11

“Related material:
Jung’s Imago and Solomon’s Cube.”

See also

The image “http://www.log24.com/log/pix05B/051209-Douglas1.jpg” cannot be displayed, because it contains errors.

Diamonds

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

Saturday, November 25, 2006

Saturday November 25, 2006

Filed under: General — Tags: — m759 @ 1:25 pm

Today's birthdays:

Bob Lind, composer of "Elusive Butterfly," and General Augusto Pinochet.

Also today:

Stones tour rolls into Vancouver.

These events prompt fond memories of a Log24 entry from the Feast of the Transfiguration in 2002 and of more recent entries from this date last year– Buckley and Pinochet and Rehearsing Hell.

Perhaps the afterlife will include, for some, a Mick Jagger rendition of the Lind tune (along with the Percy Faith rendition of "Satisfaction" mentioned in The Last Samurai.)

Monday, October 9, 2006

Monday October 9, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:00 am
ART WARS:
To Apollo

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

“This is the garden of Apollo,
the field of Reason….”
John Outram, architect

To Apollo (10/09/02)
Art Wars: Apollo and Dionysus
(10/09/02)
Balanchine’s Birthday
(01/09/03)
Art Theory for Yom Kippur
(10/05/03)
A Form
(05/22/04)
Ineluctable
(05/27/04)
A Form, continued
(06/05/04)
Parallelisms
(06/06/04)
Ado
(06/25/04)
Deep Game
(06/26/04)
Gameplayers of Zen
(06/27/04)
And So To Bed
(06/29/04)
Translation Plane for Rosh Hashanah
(09/15/04)
Derrida Dead
(10/09/04)
The Nine
(11/09/04)
From Tate to Plato
(11/19/04)
Art History
(05/11/05)
A Miniature Rosetta Stone
(08/06/05)
High Concept
(8/23/05) 
High Concept, Continued
(8/24/05)
Analogical Train of Thought
(8/25/05)
Today’s Sermon: Magical Thinking
(10/09/05)
Balance
(10/31/05)
Matrix
(11/01/05)
Seven is Heaven, Eight is a Gate
(11/12/05)
Nine is a Vine
(11/12/05)
Apollo and Christ
(12/02/05)
Hamilton’s Whirligig
(01/05/06)
Cross
(01/06/06)
On Beauty
(01/26/06)
Sunday Morning
(01/29/06)
Centre
(01/29/06)
New Haven
(01/29/06) 
Washington Ballet
(02/05/06)
Catholic Schools Sermon
(02/05/06)
The Logic of Apollo
(02/05/06)
Game Boy
(08/06/06)
Art Wars Continued: The Krauss Cross
(09/13/06)
Art Wars Continued: Pandora’s Box
(09/16/06)
The Pope in Plato’s Cave
(09/16/06)
Today’s Birthdays
(09/26/06)
Symbology 101
(09/26/06)

Tuesday, October 3, 2006

Tuesday October 3, 2006

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

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

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

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

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, design theory, duads, duality, error correcting codes, exceptional groups, 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 schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Sunday, February 19, 2006

Sunday February 19, 2006

Filed under: General — m759 @ 2:04 pm

But seriously…

Raiders of the Lost Matrix
(continued)

The Matrix:

The image “http://www.log24.com/log/pix06/060219-Bingo2.jpg” cannot be displayed, because it contains errors.

Click on pictures for details.

In memory of George T. Davis,
who died on February 4,
a Hollywood ending:

Santa Claus rides alone.”
Clint Eastwood  

Monday, February 6, 2006

Monday February 6, 2006

Filed under: General,Geometry — m759 @ 10:00 am
The Diamond Theory of Truth


“Legend says that when the stones
are brought together the diamonds
inside of them will glow.”

— Harrison Ford in
“Indiana Jones and the
Temple of Doom”

In today’s online New York Times:

(1) A review of pop-archaeology TV,
     “Digging for the Truth,”
(2) a Sunday news story,
     “Looking for the Lie,”
(3) and a profile,
     “Storyteller in the Family.”

From (1):
“The season premiere ‘Digging for the Truth: The Real Temple of Doom,’ showed Mr. Bernstein in South America, exploring tunnels….”
From (2):
“… scientists are building a cognitive theory of deception to show what lying looks like….”
From (3):
“I did feel one had to get not just the facts, but the emotional underpinnings.”

Related material:

Log24 on

Harrison Ford’s birthday
last July–

The image “http://www.log24.com/log/pix05A/050713-Ford.jpg” cannot be displayed, because it contains errors.

— and Mathematics and Narrative.

See also Saturday’s entry,
Raiders of the Lost Matrix,
for logic as an aid in
detecting lies.

Saturday, February 4, 2006

Saturday February 4, 2006

Filed under: General — Tags: , — m759 @ 10:00 am
Raiders of
the Lost Matrix

(continued)

The Archaeologist
with a Thousand Faces

"From often humble beginnings, and often with a childhood fascination for antiquity, the archaeologist leaves familiar surroundings to undergo exacting professional training under a series of mentors and when armed, at last, with the intellectual weapons of the profession, sets off for unfamiliar or exotic realms, braving opposition and danger to solve an ancient mystery.  The lives of… real-life archaeologists… have lent themselves to this style of retelling… as have such fictional heroes as John Cullinane (Michener 1965) and Indiana Jones."

— From "Promised Lands and Chosen Peoples: The Politics and Poetics of Archaeological Narrative," by Neil Asher Silberman, pp. 249-262 in Nationalism, Politics, and the Practice of Archaeology, edited by Philip L. Kohl and Clare Fawcett, Cambridge University Press, paperback, published Feb. 8, 1996.

From Today in History,
by the Associated Press:

Thought for Today:
"Character consists of what you do
on the third and fourth tries."

James Michener,
American author (1907-1997),
attributed by
Simpson's Contemporary Quotations
to Chesapeake, Random House, 78.

The Matrix:

First try:

On Linguistic Creation
June 25, 1999

Second try:
Art Wars: Picasso's Birthday,
Oct. 25, 2002

Third try:
Matrix of the Death God,
May 25, 2003

Fourth try:
Happy Birthday,
July 26, 2004

Tuesday, December 6, 2005

Tuesday December 6, 2005

Filed under: General — Tags: — m759 @ 1:00 pm
Headline in today's New York Times:

'Year of Magical Thinking'
Headed for Broadway

which suggests…

Heaven, Hell,
and Hollywood

 
(continued)

"This could be Heaven
or this could be Hell."

The Eagles, Hotel California

"There are no facts,
there is no truth–
just data to be manipulated
."

Don Henley, The Garden of Allah

Data:

The New York Lottery numbers
on Joan Didion's birthday,
Monday, Dec. 5, 2005, were

Mid-day 729,
Evening 439.

Since that day's Log24 entry,
Magical Thinking, interpreted
the previous day's (Sunday's)
NY lottery numbers as a date
and a page number, it seems
appropriate to do a follow-up.

Date 7/29:

See Log24, 7/29/05,
Anatomy of a Death:

The image “http://www.log24.com/log/pix05A/050729-Bass5.jpg” cannot be displayed, because it contains errors.


Page 439:

See Bartlett's Familiar
Quotations
, 1919, p. 439
:

A man’s ingress into the world
is naked and bare,
His progress through the world
is trouble and care;
And lastly,
his egress out of the world,
is nobody knows where.

— John Edwin (1749-1790)

Related material:

The Log24 version of
"This Way to the Egress,"
Directions Out,
linked to in yesterday's
  Magical Thinking.

Saturday, July 30, 2005

Saturday July 30, 2005

Filed under: General — Tags: , , — m759 @ 11:21 am

Born today: Laurence Fishburne

Matrix

“The nine-fold square has centre, periphery, axes and diagonals. But all are present only in their bare essentials. It is also a sequence of eight triads. Four pass through the centre and four do not. This is the garden of Apollo, the field of Reason, sheltered by the Gate from the turmoil of the Delta, with its endless cycles of erasure and reinscription. This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale….”– Architects John Outram Associates
on work at Rice University

Yale Daily News, Jan. 11, 2001:  

“When New Haven was founded, the city was laid out into a grid of nine squares surrounded by a great wilderness.
Last year History of Art Professor Emeritus Vincent Scully said the original town plan reflected a feeling that the new city should be sacred.
Scully said the colony’s founders thought of their new Puritan settlement as a ‘nine-square paradise on Earth, heaven on earth, New Haven, New Jerusalem.'”

“Real and unreal are two in one:
New Haven
Before and after one arrives….”

— Wallace Stevens,
An Ordinary Evening in New Haven,’ XXVIII

Related material:
Log24 entries on
St. Peter’s Day, 2004

Thursday, June 23, 2005

Thursday June 23, 2005

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

Mathematics and Metaphor

The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles.  The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible.  The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history  and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
 
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.

Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated — 

“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182

This is false.  See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.

It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —

From Peter Woit’s weblog, Feb. 10, 2005:

“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'” 

In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that

“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two.  And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….

On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”

Here is another statement of the Rosetta-stone metaphor, from Weil’s translator, Martin H.  Krieger, in the A.M.S. Notices of November 2004,  “Some of What Mathematicians Do” (pdf):

“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’  Given an ability to read one column, can you find its translation in the other columns?  In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry.  In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”

For greater clarity, see  Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraico-geometric,” over finite fields), and arithmetic (i.e., number-theoretic).
 
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:

Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world.  Both these branches of mathematics had been studied intensively but separately…. Than along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds.  Mathematicians love to build bridges.”

Simon Singh: “The value of mathematical bridges is enormous.  They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s  creations…. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time.  Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur.  ‘But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'”

If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.

Monday, June 6, 2005

Monday June 6, 2005

Filed under: General — m759 @ 1:00 pm

Order and Disorder

From “Connoisseur of Chaos,”
by Wallace Stevens, in
Parts of a World, 1942:


I

A.  A violent order is a disorder; and
B.  A great disorder is an order.
    These Two things are one. (Pages of illustrations.)               

IV  

A.  Well, an old order is a violent one. This proves nothing.
    Just one more truth, one more
    Element in the immense disorder of truths.

B.  It is April as I write. The wind
    Is blowing after days of constant rain.
    All this, of course, will come to summer soon.
    But suppose the disorder of truths should ever come
    To an order, most Plantagenet, most fixed. . . .
    A great disorder is an order.
    Now, A And B are not like statuary, posed
    For a vista in the Louvre. They are things chalked
    On the sidewalk so that the pensive man may see.

V

    The pensive man . . . He sees that eagle float
    For which the intricate Alps are a single nest.

Related material:
“Derrida on Plato on writing says ‘In order for these contrary values (good/evil, true/false, essence/appearance, inside/outside, etc.) to be in opposition, each of the terms must be simply EXTERNAL to the other, which means that one of these oppositions (the opposition between inside and outside) must already be accredited as the matrix of all possible opposition.’ “

Peter J. Leithart

See also

Skewed Mirrors,
Sept. 14, 2003


“Evil did not  have the last word.”
Richard John Neuhaus, April 4, 2005

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


PARIS,
1922-1939

“There is never any ending to Paris.”
— Ernest Hemingway

Tuesday, April 5, 2005

Tuesday April 5, 2005

Filed under: General,Geometry — Tags: — m759 @ 3:17 pm
Art History:
The Pope of Hope

At the Vatican on
Shakespeare's Birthday
(See Log24.net,
Oct. 4, 2002)

See also the iconology
what Dan Brown in
The Da Vinci Code
  calls "symbology" —
of Pandora's Box
at Log24.net,
March 10, 2005:

The image “http://www.log24.com/log/pix05/050310-Nell2.jpg” cannot be displayed, because it contains errors.

 

"Man and woman are a pair of locked caskets,
each containing the key to the other."

Baroness Karen Blixen

"Karol Wojtyla had looked into
the heart of darkness–
and at the heart of darkness
discovered reason
for an indomitable hope.

He lived on the far side of
the greatest catastrophe
in human history,
the death of the Son of God,
and knew that evil
did not have the last word.
This is the key…."

Richard John Neuhaus,
April 4, 2005

The image “http://www.log24.com/log/pix05/050405-JoyceGeometry.gif” cannot be displayed, because it contains errors.

Finnegans Wake, p. 293,
"the lazily eye of his lapis"

 

The image “http://www.log24.com/log/pix05/050403-StPetersSq3.jpg” cannot be displayed, because it contains errors.

 

Perette Elizabeth Michelli on the Ovato Tondo:

 

"Notice how the Pope turns out to be
at the center of the breaking and
redefining of the Classical system."

"Derrida on Plato on writing says 'In order for these contrary values (good/evil, true/false, essence/appearance, inside/outside, etc.) to be in opposition, each of the terms must be simply EXTERNAL to the other, which means that one of these oppositions (the opposition between inside and outside) must already be accredited as the matrix of all possible opposition.' "

Peter J. Leithart

See also


Skewed Mirrors
,
Sept. 14, 2003

"Evil did not  have the last word."
Richard John Neuhaus, April 4, 2005

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

PARIS,
1922-1939

"There is never any ending to Paris."
— Ernest Hemingway

For the first word, see Louis Armand on
Lethe, erinnerung, and riverrun.

See also the following passage,
linked to on the Easter Vigil, 2005:

  You will find to the left of the House of Hades
    a spring,
  And by the side thereof standing
    a white cypress.
  To this spring approach not near.
  But you shall find another,
    from the lake of Memory
  Cold water flowing forth, and there are
    guardians before it.
  Say, "I am a child of Earth and starry Heaven;
  But my race is of Heaven alone.
    This you know yourselves.
  But I am parched with thirst and I perish.
    Give me quickly
  The cold water flowing forth
    from the lake of Memory."

Wednesday, March 30, 2005

Wednesday March 30, 2005

Filed under: General — m759 @ 1:28 pm

Logocentric Theology

Logic is all about the entertaining of possibilities.”

— Colin McGinn,
Mindsight: Image, Dream, Meaning,
Harvard U. Press (See yesterday’s entry.)

“God is the sum of all possibilities.”

— Isaac Bashevis Singer, according to
the Associated Press “Today in History
feature for today, March 30, 2005

“A probability space is a measure space with total measure one.”

Gregory F. Lawler, Probability Notes

“Hear, O Israel: The Lord our God, the Lord is one.”

Deuteronomy 6:4

For other illustrations of logocentric theology, see

Matrix of the Death God (May 25, 2003),

Transcendental Meditation (July 30, 2003),

and, for Warren Beatty’s birthday today,

Graphical Password
(April 27, 2003).

Thursday, March 3, 2005

Thursday March 3, 2005

Filed under: General — Tags: — m759 @ 3:26 pm
Necessity, Possibility, Symmetry

The image “http://www.log24.com/log/pix05/050303-Symmetry.gif” cannot be displayed, because it contains errors.


Matrix group actions,
March 26, 1985

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

Keith Allen Korcz,
(Log24.net, 1/25/05)

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

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Thursday, October 7, 2004

Thursday October 7, 2004

Filed under: General — m759 @ 6:30 pm

Absolute Full Spin Mode

In today’s news:

Kerry… said Bush and Cheney
were in ‘absolute full spin mode.'”

Today’s midday
California lottery number:

525.
Related material–
5/25, 2003:

Star Wars and
Matrix of the Death God

For similar theological remarks,
see yesterday’s
Spin the Numbers.

Friday, September 17, 2004

Friday September 17, 2004

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

3:57:09…
Time is a Weapon

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

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

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

— Jesse Baumann,
    The Matrix: The Magic Bullet 

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

Friday, August 01, 2003:

Fearful Meditation 

Ray Price - Time

TIME, Aug. 4, 2003

Ray Price — Time

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

— Lisa Berg, NashvilleCountry.com

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

— Shakespeare, Sonnet 65

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

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

— Hermann Hesse, The Glass Bead Game

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

— Cole Porter, “Every Time We Say Goodbye

Tuesday, August 17, 2004

Tuesday August 17, 2004

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

The Zen of Abraham

Today’s Zen Chautauqua, prompted by the fact that this is Abrahamic week at the real Chautauqua, consists of links to

The Matrix of Abraham,

Matrix of the Death God, and

Happy Birthday, Kate and Kevin.

The real Chautauqua’s program this week is, of course, Christian rather than Zen.  Its theme is “Building a Global Neighborhood: The Abrahamic Vision 2004.”  One of the featured performers is Loretta Lynn; in her honor (and, of course, that of Sissy Spacek), I will try to overcome the fear and loathing that the Semitic (i. e., “Abrahamic”) religions usually inspire in me.

To a mathematician, the phrase “global neighborhood” sounds like meaningless politico-religious bullshit —  a phrase I am sure accurately characterizes most of the discourse at Chautauqua this week.  But a Google search reveals an area of research — “particle swarm optimization” in which the phrase “global neighborhood” actually means something.  See

A Hybrid Particle Swarm
and Neural Network Approach
for Reactive Power Control,
by Paulo F. Ribeiro and
W. Kyle Schlansker
(pdf).

This article includes the following:

Given the sophistication of his writing, I am surprised at Schlansker’s Christian background:

A good omen for the future is the fact that Schlansker balances the looney Semitic (or “Abrahamic”) teachings of Christianity with good sound Aryan religion, in the form of the goddess Themis.

 Themis, often depicted as “Justice”

For those who must have an Abraham, Schlansker’s paper includes the following:

A Themis figure I prefer to the above:

For more on religious justice
at midnight in the garden of
good and evil, see the Log24
entries of Oct. 1-15, 2002.

For material on Aryan religion that is far superior to the damned nonsense at Chautauqua, New York, this week, see

Jane Ellen Harrison’s Themis: a Study of the Social Origins of Greek Religion, with an excursus on the ritual forms preserved in Greek tragedy by Gilbert Murray and a chapter on the origin of the Olympic games by F. M. Cornford.  Rev. 2nd ed., Cambridge, Cambridge U.P., 1927.

Those who prefer the modern religion of Scientism will of course believe that Themis is purely imaginary, and that truth is to be found in modern myths like that of Carl Sagan’s novel Contact, illustrated below.

Jodie Foster (an admirer of
Leni Riefenstahl) and the
opening of the 1936 Olympics

“Heraclitus…. says: ‘The ruler whose prophecy occurs at Delphi oute legei oute kryptei, neither gathers nor hides, alla semainei, but gives hints.'”
An Introduction to Metaphysics, by Martin Heidegger, Yale University Press paperback, 1959, p. 170

“The lord whose oracle is in Delphi neither indicates clearly nor conceals, but gives a sign.”
Adolf Holl, The Left Hand of God, Doubleday, 1998, p. 50

Monday, July 26, 2004

Monday July 26, 2004

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

Happy Birthday

to Kate Beckinsale
(star of Cold Comfort Farm)

and Kevin Spacey
(star of The Usual Suspects).

From a novel,
The Footprints of God,
published August 12, 2003

A tour guide describes
stations of the cross in Jerusalem:

"Ibrahim pointed down the cobbled street to a half circle of bricks set in the street.  'There is where Jesus began to carry the cross.  Down the street is the Chapel of Flagellation, where the Roman soldiers whipped Jesus, set on him a crown of thorns, and said, "Hail, King of the Jews!" Then Pilate led him to the crowd and cried, "Ecce homo!  Behold the man!" '

Ibrahim delivered this information with the excitement of a man reading bingo numbers in a nursing home."

In keeping with this spirit of religious fervor and with the spirit of Carl Jung, expositor of the religious significance of the mandala,

Behold —

The Mandala of Abraham

For the religious significance of this mandala,
see an entry of May 25, 2003:

Matrix of the Death God.

Wednesday, July 7, 2004

Wednesday July 7, 2004

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

Beyond Geometry

(Title of current L. A. art exhibit)

John Baez:

What is the difference between topology and geometry?

Geometry you learn in high school; topology in college. So, topology costs more.

A bit more seriously….

“The greatest obstacle to discovery
is not ignorance —
  it is the illusion of knowledge.”

— Daniel J. Boorstin,
American historian, educator, writer.
Source: The Washington Post,
“The Six O’Clock Scholar,”
by Carol Krucoff (29 Jan. 1984)

For the illusion of knowledge,
see (for instance)
The Importance of Being Nothingness,
by Craig J. Hogan
(American Scientist, Sept.-Oct. 2001).

A bit more seriously…

“These cases are
neither harmless nor amusing.”
— Craig J. Hogan, op. cit.

For example:

“Thanks to Dr. Matrix
for honouring this website
with the Award for Science Excellence
on May 14, 2002 and selecting it
for prominent display in the categories
of Mathematics and Creative Minds.”

See also my notes
On Dharwadker’s Attempted Proof,
November 28, 2000, and
The God-Shaped Hole,
February 21, 2001.

Friday, April 30, 2004

Friday April 30, 2004

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

Notes

  

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

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

Michael Bryson 

From a Beethoven’s Birthday entry:

Moulin Bleu

  

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

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

Song for the
Unification of Europe
(Blue 1)

From today’s news:

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

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

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

Tuesday, December 16, 2003

Tuesday December 16, 2003

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

Moulin Bleu

  

Kaleidoscope turning…
Shifting pattern
within unalterable structure…

— Roger Zelazny, Eye of Cat   

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

Song for the
Unification of Europe
(Blue 1)

Friday, December 12, 2003

Friday December 12, 2003

Filed under: General — m759 @ 4:07 pm

For Sinatra’s Birthday

Sinatra made as good use as anyone in the past century of the 12-note tempered scale, so the above seems a reasonably apt tribute on this, his 12/12 birthday.

Friday, November 7, 2003

Friday November 7, 2003

Filed under: General — m759 @ 1:28 pm

Today in History:
The Comeback Kid

(Courtesy of Dr. Mac’s Cultural Calendar)

On this date:

In 1962, having lost the California governor’s race, Richard Nixon said to the press, “You won’t have Nixon to kick around any more.”

In 1972, Republican incumbent President Richard Nixon was re-elected, defeating Democratic candidate George McGovern, 520 electoral votes to 17.

From the archives of singer/songwriter Shannon Campbell (“voice of an angel, mouth of a truckdriver”)–

Feb. 6, 2002: The Essential Matrix

NEO: (whines) Who am I?

TRINITY: You are The One.

EVERYONE ELSE: Eh, he might be The One.

TRINITY: He is The One.

NEO: I am not The One.

TRINITY: You are The One.

THE ORACLE: You are not The One, but you can’t tell anybody.

NEO: (whines) But I wanted to be The One. I want to go home….

TRINITY: Fuck. He’s not The One.

EVERYONE ELSE: Told you so.

MORPHEUS: Sure wish someone was The One. I’m in deep shit.

Thursday, November 6, 2003

Thursday November 6, 2003

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

Legacy Codes:

The Most Violent Poem

Lore of the Manhattan Project:

From The Trinity Site

“I imagined Oppenheimer saying aloud,
‘Batter my heart, three person’d God,”
unexpectedly recalling John Donne’s ‘Holy Sonnet [14],’
and then he knew, ‘ “Trinity” will do.’
Memory has its reasons.

‘Batter my heart’ — I remember these words.
I first heard them on a fall day at Duke University in 1963.
Inside a classroom twelve of us were
seated around a long seminar table
listening to Reynolds Price recite this holy sonnet….

I remember Reynolds saying, slowly, carefully,
‘This is the most violent poem in the English language.’ ”

Related Entertainment

Today’s birthday:
director Mike Nichols

From a dead Righteous Brother:

“If you believe in forever
Then life is just a one-night stand.”

Bobby Hatfield, found dead
in his hotel room at
7 PM EST Wednesday, Nov. 5, 2003,
before a concert scheduled at
Western Michigan University, Kalamazoo
.

From a review of The Matrix Revolutions:

“You’d have to be totally blind at the end
to miss the Christian symbolism….
Trinity gets a glimpse of heaven…. And in the end…
God Put A Rainbow In The Clouds.”

Moral of the
Entertainment:

According to Chu Hsi [Zhu Xi],

“Li” is
“the principle or coherence
or order or pattern
underlying the cosmos.”

— Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990

Related Non-Entertainment

Symmetry and a Trinity
(for the dotting-the-eye symbol above)

Introduction to Harmonic Analysis
(for musical and historical background)

Mathematical Proofs
(for the spirit of Western Michigan
University, Kalamazoo)

Moral of the
Non-Entertainment:

“Many kinds of entity
become easier to handle
by decomposing them into
components belonging to spaces
invariant under specified symmetries.”

The importance of
mathematical conceptualisation

by David Corfield,
Department of History and
Philosophy of Science,
University of Cambridge

See, too,
Symmetry of Walsh Functions and
Geometry of the I Ching.

Wednesday, November 5, 2003

Wednesday November 5, 2003

Filed under: General — Tags: , — m759 @ 2:23 pm

Game Over

 "Everything that has a beginning
     has an end."

— The Matrix Revolutions

Matrix, by Knots, Inc., 1979.

"Easy to master — A lifetime to enjoy!"

The object for 2 players (8-adult)
is to be the first to form a line
consisting of 4 different
colored chips.

Imagist Poem

Digital 'tears in the rain'

Image suggesting the 'Go chip' in 'Wild Palms'

(Recall the Go-chip
in Wild Palms.)

Wednesday, July 30, 2003

Wednesday July 30, 2003

Filed under: General — m759 @ 2:45 am

Transcendental Meditation

This week’s
 New Yorker
:

Transcendental Man
New books on
Ralph Waldo Emerson
for his bicentennial.
by John Updike

This week’s
 Time cover
:

The bicentennial of Ralph Waldo Emerson was on May 25, 2003.  For a commemoration of Emerson on that date, click on the picture below of Harvard University’s Room 305, Emerson Hall.

 

This will lead you to a discussion of the properties of a 5×5 array, or matrix, with a symbol of mystical unity at its center.  Although this symbol of mystical unity, the number “1,” is not, pace the Shema, a transcendental number, the matrix is, as perhaps a sort of Emersonian compensation, what postmodernists would call phallologocentric.  It is possible that Emerson is a saint; if so, his feast day (i.e., date of death), April 27, might reveal to us the sort of miraculous fact hoped for by Fritz Leiber in my previous entry.  A check of my April 27 notes shows us, lo and behold, another phallologocentric 5×5 array, this one starring Warren Beatty.  This rather peculiar coincidence is, perhaps, the sort of miracle appropriate to a saint who is, as this week’s politically correct New Yorker calls him, a Big Dead White Male.

 Leiber’s fiction furnishes “a behind-the-scenes view of the time change wars.”

“It’s quarter to three…” — St. Frank Sinatra

Tuesday, July 1, 2003

Tuesday July 1, 2003

Filed under: General — Tags: — m759 @ 5:37 pm

Jew’s on First

This entry is dedicated to those worshippers of Allah who have at one time or another cried
Itbah al-Yahud!” … Kill the Jew!
(See June 29 entries).

Dead at 78

Comedian Buddy Hackett died on Tuesday, July First, 2003, according to the New York Times.  According to Bloomberg.com, he died Sunday or Monday.

Associated Press

Buddy Hackett,
on the set of
“It’s a Mad, Mad,
Mad, Mad
World”
in 1962.

Whatever.  We may imagine he has now walked, leading a parade of many other stand-up saints, into a bar.


Hepburn at Chaillot

MIDRASH
for Buddy Hackett

From my May 25 entry,

Matrix of the Death God:

R. M. Abraham’s Diversions and Pastimes, published by Constable and Company, London, in 1933, has the following magic square:

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web include:

On Linguistic Creation

Picasso’s Birthday

1991 Yearbook
Rolling Stone



Hackett

In the Picasso’s Birthday version, 22 of the 25 magic square cells are correlated with pictures on the “Class of ’91” cover of Rolling Stone magazine.  Number 7 is Rod Stewart.  In accordance with the theological rhyme “Seven is heaven, eight is a gate,” our site music for today is “Forever Young,” a tune made famous by Stewart.

Roderick, actually   the name of the hero in “Madwoman of Chaillot”

Thursday, June 26, 2003

Thursday June 26, 2003

Filed under: General — Tags: , — m759 @ 5:00 am

ART WARS:
Art at the Vanishing Point

From the web page Art Wars:

"For more on the 'vanishing point,'
or 'point at infinity,' see
Midsummer Eve's Dream."

On Midsummer Eve, June 23, 2003, minimalist artist Fred Sandback killed himself.

Sandback is discussed in The Dia Generation, an April 6, 2003, New York Times Magazine article that is itself discussed at the Art Wars page.

Sandback, who majored in philosophy at Yale, once said that

"Fact and illusion are equivalents."

Two other references that may be relevant:

The Medium is
the Rear View Mirror
,

which deals with McLuhan's book Through the Vanishing Point, and a work I cited on Midsummer Eve  …

Chapter 5 of Through the Looking Glass:

" 'What is it you want to buy?' the Sheep said at last, looking up for a moment from her knitting.

'I don't quite know yet,' Alice said very gently.  'I should like to look all round me first, if I might.'

'You may look in front of you, and on both sides, if you like,' said the Sheep; 'but you ca'n't look all round you — unless you've got eyes at the back of your head.'

But these, as it happened, Alice had not got: so she contented herself with turning round, looking at the shelves as she came to them.

The shop seemed to be full of all manner of curious things — but the oddest part of it all was that, whenever she looked hard at any shelf, to make out exactly what it had on it, that particular shelf was always quite, empty, though the others round it were crowded as full as they could hold.

'Things flow about so here!' she said at last in a plaintive tone…."

 "When Alice went
     through the vanishing point
"
 

Tuesday, June 17, 2003

Tuesday June 17, 2003

Filed under: General — Tags: , — m759 @ 5:20 pm

Claves Regni Caelorum

On actor Gregory Peck, who died Thursday, June 12, 2003:

"He had early success in 'The Keys of the Kingdom,' in which he played a priest."

As Peck noted in a videotape played at his memorial service June 16,

"As a professional," he added, "I think I'd like to be thought of as a good storyteller; that's what's always interested me."

June 16, besides being the day of Peck's memorial, was also Bloomsday.  My entry for 1 PM on Bloomsday, a day celebrating the Ulysses of James Joyce, consists of the three words "Hickory, Dickory, Dock."  A comment on that entry:

"I prefer the Wake."

The following, from the Discordian Scriptures, provides a connection between the Bloomsday mouse and the Wake of patriarch Gregory Peck.

Hickory Dickory Dock

Hickory, dickory, dock!

Here we are on higher ground at once. The clock symbolizes the spinal column, or if you prefer it, Time, chosen as one of the conditions of normal consciousness. The mouse is the Ego; "Mus", a mouse, being only "Sum", "I am", spelt Qabalistically backwards.  This Ego or Prana or Kundalini force being driven up the spine, the clock strikes one, that is, the duality of consciousness is abolished. And the force again subsides to its original level. "Hickory, dickory, dock!" is perhaps the mantra which was used by the adept who constructed this rime, thereby hoping to fix it in the minds of men; so that they might attain to Samadhi by the same method. Others attribute to it a more profound significance — which is impossible to go into at this moment, for we must turn to:
 
Humpty Dumpty sat on a wall….

The Bloom of Ulysses has a certain philosophical kinship with Yale literary critic Harold Bloom.  For material related to the latter Bloom's study of Gnosticism, see Chaos Matrix.  For the conflict between Gnostic and Petrine approaches to religion, see Poul Anderson's Operation Chaos.

From an account of Peck's memorial service:

"Mourners included… Piper Laurie…."

OK, he's in.

 

Sunday, June 15, 2003

Sunday June 15, 2003

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

Readings for Trinity Sunday

  1. Triune knot:
    Problems in Combinatorial Group Theory, 7 and 8, in light of the remark in Section 8.3 of Lattice Polygons and the Number 12 
  2. Cardinal Newman:
    Sermon 24
  3. Simon Nickerson:
    24=8×3.

For more on the structure
discussed by Nickerson, see

Raiders of the Lost Matrix:

For theology in general, see

Jews Telling Stories.

Confession in 'The Seventh Seal'

Monday, May 26, 2003

Monday May 26, 2003

Filed under: General,Geometry — m759 @ 4:25 am

Mental Health Month, Day 26:

Many Dimensions,
Part II
— The Blue Matrix 

But seriously…

John Baez in July 1999:

"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."

Well, maybe.  I certainly hope so.  If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light.  Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory.  See the following from my journal:

Evariste Galois and the Rock that Changed Things, and

A Time to Gather Stones Together: Readings for Yom Kippur.

See, too, this web page on Marion Zimmer Bradley's fictional

Matrices, or Blue Star-Stones, and

the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.
 

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — Tags: — m759 @ 7:11 pm

ART WARS

Mental Health Month, Day 25:

Matrix of the Death God

Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham.  (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.)  For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.

 

See, too, The Matrix of Abraham, illustrated below.  This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web:

On Linguistic Creation

Picasso’s Birthday

A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).

This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.

Saturday, May 24, 2003

Saturday May 24, 2003

Filed under: General — m759 @ 1:06 am

Mental Health Month, Day 24:

The Sacred Day of
Kali, the Dark Lady

On this day, Gypsies from all over Europe gather in Provence for the sacred day of St. Sarah, also known as Kali.

Various representations of Kali exist; there is a novel about the ways men have pictured her:

From the prologue to
The Dark Lady,

 by Mike Resnick.

She was old when the earth was young.

She stood atop Cemetery Ridge when Pickett made his charge, and she was there when the six hundred rode into the Valley of Death.  She was at Pompeii when Mount Vesuvius blew, and she was in the forests of Siberia when the comet hit.  She hunted elephant with Selous and buffalo with Cody, and she was there the night the high wire broke beneath the Flying Wallendas.  She was at the fall of Troy and the Little Bighorn, and she watched Manolete and Dominguez face the brave bulls in the bloodstained arenas of Madrid….

She has no name, no past, no present, no future.  She wears only black, and though she has been seen by many men, she is known to only a handful of them.  You’ll see her — if you see her at all — just after you’ve taken your last breath.  Then, before you exhale for the final time, she’ll appear, silent and sad-eyed, and beckon to you.

She is the Dark Lady, and this is her story.

The above is one of the best descriptions of Kali I know of in literature; another is in a short story by Fritz Leiber, “Damnation Morning.”   It is not coincidental that one collection of Leiber’s writings is called “Dark Ladies.”

My journal note “Biblical Proportions” was in part inspired by Leiber.

Frank Sinatra may have pictured her as Ava Gardner.  I think I saw her the night Sinatra died… hence my entries of March 31 and April 2, 2003. 

It is perhaps not irrelevant that Kali is, among other things, a mother goddess, and that my entry “Raiders of the Lost Matrix” of May 20 deals with this concept and with the number 24.

The above religious symbol (see “Damnation Morning“) pictures both the axes of symmetry of the square¹ and a pattern with intriguing combinatorial properties².  It also is the basis of a puzzle³ I purchased on August 29, 1997 — Judgment Day in Terminator 2.  Linda Hamilton as Sarah Connor in that film is an excellent representation of the Dark Lady, both as mother figure and as Death Goddess.

 
Sarah Connor

Background music: “Bit by bit…” — Stephen Sondheim… See Sondheim and the Judgment Day puzzle in my entry of May 20. The Lottery Covenant.

¹ A. W. Joshi, Elements of Group Theory for Physicists, Third Edition, Wiley, 1982, p. 5

² V. K. Balakrishnan, Combinatorics, McGraw-Hill, 1995, p. 180

³ The Izzi Puzzle

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