Monday, April 20, 2020

Wheel Turnin’ ’Round and ’Round

Filed under: General — m759 @ 11:54 PM

Friday, January 24, 2020

Wheel Turnin’ ’Round and ’Round

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


Monday, August 20, 2018

A Wheel for Ellmann

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

The title was suggested by Ellmann's roulette-wheel analogy
in the previous post, "The Perception of Coincidence."

I Ching hexagrams as a Singer 63-cycle, plus zero

Sunday, December 17, 2017

Wheelwright and the Wheel

Filed under: General — m759 @ 9:00 PM

Wheelwright on 'the still point' at the center of a turning wheel, in 'The Burning Fountain'

From the 1968 "new and revised edition" —

See also the previous post.

For the phrase "burning fountain," see Shelley's "Adonais,"
as well as Logos (a post of Dec. 4) and The Crimson Abyss.

Saturday, April 25, 2020

The Grandfather Clock

Filed under: General — m759 @ 5:16 PM

“Well, I’ve helped to wind up the clock —
I might as well hear it strike!”

— Said to be a quotation from the grandfather
of a “pirate radio” founder who reportedly died
at 79 on April 20.

See as well this journal on the night of April 20 —

Wheel Turnin’ ’Round and ’Round

and a search for “Wheel of Time.”

Wednesday, March 4, 2020

Language Game

Filed under: General — Tags: — m759 @ 10:58 PM

"Wheel Turnin’ ’Round and ’Round" 

Steely Dan

Thursday, February 20, 2020

Once Upon a Time in Las Vegas

Filed under: General — Tags: — m759 @ 4:25 PM

Wheel Turnin’ ’Round and ’Round

Monday, August 20, 2018

The Perception of Coincidence

Filed under: General — Tags: — m759 @ 2:15 AM

Ellmann on Joyce and 'the perception of coincidence' —

"Samuel Beckett has remarked that to Joyce reality was a paradigm,
an illustration of a possibly unstatable rule. Yet perhaps the rule
can be surmised. It is not a perception of order or of love; more humble
than either of these, it is the perception of coincidence. According to
this rule, reality, no matter how much we try to manipulate it, can only
assume certain forms; the roulette wheel brings up the same numbers
again and again; everyone and everything shift about in continual
movement, yet movement limited in its possibilities."

— Richard Ellmann, James Joyce , rev. ed.. Oxford, 1982, p. 551

Sunday, August 19, 2018

Possible Permutations

Filed under: General,Geometry — Tags: — m759 @ 9:18 PM

John Calder, an independent British publisher who built a prestigious list
of authors like Samuel Beckett and Heinrich Böll and spiritedly defended
writers like Henry Miller against censorship, died on Aug. 13 in Edinburgh.
He was 91.

— Richard Sandomir in the online New York Times  this evening

On Beckett —


Also on August 13th


Thursday, October 12, 2017

East Meets West

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:09 PM

Wednesday, May 10, 2017

In the Park with Yin and Yang

Filed under: General,Geometry — Tags: , — m759 @ 10:35 PM

In memory of an art dealer who 
reportedly died on Sunday, May 7—

Decorations for a Cartoon Graveyard

Saturday, April 30, 2016

A Lucifer for Walpurgisnacht

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

A politician as 'Lucifer in the flesh'

A more impressive Lucifer —

The late theoretical physicist John Archibald Wheeler,
author of the phrase "it from bit."

Related material —

"The Thing and I" (April 17, 2016) and an essay by
Julian Barbour, "Bit from It."

Thursday, April 14, 2016

Strange Awards

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

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

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

Compare and contrast —

A circular I Ching

I of course prefer the Galois I Ching .

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

Wednesday, April 13, 2016

Black List

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

A search for "Max Black" in this journal yields some images
from a post of August 30, 2006 . . .

A circular I Ching

The image “http://www.log24.com/log/pix06A/060830-SeventhSymbol.jpg” cannot be displayed, because it contains errors.

"Jackson has identified the seventh symbol."
— Stargate

The "Jackson" above is played by the young James Spader,
who in an older version currently stars in "The Blacklist."

"… the memorable models of science are 'speculative instruments,'
to borrow I. A. Richards' happy title. They, too, bring about a wedding
of disparate subjects, by a distinctive operation of transfer of the
implications  of relatively well-organized cognitive fields. And as with
other weddings, their outcomes are unpredictable."

Max Black in Models and Metaphors , Cornell U. Press, 1962

Wednesday, February 18, 2015

Noon on Ash Wednesday

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

Text and Context.

Thursday, February 12, 2015

Dead Reckoning

Filed under: General,Geometry — Tags: — m759 @ 12:48 PM

Continued from yesterday.

The passage on Claude Chevalley quoted here
yesterday in the post Dead Reckoning was, it turns out,
also quoted by Peter Galison in his essay "Structure of Crystal,
Bucket of Dust" in Circles Disturbed: The Interplay of 
Mathematics and Narrative  
(Princeton University Press, 2012,
ed. by Apostolos Doxiadis and Barry Mazur).

Galison gives a reference to his source:

"From 'Claude Chevalley Described by His Daughter (1988),' 
in Michèle Chouchan, Nicolas Bourbaki: Faits et légendes
(Paris: Éditions du Choix, 1995), 36–40, translated and cited
in Marjorie Senechal, 'The Continuing Silence of Bourbaki:
An Interview with Pierre Cartier, June 18, 1997,' 
Mathematical Intelligencer  1 (1998): 22–28."

Galison's essay compares Chevalley with the physicist
John Archibald Wheeler. His final paragraph —

"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginning-end of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.'"

See also posts tagged Wheeler.

Friday, February 8, 2013


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

A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields 
the following.

Wightman at Wikipedia:
"His graduate students include
 Arthur Jaffe,  Jerrold Marsden, and Alan Sokal."

"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."

— Arthur Jaffe in News at Princeton , Jan. 30

Marsden at Wikipedia
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)

A Wikipedia reference in the symplectic geometry article leads to

Symplectic Geometry Lies at the Very
Foundations of Physics and Mathematics

Mark J. Gotay
Department of Mathematics
University of Hawai‘i

James A. Isenberg
Institute of Theoretical Science and Department of Mathematics
University of Oregon

February 18, 1992


We would like to thank Jerry Marsden and Alan Weinstein
for their comments on previous drafts.

Published in: Gazette des Mathématiciens  54, 59-79 (1992).


"Physics is geometry .  This dictum is one of the guiding
principles of modern physics. It largely originated with
Albert Einstein…."

A different account of the dictum:

The strange term Geometrodynamics 
is apparently due to Wheeler.

Physics may or may not be geometry, but
geometry is definitely not physics.

For some pure geometry that has no apparent 
connection to physics, see this journal
on the date of Wightman's death.

Wednesday, November 28, 2012


Filed under: General — Tags: , — m759 @ 2:00 PM

From a New York Times  weblog last night—

The Reconstruction of Rome

The New York Times , Nov. 27, 2012, 9:00 PM

Logic Pro software enables us to layer complex
technolike tracks and simulate meta, sampled orchestras—
fake orchestras that have on more than one occasion
fooled a jury of the most discriminating composers
into thinking it was the real thing. …

Several decades of cultural relativism has helped to
hasten the decline of the dominance of Western canon….

This next generation is becoming adept at taking small
bits of information, unformed, and assembling it
into larger asynchronous maps, of nonlinear order. 

IT from BITS*

These failures of number agreement—
orchestras… it,  decades… has,  bits… it —
suggest a look at synesis.

Synesis is a traditional grammatical/rhetorical term
derived from Greek σύνεσις (originally meaning "unification,
meeting, sense, conscience, insight, realization, mind, reason").
A constructio kata synesin  (or constructio ad sensum  in Latin)
means a grammatical construction in which a word takes
the gender or number not of the word with which it should
regularly agree, but of some other word implied in that word.
It is effectively an agreement of words with the sense,
instead of the morphosyntactic form. Example:
"If the band are popular, they will play next month." —Wikipedia

The conclusion of Wikipedia's synesis article is of particular interest:

See also…. Elohim , a Hebrew word whose number varies.

* A nod to the late John Archibald Wheeler

Saturday, June 16, 2012

Chiral Problem

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

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post


Detail from Knowles's obituary in this  morning's New York Times

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"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

From Pilate Goes to Kindergarten

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube

Problem: Is there a different coordinatization
 that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

"Chiral Tetrahedrons as Unitary Quaternions"—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

Friday, February 17, 2012

Pregeometry and Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 7:35 PM

Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .

From a paper on that topic—

"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry  (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."

— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science  2 (1): 137-176

Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.

Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.

Some related material from pure mathematics—


Click image for further details.

Friday, February 11, 2011

Brightness at Noon (continued)

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

From The Seventh Symbol

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

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

Monday, July 27, 2009

Monday July 27, 2009

Filed under: General,Geometry — Tags: — m759 @ 2:29 PM
Field Dance

The New York Times
on June 17, 2007:

 Design Meets Dance,
and Rules Are Broken

Yesterday's evening entry was
on the fictional sins of a fictional
mathematician and also (via a link
to St. Augustine's Day, 2006), on
the geometry of the I Ching* —

The eternal
combined with
the temporal:

Circular arrangement of I Ching hexagrams based on Singer 63-cycle in the Galois field GF(64)

The fictional mathematician's
name, noted here (with the Augustine-
I Ching link as a gloss) in yesterday's
evening entry, was Summerfield.

From the above Times article–
"Summerspace," a work by
 choreographer Merce Cunningham
and artist Robert Rauschenberg
that offers a competing
 vision of summer:

Summerspace — Set by Rauschenberg, choreography by Cunningham

Cunningham died last night.

John Cage, Merce Cunningham, Robert Rauschenberg in the 1960's

From left, composer John Cage,
choreographer Merce Cunningham,
and artist Robert Rauschenberg
in the 1960's

"When shall we three meet again?"

* Update of ca. 5:30 PM 7/27– today's online New York Times (with added links)– "The I Ching is the 'Book of Changes,' and Mr. Cunningham's choreography became an expression of the nature of change itself. He presented successive images without narrative sequence or psychological causation, and the audience was allowed to watch dance as one might watch successive events in a landscape or on a street corner."

Saturday, December 20, 2008

Saturday December 20, 2008

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

Cheap* Epiphanieshttp://www.log24.com/images/asterisk8.gif
for the Church of
the Forbidden Planet

Mid-day lotteries Dec. 19:
* NY 198  http://www.log24.com/images/asterisk8.gif PA  918

From 9/18:

O the mind, mind has mountains,
   cliffs of fall
Frightful, sheer, no-man fathomed.
   Hold them cheap
May who ne'er hung there.


Evening lotteries Dec. 19:
* NY 198  http://www.log24.com/images/asterisk8.gif PA 414

From 4/14:

"minds blazing, to the barricades"

The New York Times
    on the Wheeler effect

See also
Bloomsday for Nash:
The Revelation Game —

      Black disc from end of Ch. 17 of Ulysses

For details,
click on the
black hole.

Thursday, December 18, 2008

Thursday December 18, 2008

Filed under: General — Tags: — m759 @ 1:00 PM
Polar Opposites

Susan Sontag in
this week's New Yorker:
"The mind is a whore."

Embedded in the Sontag
article is the following:

The New Yorker on Santa's use of the word 'ho'

I Ching hexagrams as a Singer 63-cycle, plus zero

Act One

South Pole:

David Mamet's book 'A Whore's Profession'

Hexagram 21 in the King Wen sequence

Shi Ho

Act Two

North Pole:

Susan Sontag

Hexagram 2 in the King Wen sequence


"If baby I'm the bottom,
you're the top."
Cole Porter   

Happy birthday,
Steven Spielberg.

Tuesday, December 16, 2008

Tuesday December 16, 2008

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

From The n-Category Cafe today:

David Corfield at 2:33 PM UTC quoting a chapter from a projected second volume of a biography:

"Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty… was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation."

John Baez at 7:14 PM UTC on research:

"I just don’t want to reinvent a wheel, or waste my time inventing a square one."

For the adoption and embracing of such a problematic phenomenon, see The Square Wheel (this journal, Sept. 14, 2004).

For a connection of the square wheel with yesterday's entry for Julie Taymor's birthday, see a note from 2002:

Wolfram's Theory of Everything
and the Gameplayers of Zan

Related pictures–

From Wolfram:


A Square

From me:


A Wheel

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General — Tags: , — m759 @ 8:28 AM
"Caught up 
    in circles…"

— Song lyric,  
Cyndi Lauper


Alethiometer from
"The Golden Compass"


The 64 hexagrams of the I Ching in a circular arrangement suggested by a Singer 63-cycle

The I Ching
as Alethiometer


See also this morning's
later entry, illustrating
the next line of Cyndi
Lauper's classic lyric
"Time After Time" —

"… Confusion is    
  nothing new."

Wednesday, April 16, 2008

Wednesday April 16, 2008

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

Poetry for Physicists:
The Gates of Hell

From the obituary of physicist John Archibald Wheeler at Princeton:

In the fall of 1967, he was invited to give a talk…. As he spoke, he… [mentioned] something strange… what he called a gravitationally completely collapsed object. But such a phrase was a mouthful, he said, wishing aloud for a better name. "How about black hole?" someone shouted from the audience.

That was it. "I had been searching for just the right term for months, mulling it over in bed, in the bathtub, in my car, wherever I had quiet moments," he later said. "Suddenly this name seemed exactly right." He kept using the term, in lectures and on papers, and it stuck.

From Log24 last year on this date ("Happy Birthday, Benedict XVI"):

"Know the one about the Demiurge and the Abridgment of Hope?"

— Robert Stone, A Flag for Sunrise, Knopf, 1981, the final page, 439

From Dante, The Inferno, inscription on the gates of Hell:

"Abandon all hope, ye who enter."

From Psychoshop, an unfinished novel by Alfred Bester completed by Roger Zelazny:

His manner was all charm and grace; pure cafe society….

He purred a chuckle. "My place. If you want to come, I'll show you."

"Love to. The Luogo Nero? The Black Place?"

"That's what the locals call it. It's really Buoco Nero, the Black Hole."

"Like the Black Hole of Calcutta?"

"No. Black Hole as in astronomy. Corpse of a dead star, but also channel between this universe and its next-door neighbor."

"Here? In Rome?"

"Sure. They drift around in space until they run out of gas and come to a stop. This number happened to park here."

"How long ago?"

"No one knows," he said. "It was there six centuries before Christ, when the Etruscans took over a small town called Roma and began turning it into the capital of the world."


Related material:

Log24 on

Life of the Party
(March 24, 2006),
(March 26, 2006)

Monday, April 14, 2008

Monday April 14, 2008

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

Classical Quantum

From this morning's
New York Times:

Physicist John A. Wheeler with diagrams of classical and quantum ways to get from point A to point B

"John A. Wheeler, a visionary physicist… died Sunday morning [April 13, 2008]….

… Dr. Wheeler set the agenda for generations of theoretical physicists, using metaphor as effectively as calculus to capture the imaginations of his students and colleagues and to pose questions that would send them, minds blazing, to the barricades to confront nature….

'He rejuvenated general relativity; he made it an experimental subject and took it away from the mathematicians,' said Freeman Dyson, a theorist at the Institute for Advanced Study….

… he [Wheeler] sailed to Copenhagen to work with Bohr, the godfather of the quantum revolution, which had shaken modern science with paradoxical statements about the nature of reality.

'You can talk about people like Buddha, Jesus, Moses, Confucius, but the thing that convinced me that such people existed were the conversations with Bohr,' Dr. Wheeler said….

… Dr. Wheeler was swept up in the Manhattan Project to build an atomic bomb. To his lasting regret, the bomb was not ready in time to change the course of the war in Europe….

Dr. Wheeler continued to do government work after the war, interrupting his research to help develop the hydrogen bomb, promote the building of fallout shelters and support the Vietnam War….

… Dr. Wheeler wondered if this quantum uncertainty somehow applied to the universe and its whole history, whether it was the key to understanding why anything exists at all.

'We are no longer satisfied with insights only into particles, or fields of force, or geometry, or even space and time,' Dr. Wheeler wrote in 1981. 'Today we demand of physics some understanding of existence itself.'

At a 90th birthday celebration in 2003, Dr. Dyson said that Dr. Wheeler was part prosaic calculator, a 'master craftsman,' who decoded nuclear fission, and part poet. 'The poetic Wheeler is a prophet,' he said, 'standing like Moses on the top of Mount Pisgah, looking out over the promised land that his people will one day inherit.'"

Dennis Overbye, The New York Times,
    Monday, April 14, 2008

As prophets go, I prefer
 the poet Wallace Stevens:

"point A / In a perspective
that begins again / At B"

— Wallace Stevens,
"The Rock"

Saturday, February 23, 2008

Saturday February 23, 2008

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

"An acute study of the links
between word and fact"
Nina daVinci Nichols

Thanks to a Virginia reader for a reminder:
Virginia /391062427/item.html? 2/22/2008 7:37 PM
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.


The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure–


The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Thursday, February 1, 2007

Thursday February 1, 2007

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

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

The above is from
Feb. 15, 2006.

"I don't believe in an afterlife, so I think this is it, and I'm trying to spend my time as best I can, and I'm trying to spend my time so I'm proud of what I've done, and I try not to do any things that I'm not proud of."

Jim Gray, 2002 interview (pdf)

Commencement Address (doc)
to Computer Science Division,
College of Letters and Science,
University of California, Berkeley,
by Jim Gray,
May 25, 2003:

"I was part of Berkeley's class of 1965. Things have changed a lot since then….

So, what's that got to do with you? Well, there is going to be MORE change…. Indeed, change is accelerating– Vernor Vinge suggests we are approaching singularities when social, scientific and economic change are so rapid that we cannot imagine what will happen next.  These futurists predict humanity will become post-human. Now, THAT! is change– a lot more than I have seen.

If it happens, the singularity will happen in your lifetime– and indeed, you are likely to make it happen."


I Ching, Hexagram 39

For other singular
sci-fi tales, click on
the above hexagram.

More from Gray's speech:

"I am an optimist. Science is a Faustian bargain– and I am betting on mankind muddling through. I grew up under the threat of atomic war; we've avoided that so far. Information Technology is a Faustian bargain. I am optimistic that we can have the good parts and protect ourselves from the worst part– but I am counting on your help in that."

"Not fare well,
But fare forward, voyagers."

— T. S. Eliot,
"The Dry Salvages"

Wednesday, August 30, 2006

Wednesday August 30, 2006

Filed under: General,Geometry — Tags: — m759 @ 10:07 AM
The Seventh Symbol:

A Multicultural Farewell

to a winner of the
Nobel Prize for Literature,
the Egyptian author of
The Seventh Heaven:
Supernatural Stories

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06A/060830-SeventhSymbol.jpg” cannot be displayed, because it contains errors.

"Jackson has identified
the seventh symbol."

Other versions of
the seventh symbol —

Chinese version:

The image “http://www.log24.com/log/pix06A/060830-hexagram20.gif” cannot be displayed, because it contains errors.

pictorial version:

The image “http://www.log24.com/log/pix06A/060830-Box.jpg” cannot be displayed, because it contains errors.

algebraic version:

The image “http://www.log24.com/log/pix06A/060830-Algebra.jpg” cannot be displayed, because it contains errors.

"… Max Black, the Cornell philosopher, and others have pointed out how 'perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra' …."

— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25

Monday, August 28, 2006

Monday August 28, 2006

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM
Today's Sinner:

Augustine of Hippo, who is said to
have died on this date in 430 A.D.

"He is, after all, not merely taking over a Neoplatonic ontology, but he is attempting to combine it with a scriptural tradition of a rather different sort, one wherein the divine attributes most prized in the Greek tradition (e.g. necessity, immutability, and atemporal eternity) must somehow be combined with the personal attributes (e.g. will, justice, and historical purpose) of the God of Abraham, Isaac, and Jacob."

Stanford Encyclopedia of Philosophy on Augustine

Here is a rather different attempt
to combine the eternal with the temporal:


The Eternal

Symbol of necessity,
immutability, and
atemporal eternity:

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

For details, see
finite geometry of
the square and cube

The Temporal

Symbol of the
God of Abraham,
Isaac, and Jacob:

The image “http://www.log24.com/log/pix06A/060828-Cloud.jpg” cannot be displayed, because it contains errors.

For details, see
Under God
(Aug. 11, 2006)

The eternal
combined with
the temporal:


Singer 63-cycle in the Galois field GF(64) used to order the I Ching hexagrams

Related material:

Hitler's Still Point and
the previous entry.

Tuesday, July 18, 2006

Tuesday July 18, 2006

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

Sacred Order

In memory of Philip Rieff, who died on July 1, 2006:

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Related material:

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The image ?http://www.log24.com/theory/images/MySpace.jpg? cannot be displayed, because it contains errors.

For details, see the
five Log24 entries ending
on the morning of
Midsummer Day, 2006.

Thanks to University Diaries for pointing out the essay on Rieff.
That essay says Rieff had "a dense, knotty, ironic style designed to warn off impatient readers. You had to unpack his aphorisms carefully. And this took a while. As a result, his thinking had a time-release effect." Good for him.  For a related essay (time-release effect unknown), see Hitler's Still Point: A Hate Speech for Harvard.

Friday, June 23, 2006

Friday June 23, 2006

Filed under: General — Tags: — m759 @ 9:00 PM
Go with the Flow

Review of a
Feb. 15, 2006, entry:

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The image ?http://www.log24.com/images/IChing/WilhelmHellmut.gif? cannot be displayed, because it contains errors.

Wednesday, February 15, 2006

Wednesday February 15, 2006

Filed under: General,Geometry — Tags: — m759 @ 11:07 AM
Anthony Hopkins
Writes Screenplay
About God, Life & Death

These topics may be illuminated
by a study of the Chinese classics.

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

If we replace the Chinese word "I"
(change, transformation) with the
word "permutation," the relevance
of Western mathematics (which
some might call "the Logos") to
the I Ching ("Changes Classic")
beomes apparent.

Related material:

Hitler's Still Point,
Jung's Imago,
Solomon's Cube,
Geometry of the I Ching,
and Globe Award.

Yesterday's Valentine
may also have some relevance.

Monday, October 31, 2005

Monday October 31, 2005

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


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“An asymmetrical balance is sought since it possesses more movement. This is achieved by the imaginary plotting of the character upon a nine-fold square, invented by some ingenious writer of the Tang dynasty. If the square were divided in half or in four, the result would be symmetrical, but the nine-fold square permits balanced asymmetry.”– Chiang Yee, Chinese Calligraphy,
quoted in Aspen no. 10, item 8“‘Burnt Norton’ opens as a meditation on time. Many comparable and contrasting views are introduced. The lines are drenched with reminiscences of Heraclitus’ fragments on flux and movement….  the chief contrast around which Eliot constructs this poem is that between the view of time as a mere continuum, and the difficult paradoxical Christian view of how man lives both ‘in and out of time,’ how he is immersed in the flux and yet can penetrate to the eternal by apprehending timeless existence within time and above it. But even for the Christian the moments of release from the pressures of the flux are rare, though they alone redeem the sad wastage of otherwise unillumined existence. Eliot recalls one such moment of peculiar poignance, a childhood moment in the rose-garden– a symbol he has previously used, in many variants, for the birth of desire. Its implications are intricate and even ambiguous, since they raise the whole problem of how to discriminate between supernatural vision and mere illusion. Other variations here on the theme of how time is conquered are more directly apprehensible. In dwelling on the extension of time into movement, Eliot takes up an image he had used in ‘Triumphal March’: ‘at the still point of the turning world.’ This notion of ‘a mathematically pure point’ (as Philip Wheelwright has called it) seems to be Eliot’s poetic equivalent in our cosmology for Dante’s ‘unmoved Mover,’ another way of symbolising a timeless release from the ‘outer compulsions’ of the world. Still another variation is the passage on the Chinese jar in the final section. Here Eliot, in a conception comparable to Wallace Stevens’ ‘Anecdote of the Jar,’ has suggested how art conquers time:

       Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

— F. O. Matthiessen,
The Achievement of T.S. Eliot,
Oxford University Press, 1958,
as quoted in On “Burnt Norton”

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:25 PM

The Fugue

   "True joy is a profound remembering, and true grief is the same.
    Thus it was, when the dust storm that had snatched Cal up finally died, and he opened his eyes to see the Fugue spread out before him, he felt as though the few fragile moments of epiphany he'd tasted in his twenty-six years– tasted but always lost– were here redeemed and wed. He'd grasped fragments of this delight before. Heard rumour of it in the womb-dream and the dream of love; known it in lullabies. But never, until now, the whole, the thing entire.
    It would be, he idly thought, a fine time to die.
    And a finer time still to live, with so much laid out before him."

— Clive Barker,
 Book Two:
The Fugue

From Monday:

Book Three:
Out of the
Empty Quarter

"The wheels of its body rolled,
the visible mathematics
    of its essence turning on itself…."

From Friday:

The image “http://www.log24.com/log/pix05B/050806-Square.bmp” cannot be displayed, because it contains errors.

  For the meaning
of this picture, see
Geometry of the
4×4 Square.

For graphic designs
based on this geometry,
see Theme and Variations
and Diamond Theory.

For these designs in the
context of a Bach fugue,
see Timothy A. Smith's
essay (pdf) on

Fugue No. 21 in B-Flat Major
from Book II of
The Well-Tempered Clavier
by Johann Sebastian Bach.

Smith also offers a
Shockwave movie
that uses diamond theory
to illustrate this fugue.

Tuesday, August 2, 2005

Tuesday August 2, 2005

Filed under: General — Tags: — m759 @ 5:24 AM

Final Arrangements, continued


From yesterday's Log24
Clive Barker's Weaveworld:

Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.

If I stand starry-eyed
That's a danger in paradise
For mortals who stand beside
  An angel like you.

Robert Wright and George Forrest

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Monday, August 1, 2005

Monday August 1, 2005

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

    "Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
"Empty" is a misnomer…  the Rub' al-Khali contains many hidden riches.'"

Maps from the Sky,
   Saudi Aramco World, March/April 1995

From Weaveworld

Book Three:
Out of the Empty Quarter,
 by Clive Barker, 1987:

… As a child he'd learned the names of all the angels and archangels by heart: and among the mighty, Uriel was of the mightiest.  The archangel of salvation: called by some the flame of God…. What had he done, stepping into the presence of such power?  This was Uriel, of the principalities….
    Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.


"The serpent's eyes shine
As he wraps around the vine
In the Garden of Allah."

Don Henley, 1995  

Tuesday, September 14, 2004

Tuesday September 14, 2004

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

The Square Wheel

Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions.  George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel.  Harmonic analysis based on square functions– the "square wheel," as it were– is also not without its importance.

For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002).  Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986.  A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.

That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh functions– which have, in fact, properties very like those of the trig functions.  For details, see Symmetry of Walsh Functions, updated today.

"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake.  Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."

From Error-Correcting Codes
Through Sphere Packings
To Simple Groups
by Thomas M. Thompson,
Mathematical Association of America, 1983

Sunday, September 12, 2004

Sunday September 12, 2004

Filed under: General — m759 @ 1:13 PM

The Turning

A way a lone
a last a loved
a long the


Click on pictures
for further details.

For the essence and the end
Of his labor is beauty… 
one beauty, the rhythm of that Wheel

Robinson Jeffers,
“Point Pinos and Point Lobos”

Saturday, December 13, 2003

Saturday December 13, 2003

Filed under: General,Geometry — Tags: — m759 @ 2:02 PM

We Are the Key:
The Shining of December 13

For James and Lucia Joyce

In the Orbit of Genius —
TIME, Dec. 1, 2003

"Once, when her mother asked if Joyce should visit her in the sanatorium, Lucia said, 'Tell him I am a crossword puzzle, and if he does not mind seeing a crossword puzzle, he is to come out.' "

Compare and contrast
with Finnegans Wake

From Roger Zelazny's Eye of Cat:

"A massive, jaguarlike form with a single, gleaming eye landed on the vehicle's hood forward and to the front.  It was visible for but an instant, and then it sprang away. The car tipped, its air cushion awry, and it was already turning onto its side before he left the trail.  He fought with the wheel and the attitude control, already knowing that it was too late.  There came a strong shock accompanied by a crunching noise, and he felt himself thrown forward.

Kaleidoscope turning… Shifting  pattern within unalterable structure… Was it a mistake? There is pain with the power…  Time's friction at the edges…  Center loosens, forms again elsewhere…  Unalterable?  But – Turn outward.  Here songs of self erode the will till actions lie stillborn upon night's counterpane.  But – Again the movement…  Will it hold beyond a catch of moment?  To fragment…  Not kaleidoscope.  No center.  But again… To form it will.  To will it form.  Structure… Pain…  Deadly, deadly…  And lovely.  Like a sleek, small dog… A plastic statue… The notes of an organ, the first slug of gin on an empty stomach… We settle again, farther than ever before… Center. The light!… It is difficult being a god. The pain. The beauty. The terror of selfless –  Act!  Yes. Center, center, center… Here? Deadly…

necess yet again from bridge of brainbow oyotecraven stare decesis on landaway necessity timeslast the arnings ent and tided turn yet beastfall nor mindstorms neither in their canceling sarved cut the line that binds ecessity towarn and findaway twill open pandorapack wishdearth amen amenusensis opend the mand of min apend the pain of durthwursht vernichtung desiree tolight and eadly dth cessity sesame

We are the key."

Tuesday, August 19, 2003

Tuesday August 19, 2003

Filed under: General — Tags: — m759 @ 10:23 PM

O'Hara's Fingerpost

In The New York Times Book Review of next Sunday (August 24, 2003), Book Review editor Charles McGrath writes that author John O'Hara

"… discovered a kind of story… in which a line of dialogue or even a single observed detail indicates that something crucial has changed."

From the Online Etymology Dictionary:

crucial – 1706, from Fr. crucial… from L. crux (gen. crucis) "cross." The meaning "decisive, critical" is extended from a logical term, Instantias Crucis, adopted by Francis Bacon (1620); the notion is of cross fingerboard signposts at forking roads, thus a requirement to choose.

The remainder of this note deals with the "single observed detail" 162.



Instantias Crucis

Francis Bacon says

"Among Prerogative Instances I will put in the fourteenth place Instances of the Fingerpost, borrowing the term from the fingerposts which are set up where roads part, to indicate the several directions. These I also call Decisive and Judicial, and in some cases, Oracular and Commanding Instances. I explain them thus. When in the investigation of any nature the understanding is so balanced as to be uncertain to which of two or more natures the cause of the nature in question should be assigned on account of the frequent and ordinary concurrence of many natures, instances of the fingerpost show the union of one of the natures with the nature in question to be sure and indissoluble, of the other to be varied and separable; and thus the question is decided, and the former nature is admitted as the cause, while the latter is dismissed and rejected. Such instances afford very great light and are of high authority, the course of interpretation sometimes ending in them and being completed. Sometimes these instances of the fingerpost meet us accidentally among those already noticed, but for the most part they are new, and are expressly and designedly sought for and applied, and discovered only by earnest and active diligence."

The original:

Inter praerogativas instantiarum, ponemus loco decimo quarto Instantias Crucis; translato vocabulo a Crucibus, quae erectae in biviis indicant et signant viarum separationes. Has etiam Instantias Decisorias et Judiciales, et in casibus nonnullis Instantias Oraculi et Mandati, appellare consuevimus. Earum ratio talis est. Cum in inquisitione naturae alicujus intellectus ponitur tanquam in aequilibrio, ut incertus sit utri naturarum e duabus, vel quandoque pluribus, causa naturae inquisitae attribui aut assignari debeat, propter complurium naturarum concursum frequentem et ordinarium, instantiae crucis ostendunt consortium unius ex naturis (quoad naturam inquisitam) fidum et indissolubile, alterius autem varium et separabile ; unde terminatur quaestio, et recipitur natura illa prior pro causa, missa altera et repudiata. Itaque hujusmodi instantiae sunt maximae lucis, et quasi magnae authoritatis; ita ut curriculum interpretationis quandoque in illas desinat, et per illas perficiatur. Interdum autem Instantiae Crucis illae occurrunt et inveniuntur inter jampridem notatas; at ut plurimum novae sunt, et de industria atque ex composito quaesitae et applicatae, et diligentia sedula et acri tandem erutae.

— Francis Bacon, Novum Organum, Book Two, "Aphorisms," Section XXXVI

A Cubist Crucifixion

An alternate translation:

"When in a Search of any Nature the Understanding stands suspended, the Instances of the Fingerpost shew the true and inviolable Way in which the Question is to be decided. These Instances afford great Light…"

From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:

"The picture, viewed as a whole, is a cubist description, where each portrait looks strikingly different; the failings of each character's vision are obvious. However, in a cubist painting the viewer often can envision the subject in reality. Here, even after turning the last page, we still have a fuzzy view of what actually transpired. Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way. The final narrative is also titled An Instance of the Fingerpost, perhaps implying that we are to see truth and clarity in this version. But the biggest mystery of this book is that we have actually have no reason to credit the final narrative more than the previous three and so the story remains an enigma, its truth still uncertain."

For the "162" enigma, see


The Matthias Defense, and

The Still Point and the Wheel.

See also the December 2001 Esquire and

the conclusion of my previous entry.

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