Friday, November 30, 2012


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

"….mirando il punto  
a cui tutti li tempi son presenti"

— Dante, Paradiso , XVII, 17-18

 For instance

IMAGE- Three films from Christmas 1963 (IMDb): Captain Newman, MD; The Prize; Love with the Proper Stranger

Click image for higher quality.


Filed under: General — m759 @ 12:26 AM

The title of yesterday's 11:22 PM post was "The Place of the Lion."

This is also the title of a novel by Charles Williams.

See, too, Midsummer Eve's Dream and Midsummer Night 2007.

Thursday, November 29, 2012

The Place of the Lion

Filed under: General — m759 @ 11:22 PM

For C. S. Lewis, who was born on this date in 1898,
and Natalie Wood, who died on this date in 1981

"He was accustomed to receiving manuscripts from strangers…."
— C. P. Snow on mathematician G. H. Hardy

"Whoever you are— I have always depended on
the kindness of strangers." — A Streetcar Named Desire

From this journal on September 24, 2012

"A single self-transcendence" — Aldous Huxley

From an anonymous author at the website Kill Devil Hill

"This little story… has that climactic moment of 
heightened awareness…. This is a moment where
two individuals become one, empowering them
to transcend the limitations of their own individual
frailty and society. It's an epiphany, an almost
divine spark. It is an experience when one plus one
don't equal two, but something far greater."

Kill Devil Hills also appears in a 1983 film—

"Suppose it were possible to transfer
from one mind to another
the experience of another person."

— Trailer for "Brainstorm" (1983),
the last film of Natalie Wood

Lines of Symbols

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

C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology

"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."

Some related material (click to enlarge)—

The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.

See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—

IMAGE- Image from Wikipedia articles 'Finite geometry' and 'Hesse configuration.'

The Hesse here is not Hermann, but Otto.

Conceptual Art

Filed under: General,Geometry — m759 @ 12:09 PM

Quotes from the Bremen site

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

" 'compArt | center of excellence digital art' is a project
at the University of Bremen, Germany. It is dedicated
to research and development in computing, design,
and teaching. It is supported by Rudolf Augstein Stiftung,
the University of Bremen, and Karin und Uwe Hollweg Stiftung."

See also Stiftung in this journal.

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

Annals of Religion

Filed under: General — Tags: — m759 @ 2:56 AM

IMAGE- 'Candlestick of Death' in Michener's 'The Source'
IMAGE- An Oscar for 'Chicago'

Diamond Theory

Filed under: General,Geometry — Tags: — m759 @ 2:18 AM

A pdf of a 1977 three-page article with this title
has been added at finitegeometry.org/sc.

Tuesday, November 27, 2012

Raiders of the Lost Lottery

Filed under: General — m759 @ 2:06 PM

Today's midday New York Lottery: 444 and 5222.

See Deuteronomy  4:44 and 5:2-22.


Filed under: General,Geometry — m759 @ 12:25 PM

The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."

The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.

Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .

(Naturally, Mitchell's definition of "reflection" and his statement that

"The discussion of the binary groups
applies also to the case p = 2."

should be approached with care.)

A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—

"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rn which contain the origin."

For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.

Monday, November 26, 2012

“The Eight”…

Filed under: General — Tags: — m759 @ 2:18 PM

Meets "The Master"—

IMAGE- Joaquin Phoenix, corridor scene in 'The Master'

Today's midday NY Lottery: 333 and 5885.

"Continue a search for thirty-three and three." — The Eight  (1988)

"Make me young." — Kilgore Trout in
Breakfast of Champions . Trout was modeled after
author Theodore Sturgeon who died on 5/8/85.

(An example of Sturgeon's work: The Dreaming Jewels  (1950).)

Related illustrations from the eighth day of 2012—

See also "I'm sorry to be catechizing you like this."

Sunday, November 25, 2012

By the Numbers

Filed under: General — m759 @ 9:18 PM

The New York Lottery:

Draw Day: Twice Daily
Draw Time: Midday: 12:20 p.m. – 12:30 p.m.
Evening: 7:30 p.m. – 7:40 p.m. 

NY Lottery this evening: 674 and 1252 —


Contrapuntal themes:

Related Log24 posts today—
at 11:30 AM ET  (see post 674 ) and
at 7 PM ET (see post 1252).

The Devil at Midday:

Interpreting the midday numbers,
172 and 7817, is more difficult. Perhaps 172
refers to a Zen page number in a post from 
the Feast of Saint Louis in 2003, and perhaps,
in a less saintly manner, 7817 refers to
two posts in which these four digits appear
in product numbers within links— namely,
the Garden Party  "Background" link and 
the Seven Bridges  "wild" link.

Then again, perhaps not.

Grim Joke

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

"It's a grim joke." — Amy Adams in "The Master"

When Irish Eyes Are Smiling

Geometry lesson

Click diagram for some background from 3/17.
See, too, some background on Amy Adams and on Leap Day.

For related Harvard humor, see Venn Diagram.


Filed under: General — Tags: — m759 @ 11:30 AM

… For the Boston Church of the Advent coffee hour:

Suggested topic: 

The Klee picture on the above cup— a graphic rendition of the poem
Once Emerged from the Gray of Night — as presented on the cover
and discussed in the text of Martha B. Helfer's The Word Unheard .

Talk amongst yourselves.


Filed under: General — m759 @ 11:00 AM

Today's sermon is for Martha B. Helfer, author of
the treatise on Darstellung  in today's previous post
and of the following—

(Click for clearer image.)

Helfer's The Word Unheard  was published by Northwestern University Press
on St. Andrew's Day, 2011. Log24 posts on that day—

Lines, Grids, and Fatuity for St. Andrew's Day.

The last of these warned of an upcoming Jewish Book Week event
on February 22, 2012.

That date turned out to be Ash Wednesday. See a Log24 post on that topic
that quotes a poet, T.S. Eliot, with anti-Semitic proclivities—

"And the light shone in darkness and
Against the Word the unstilled world still whirled
About the centre of the silent Word."

— T. S. Eliot, "Ash Wednesday"

This is perhaps not entirely irrelevant to Helfer's title, The Word Unheard .

* A concept of Schopenhauer and Hitler, and the first name of
   a fictional Boston mathematician.


Filed under: General — m759 @ 10:31 AM

The title of yesterday's post Will and Representation is of course
a reference to Schopenhauer's philosophical work of that name.

As the post itself indicates, the title is also a punning reference to
mathematical representation theory . 

To avoid confusion, it should be noted that Schopenhauer's
representation , in the original German, was Vorstellung .

The German for mathematical  representation theory is,
on the other hand, Darstellungstheorie . (The mathematical
use of Vorstellung  is non-technical, referring to concepts
of pedagogy. (A group presentation  is a Präsentation .))

For a discussion of the Vorstellung-Darstellung  distinction
in philosophy, not mathematics, see… 

The Retreat of Representation: The Concept of  Darstellung
in German Critical Discourse , by Martha B. Helfer,
State University of New York Press, 1996, esp. pp. 24-26.

Saturday, November 24, 2012

Will and Representation*

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

Robert A. Wilson, in an inaugural lecture in April 2008—

Representation theory

A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).

But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of 

This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.

Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.

We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.

As Poincaré said,

Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.

Par exemple

Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:

IMAGE- Fano plane transformed to eightfold cube, and partitions of the latter as points of the former

* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.

Reappearing All Over Again

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

For the title, see the phrase "reappearing number" in this journal.

Some related mathematics—

the Greek labyrinth of Borges, as well as…

IMAGE- Robert Wilson on the projective line with 24 points and its image in the MOG.

Note that "0" here stands for "23," while corresponds to today's date.

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

Who Shot J. R.?

Filed under: General — m759 @ 4:24 AM

For the answer, click here.

Connoisseurs of synchronicity may note two posts
from the date the above photo was taken:

First Draft of History and The Swedish Solution.

(The late Larry Hagman was, according to NBC News, "a longtime member 
of the Peace and Freedom Party, a minor leftist organization in California.")

Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — m759 @ 10:48 PM

(Continued from 1986)

S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.

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

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: A 4×4 array.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of a 4x4 array into two 8-sets

A representative coordinatization:


0000  0001  0010  0011
0100  0101  0110  0111
1000  1001  1010  1011
1100  1101  1110  1111


The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.

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

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of an array of 16 triangles into two 8-sets

A representative coordinatization:

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

Putting the “I” in “IT”

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

Jennifer Scott at IT Pro , Feb. 16, 2012, on Autonomy

Mike Lynch, founder of Autonomy  and vice president
of information management at HP, took to the stage
at his new parent company’s global partner conference
to impart his philosophy to the 3,000 partners gathered.

‘It is no longer about the data but about the meaning
of that data,’ he said. ‘There is a fundamental revolution
going on in information and the industry is now about
the “I” not the “T” in IT.'”

Click on the logo below for the source.

Putting the I in IT

See also today’s previous post and…

Madeleine L’Engle in
The Irrational Season
(1977), Chapter 9:

“After A Wrinkle in Time  was finally published,
it was pointed out to me that the villain,
a naked disembodied brain, was called ‘It’
because It stands for Intellectual truth
as opposed to a truth which involves the whole of us,
heart as well as mind.  That acronym had never
occurred to me.  I chose the name It intuitively,
because an IT does not have a heart or soul.
And I did not understand consciously
at the time of writing that the intellect,
when it is not informed by the heart, is evil.”

The Red Line vs. the Red Eye

Filed under: General — m759 @ 11:07 AM

Arthur C. Clarke

"Any sufficiently advanced technology is indistinguishable from magic."

"The HP/Autonomy  Debacle," by John C. Dvorak at pcmag.com on Tuesday, Nov. 20, 2012—

"The whole Autonomy  thing was weird since the company seemed to be performing magic. On co-founder Michael Richard Lynch's Wikipedia page, the company is described as 'a leader in the area of computer understanding of unstructured information, an area which is becoming known as meaning-based computing .'

I do not know how gullible HP's board of directors is, but when I see the sudden emergence of something called 'meaning-based computing,' the alarms sound and the bullcrap meter begins to tag the red line."

A story by Terence K. Huwe in Online  magazine, Sept.-Oct. 2011, defines meaning-based computing (MBC), discusses Autonomy , and llnks to…

John Markoff in The New York Times , March 4, 2011— 

"Engineers and linguists at Cataphora, an information-sifting company based in Silicon Valley, have their software mine documents for the activities and interactions of people— who did what when, and who talks to whom. The software seeks to visualize chains of events. It identifies discussions that might have taken place across e-mail, instant messages and telephone calls.

Then the computer pounces, so to speak, capturing 'digital anomalies' that white-collar criminals often create in trying to hide their activities.

For example, it finds 'call me' moments— those incidents when an employee decides to hide a particular action by having a private conversation. This usually involves switching media, perhaps from an e-mail conversation to instant messaging, telephone or even a face-to-face encounter."

For example

IMAGE- HAL reading lips in '2001: A Space Odyssey'

Wednesday, November 21, 2012

Hunting Goodwill

Filed under: General — m759 @ 1:37 PM

Jonathan Weil at Bloomberg.com yesterday
on the HP-Autonomy  story—

"The goodwill figure is especially telling."

Tuesday, November 20, 2012

Resisting the Bogus

Filed under: General — m759 @ 12:00 PM

From remarks by Wallace Stevens featured in recent Log24 posts:

"The poet finds that as between these two sources: the imagination and reality, the imagination is false, whatever else may be said of it, and reality is true; and being concerned that poetry should be a thing of vital and virile importance, he commits himself to reality, which then becomes his inescapable and ever-present difficulty and innamorata. In any event, he has lost nothing; for the imagination, while it might have led him to purities beyond definition, never yet progressed except by particulars. Having gained the world, the imaginative remains available to him in respect to all the particulars of the world. Instead of having lost anything, he has gained a sense of direction and a certainty of understanding. He has strengthened himself to resist the bogus."

— Bard College speech of 1951

Related material:

With Autonomy , H-P Bought An Old-Fashioned
Accounting Scandal. Here's How It Worked.

(Forbes , Nov. 20, 2012)

Hewlett-Packard is losing a star in Mike Lynch
(The Guardian , May 25, 2012)

The Quest for Meaning: The world's smartest search engine
took 250 years to build. Autonomy  is here.
(Wired , Feb. 2000)

The Particulars of Language

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

This post is in memory of an English church musician whose
death was noted here yesterday evening in a post titled
"The Particulars of Rapture," a phrase from Wallace Stevens.

Sir Philip Ledger, who died on Sunday, was a
"church musician who produced magical settings
of carols," according to The Telegraph  RSS feed.

It is not clear what the Telegraph  meant by "magical settings."

Perhaps the phrase "his settings of carols" in his obituary refers
to the annual service of Nine Lessons and Carols at Cambridge,
where he was for a time director of music at King's College.

If so, the settings would be the Lessons, readings from the Bible
(a "ledger," in the sense defined below).  Such readings should
not be confused with notions from the world of Harry Potter. 

Examples: The Lessons from last Christmas in Cambridge.

IMAGE- Meanings of the word 'ledger'

Monday, November 19, 2012

The Particulars of Rapture

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

"And forth the particulars of rapture come."

— Wallace Stevens, "Notes Toward a Supreme Fiction,"
     Canto IV of "It Must Change," quoted here yesterday.

A death yesterday: Sir Philip Ledger.

Poetry and Truth

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

From today's noon post

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Stevens also said at Bard that

"When Joan of Arc said: 

Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.

these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."

There are those who would dispute this.

Some related material:

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

A poetic approach to geometry—

"A surface" and "the rock," from All Saints' Day, 2012

Spaces as Hypercubes

— and from 1981—


Some mathematical background for poets in Purgatory—

"… the Klein correspondence underlies Conwell's discussion 
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M24."


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

Speech by Wallace Stevens upon accepting
an honorary degree from Bard College in 1951

(Click to enlarge.)

Transcription of conclusion:

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Sunday, November 18, 2012


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

Happy birthday to

IMAGE- Margaret Atwood, Kim Wilde, Peta Wilson

Today's sermon, by Marie-Louise von Franz

Number and Time, by Marie-Louise von Franz

For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.

Another modern physicist, Niels Bohr, died
on this date in 1962

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

The circle above is marked with a version
of the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

For the square, see the diamond theorem.

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

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


Filed under: General — m759 @ 5:24 AM


Rachel Dodes in The Wall Street Journal
on All Souls' Day, 2012

"In one of the first lines uttered by Daniel Day-Lewis, playing Abraham Lincoln in the new Steven Spielberg film opening Nov. 9, he says, 'I could be bounded in a nutshell, and count myself a king of infinite space— were it not that I have bad dreams.'

The line was ripped straight from 'Hamlet,' by Lincoln's favorite writer, William Shakespeare. Tony Kushner, the Pulitzer Prize-winning playwright ('Angels in America') who wrote the script for the film, says that Shakespeare, much like Lincoln, 'had extraordinary mastery over the darkest parts of the human spirit.'"

The above quotation omits Shakespeare's words prefacing the nutshell part— "O God."

These same words in a different tongue—  "Hey Ram"— have often been quoted as the last words of Gandhi. (See yesterday's noon post.)

"… for the Highest Essence (brahman ),
which is the core of the world, is identical
with the Highest Self (ātman ), the kernel
of man's existence."

— Heinrich Zimmer, Myths and Symbols
in Indian Art and Civilization
, Pantheon
Books, 1946, page 142 

Related material: A post linked to here on Friday night
that itself links to a different Shakespeare speech.

Saturday, November 17, 2012

Hey RAM*

Filed under: General — m759 @ 12:00 PM

Last evening's NY Lottery numbers
985 and 3274, interpreted as the
numbers of Log24 posts, suggest
a look at Joyce's nightmarehistory.

* The title refers both to a film and to
   a Log24 post, Random Access Memory.

Friday, November 16, 2012

Ananga Ranga*

Filed under: General — m759 @ 8:00 PM

The Telegraph  this evening has an obituary for
Theresa Brown, a Texas madam who reportedly 
died on September 18, 2012.

In her honor, an update has been made to a Log24 post
of that date. I hope she finds the update relevant.

* "Love Stage" — See yesterday's post.

Thursday, November 15, 2012

A Passage to India

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


On a mathematician who died on All Souls' Day 2012—

"… he enthusiastically shared with us the many stories
of Indian epics like Mahabharata." — Online tribute

This suggests a pictorial review incorporating some
images from past Log24 posts.

Best Exotic Ananga Ranga

Log24 on All Souls' Day 2012

Click images for some background.

Wednesday, November 14, 2012

Group Actions

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

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—


Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—


Tuesday, November 13, 2012

Gyre, Illustrated

Filed under: General — m759 @ 10:09 PM

The title refers to today's previous post,
"In the Widening Gyre."

"In modal logic, specifically when representing
possible worlds, @ is sometimes used as
a logical symbol to denote the actual world 
(the world we are 'at')." —Wikipedia

"What might have been and what has been
Point to one end, which is always present."

Four Quartets

IMAGE- 'The World-Time Parallel' (cover: spirals on sphere, by Escher)

Click the gyre on the book cover above for further details.

In the Widening Gyre

Filed under: General — m759 @ 3:28 PM

Tiw’s Day

Filed under: General — m759 @ 9:48 AM

All our words from loose using have lost their edge.”
— Ernest Hemingway

A newly coined term— "Spyfall"—

— is good, but not as good as…


(Click on NOR for some connotations.
 See also Norway in this journal.)

Monday, November 12, 2012

Quine’s Dagger

Filed under: General — m759 @ 10:31 AM

This post was suggested by other weblogs' 
posts on Owen Barfield* quoted here recently, by
the Apollo Guidance Computer (see below and
MIT Technology Review), and by James Joyce:

"Unsheathe your dagger definitions." — Ulysses

* See The Appearances (Sept. 20, 2009), for
   a correction of Quine by Barfield.

The Usual Suspects

Filed under: General — m759 @ 2:01 AM

Four southern authors—

Of particular interest in light of previous posts on singer Alicia Keys and on
author Octavia Butler — author Charles G. Bell, second from left above.

See videos of  Bell's Symbolic History  series.

Sunday, November 11, 2012

Desert Cross

Filed under: General — m759 @ 5:45 PM

Two news stories for Armistice Day:

Nov. 11, 2012—

Veterans to resurrect war memorial cross
in Mojave desert in Calif., capping long legal dispute

Nov. 6, 2012—

Mojave Cross, stolen two years ago,
discovered in Bay Area

See also this journal on the reported date of the cross theft.

Lincoln Porn

Filed under: General — m759 @ 2:56 PM

Tablet  magazine, November 9th, 2012,
on Spielberg's Lincoln :

"… the movie’s lone Rocky moment
of ecstatic self-congratulation
is reserved for the amendment’s climactic
passage with the victorious congressmen
spontaneously bursting into
'The Battle Hymn of the Republic.' "

IMAGE- From Wikipedia, Venn diagram for NAND operator (cf. 'Mandorla')

"Mine eyes have seen the glory…"

IMAGE- Chris Pirillo, T-Mobile ad, 'Life Without Limits'

Both images above refer to this morning's post
 Professor Lavery's Sunday School.

For other porn from Tablet  magazine, see
Minimalist Whirl.

For other porn from Lincoln's seat of government, see
Physical Poetry and All In .

For further blasphemy, see The Apotheosis of Washington:

IMAGE- Blasphemous interior of the U.S. Capitol Dome

Professor Lavery’s Sunday School–

Filed under: General — m759 @ 8:00 AM

Saving an Appearance

(Click images for background.)

IMAGE- Owen Barfield on Israel, Christ, and 'participation'

See also Isaiah

IMAGE- Isaiah Sheffer, paid death notice, first paragraph

For a founder of Symphony Space

"To assuage this space…"

IMAGE- Weblog post on Barfield, 'To assuage this space...'

Saturday, November 10, 2012

Descartes Field of Dreams

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

(A prequel to Galois Field of Dreams)

The opening of Descartes’ Dream ,
by Philip J. Davis and Reuben Hersh—

“The modern world,
our world of triumphant rationality,
began on November 10, 1619,
with a revelation and a nightmare.”

For a revelation, see Battlefield Geometry.

For a nightmare, see Joyce’s Nightmare.

Some later work of Descartes—

From “What Descartes knew of mathematics in 1628,”
by David Rabouin, CNRS-Univ. Paris Diderot,
Historia Mathematica , Volume 37, Issue 3,
Contexts, emergence and issues of Cartesian geometry,
August 2010, pages 428–459 —

Fig. 5. How to represent the difference between white, blue, and red
according to Rule XII [from Descartes, 1701, p. 34].

A translation —

The 4×4 array of Descartes appears also in the Battlefield Geometry posts.
For its relevance to Galois’s  field of dreams, see (for instance) block designs.

Battlefield Geometry

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


Click to enlarge.

Related material from Wikipedia— Baseball metaphors for sex.

Build it…

Friday, November 9, 2012

Battlefield Geometry

Filed under: General,Geometry — Tags: — m759 @ 3:28 PM

(Continued from Sept. 11, 2007)

CIA Director David Petraeus resigns, cites extramarital affair

Trouble with the curve?

Norway High Chair

Filed under: General — m759 @ 11:00 AM

This post was suggested by today's previous post,
"Bali High Chair," that links to an empty chair award for
evangelical supporters of Mitt Romney, by Bauhaus style,
and by the example of Norwegian  design shown below—

(Happy Frigg's Day to Josefine Lyche.)

Bali High Chair

Filed under: General — m759 @ 7:00 AM

The phrase "deep play" in the previous post
was a borrowing from Clifford Geertz.

From another weblog's post on Geertz and
deep play—

When family is involved, the Balinese
are much more engaged.

See also the Balinese empty chair
and Amy Adams in this journal.


Filed under: General,Geometry — m759 @ 6:21 AM

(Continued from Deconstructing Alice)

The Dream of the Expanded Field

Image-- 4x4 square and 4x4x4 cube

"Somehow it seems to fill my head
with ideas— only I don't exactly know
what they are!"

See also Deep Play.


Filed under: General — m759 @ 5:24 AM

The New York Times  top online front page story this morning—

   more »

"A version of this article appeared in print 
on November 9, 2012, on page 
B1 of
New York edition with the headline: 
An Innovator vs. a Follower." — The Times

Some related material from this  journal

The Quality of Diamond,
Log24 entries from Feb. 2004:

The Quality
with No Name

And what is good, Phaedrus,
and what is not good…
Need we ask anyone
to tell us these things?

— Epigraph to
Zen and the Art of
Motorcyle Maintenance

Thursday, November 8, 2012

Brightness at Noon

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


Click on image for a related post.


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

Sean French, in a March 1998 New Statesman
review of a book on Wittgenstein and Hitler,
discusses the author's theory that …

"… the Jew who played the decisive role
in inspiring the Holocaust also played
the decisive role in bringing it to an end.

There is something heroic about this argument
and it would be a good subject for a novel about
the dangers of creating theories out of nothing.
Nabokov should have written it. It is not just
that there are weak links in the theory. There are
no links in the theory."

"Nabokov should have written it."… See Bend Sinister.


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

Dante, Purgatorio XVI

Esce di mano a Lui, che la vagheggia
     Prima che sia, a guisa di fanciulla,
     Che piangendo e ridendo pargoleggia,

L’anima semplicetta, che sa nulla,
     Salvo che, mossa da lieto fattore,
     Volontier torna a ciò che la trastulla.

Dante on the soul in Purgatorio 16
Related material:

and, in this journal,

Wednesday, November 7, 2012


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

In several posts now tagged Chessboard
an I Ching chessboard
image in which adjacent squares 
have the Karnaugh property

— has been replaced by a picture of
the original 1989 version in which
the Karnaugh property applies only to cells
that are adjacent in a cubic, not square,

I Ching chessboard (original 1989 arrangement)

For Leonard Cohen Fans

Filed under: General — m759 @ 9:00 AM

Leading this morning's online New York Times  
obituaries list is one for a leftist Episcopal priest.

That obituary contains a link to a 1996 Times  story — 

Old Friends, New Foes:
President and a Preacher;
One 60's Activist Runs Columbia;
One Fights It

A check for background on the Columbia president
in that story yields the following from a leftist journalist's
website in a 2012 post dated October 26 (date of the
1917 October Revolution in Russia)—

Click for further details of this leftist allegation, which is
clearly false (because based on a ludicrously bad
misreading of a Rice University puff piece), and would
be libelous if its target were alive.

See a Rice University obituary for the "Nazi spy"
in question, who died on May 25, 2010. (See also
 this journal on that date.)

The same leftist webpage contains a link to another leftist's
attack on a respected charitable organization that may
or may not have, or have had, CIA connections—

The International Rescue Committee (IRC).

In the spirit of Leonard Cohen's "then we take Berlin" lyric,
here is a note on how the IRC logo

might be rendered in a way that is, though less visually appealing,
more logical— i.e., more purely an example of Bauhaus style—


See also the recent  Black October post
on a Columbia enthusiast of the October Revolution—

Tuesday, November 6, 2012

Then We Take Berlin

Filed under: General — m759 @ 3:17 PM

See Love Ghost, from the date of Anita Björk's death.

"Friends are often
really enemies,
and adversaries
sometimes secret allies."

— Wikipedia on the film
Night People  (at right)

Anita Björk and Gregory Peck in 'Night People' (1954)

Bend Sinister

Filed under: General — m759 @ 6:30 AM

This morning's New York Times  obituaries—

These suggest a look at Solving Nabokov's Lolita Riddle ,
by Joanne Morgan (Sydney: Cosynch Press, 2005).

That book discusses Lolita as a character like Lewis Carroll's Alice.

(The Red Queen and Alice of course correspond to figures in
the first two thumbnails above.)

From the obituary associated with the third thumbnail above:

"Front-page headlines combined concision and dark humor." 

The title of this post, Bend Sinister , is not unlike such a headline.
It is the title of a novel by Nabokov (often compared with Orwell's 1984 )
that is discussed in the Lolita Riddle  book.

Related material— The bend sinister found in Log24 searches
for Hexagram 14 and for the phrase Hands-On

IMAGE- Magician's hands on his wand, viewed as a diagonal of a square

Monday, November 5, 2012

Design Cubes

Filed under: General,Geometry — m759 @ 9:48 PM

Continued from April 2, 2012.

Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties

Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz  cubes (see links below).

Horowitz Design Cubes Package

Horowitz Design Cubes (1971)

1973 Horowitz Design Cubes Patent

Horowitz Biography

For the Hollow Men

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

"Remember, remember the Fifth of November."

Very well. See a post of Nov. 5, 2005, and the related posts
ShadowsCuberCube Partitions, and Cube Review.

A Very High Order

Filed under: General — m759 @ 6:29 AM

From a Washington Post  book review of
The Essays of Leonard Michaels
by Jonathan Yardley, Sunday, July 26, 2009—

"Michaels said that his literary influences included Franz Kafka, Lord Byron and Wallace Stevens (this last is frequently quoted in these essays), but reading that paragraph one can't help thinking that as a young man in the 1950s he must have spent a lot of time listening to Mort Sahl, Lenny Bruce and the other great Jewish comedians of the day, as his own comic voice and timing bear more than passing resemblance to theirs. In another essay, 'A Sentimental Memoir,' he recalls being 'saved' as a young man by a professor of English at the University of Michigan named Austin Warren, and then he meanders into an account of falling 'insanely in love' (which seems to have been a strong predilection with him) with a girl in Warren's class:

'The girl had a slender boyish figure and blondish hair. I thought nobody but me considered her striking or had noticed the subtle perfection of her beautiful face. When I told my friend Julian that the most beautiful girl in Michigan was in Warren's class, he named her. He told me that she modeled naked for art students, she had a horrible reputation for licentiousness, everybody knew who she was, and that he was in love with her too. I decided that I was ready to forgive her everything. To forgive a girl was a very popular sentiment of the day. There were plays, novels, and movies about forgiving bad girls. As for the girl I was ready to forgive, I now suppose there were a couple of hundred other men who were forgiving her at the same time, all of us subject to a sort of spiritual narcissism that has long since gone out of the world. Too bad, I think, since it had extraordinary intensity and made a man feel tortured by goodness, which is a very high order of feeling.' "

Sitting Specially

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

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date


IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Sunday, November 4, 2012

Sunday School:

Filed under: General — m759 @ 5:48 AM

Rigor  (continued from yesterday)…


IMAGE- 'Professor Current was among the historians who raised the study of Abraham Lincoln to rigorous academic standards.'

See also, from the date of Current's death, High White Noon.

Shakespeare’s Rhyme

Filed under: General — m759 @ 5:01 AM

From Saturday Night Live  last night—

IMAGE- Louis C. K. as Shakespeare on SNL

Related material from the Log24 post "Now Lens"
(March 11, 2011)—

Errol Morris in The New York Times  on March 9, 2011

"If everything is incommensurable, then everything
is seen through the lens of the present, the lens of now ."

"Borges concluded by quoting Chesterton, 'there is nothing
more frightening than a labyrinth that has no center.' [72]"

See also Borges on Shakespeare, everything, and nothing 
in a note from September 7, 2006.

Everything and nothing in Peter J. Cameron's weblog yesterday—

The existence of everything entails the existence of nothing;
indeed, the existence of anything (any set A) entails
the existence of the empty set (the set {x∈A:x≠x}).
But not the other way round.

Right, I had better put on my anorak and go out now …

Illustration added by m759—

How many miles to Babylon?*
Three score miles and ten.
Can I get there by candle-light?**
Yes, and back again.

* Suggested by the Pindar link in this journal yesterday.

** Quoted in the "Seven is Heaven" post on All Souls' Day.

Saturday, November 3, 2012


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

A New Yorker  weblog post from yesterday, All Souls' Day

"As the mathematician Terence Tao has written,
math study has three stages:
the 'pre-rigorous,' in which basic rules are learned,
the theoretical 'rigorous' stage, and, last and most intriguing,
'the post-rigorous,' in which intuition suddenly starts to play a part."

Related material— 

Rigor  in a Log24 post of Sunday evening, May 25, 2008: "Hall of Mirrors."

Note in that post the tesseract  viewed as the lattice of
the 16 subsets of a 4-element set.

Some further material related to tesseracts and time, in three stages
(roughly corresponding to Tao's, but not in chronological order): 

  1. Bakhtin
  2. Spaces as Hypercubes, and 
  3. Pindar.

See also a recent Log24 post on remarks from Four Quartets .

(The vertices of a tesseract form, in various natural ways, four quartets.)

Quarter to Three…

Filed under: General — Tags: — m759 @ 2:45 AM

Behind the Green Door

Click for image source.

Related material—

Door Prize for Larry Bloch, who died on Sunday, October 28.

Bloch founded the Wetlands Preserve rock club in TriBeCa.

Friday, November 2, 2012

By Candlelight

Filed under: General — m759 @ 7:00 PM

"Seven is Heaven, Eight is a Gate" — Memory rhyme

For Rebekah Gay

IMAGE- Book cover (horse, girl at gate) of 'Can I Get There by Candlelight?,' illus. by Ted Lewin

Click for image source.

For All Souls’ Day

Filed under: General — m759 @ 9:48 AM

From this journal yesterday (All Saints' Day)—

"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."

New Yorker  profile of tesseract
author Madeleine L'Engle

For a discussion of this issue in greater depth—

"Truth and fact are not the same thing."

— see a 1998 award acceptance speech by L'Engle.

See also a Log24 post of March 1st, 2008, on the soul.

Smart News

Filed under: General — m759 @ 4:04 AM

The late Arthur R. Jensen was an expert on IQ testing.

From a Log24 post on Oct. 22, 2012, the reported date of his death—

IMAGE- Smithsonian 'Smart News' reports the death of Paul Kurtz

Related material — "IT" in A Wrinkle in Time .

Time and Chance (continued)

Filed under: General,Geometry — m759 @ 1:20 AM

For clergymen who embrace Trudeau's
"Story Theory of Truth" (see last evening's
7:20 PM post on geometry and A Wrinkle in Time )

Here are two meditations suggested by
yesterday evening's New York Lottery :

1.  Page 141 in Philosophies of India

2.  Post 4658 in this journal— A Wrinkle in Dimensions.

Thursday, November 1, 2012

Theories of Truth

Filed under: General,Geometry — Tags: — m759 @ 7:20 PM

A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The Non-Euclidean Revolution

The Story Theory of Truth:

"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."

New Yorker  profile of tesseract
     author Madeleine L'Engle

The Diamond Theory of Truth:

(Click image for some background.)

Spaces as Hypercubes

See also the links on a webpage at finitegeometry.org.

For All Saints’ Day

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

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

The New Yorker , April 12, 2004 —

Note the black diamond at the story's end.

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