Journal Archives of Steven H. Cullinane:
Entries to be Added to Archive

Sunday, October 18, 2009  11:00 AM

ART WARS for St. Luke's Day:

A Sermon from Christchurch
in The New York Times

Related material:

Zen and the Art
For the Burning Man

Friday, October 16, 2009  11:30 AM

Noncontinuous Groups:

A page with this title has been added to my finite-geometry site.

(For the first version of that site, see a web page cached on August 15, 2000; compare with Ivars Peterson's August 28, 2000, column "Scrambled Grids." These pieces are clearly intended for two different audiences, but there is a certain similarity in the subject matter.)

Wednesday, October 14, 2009  9:29 AM


Singer 7-Cycles

Seven-cycles by R.T. Curtis, 1987

Singer 7-cycles by Cullinane, 1985

Click on images for details.

The 1985 Cullinane version
gives some algebraic background
for the 1987 Curtis version.

The Singer referred to above is James Singer. See his "A Theorem in Finite Projective Geometry and Some Applications to Number Theory," Transactions of the American Mathematical Society 43 (1938), 377-385.

For other singers, see Art Wars and today's obituaries.

Some background: the Log24 entry of this date seven years ago, and the entries preceding it on Las Vegas and painted ponies.

Tuesday, October 13, 2009  7:00 AM

Man and His Symbols:


This morning's New York Times
reports the deaths of Nuremberg interrogator Richard W. Sonnenfeldt and of avant-garde novelist and Beckett scholar Raymond Federman.

Symbols from this journal on the dates of their deaths:

For Sonnenfeldt, who died
 on Friday, Oct. 9,
a symbol from that date:

The 3x3 grid as religious symbol

For connotations of the symbol appropriate to the name Sonnenfeldt, see the link to A Sunrise for Sunrise in the entry of Saturday, Oct. 10.

For Federman, who died
 on Tuesday, Oct. 6,
a symbol from that date:

Black monolith

A quotation that appeared here on Wednesday, Oct. 7, seems relevant to Federman:

But I am a worker, a tombstone mason, anxious to pleace averyburies and jully glad when Christmas comes his once ayear. You are a poorjoist, unctuous to polise nopebobbies....

-- James Joyce, Finnegans Wake

Monday, October 12, 2009  7:00 AM

Annals of Aesthetics:

Happy Columbus Day

Part I and Part II

Sunday, October 11, 2009  7:00 PM


Concepts of Space

Today I revised the illustrations
in Finite Geometry of the
Square and Cube

for consistency in labeling
the eightfold cube.

Related material:

Inside the White Cube:
The Ideology of
the Gallery Space

Dagger Definitions

Saturday, October 10, 2009  7:00 AM

ART WARS review:

A Sunrise
for Sunrise

Related material:

This morning's obituaries

(click to enlarge)

and Zen and Language Games

Friday, October 9, 2009  9:00 AM



The 3x3 grid as religious symbol

"...strict grids of nine pictures
    establish an egalitarian

-- Christopher Knight

Some are more
than others.

Thursday, October 8, 2009  10:30 AM

Aesthetics continued...

Knight Moves

Deborah Solomon, New York Times Magazine, Sunday, June 27, 1999:

"While modern art began as an assault on the academy, post-modern art might be described as a return to the academy. Instead of the old academy of rules, now we have the Academy of Cool, schools that treat avant-garde rebellion as a learned occupation."

Christopher Knight, LA Times art critic, on Solomon:

"Back in the day, Solomon interviewed Knight for a Times Magazine story on Los Angeles art schools. 'Having been a journalist (at that time) for almost two decades, I also did my homework,' Knight writes [in a letter to the New York Press]. 'I prepared a couple of quotable quotes on the subject, which might encapsulate larger ideas.' One of Knight's pearls of wisdom, 'Modern art began as an assault on the academy, but post-modern art might be described as a return to the academy,' excited Solomon so much that, according to Knight, she printed it as her own observation in her final piece, which bore no mention of the Knight interview. In the final story, a seriously bitter Knight writes, 'It was not a quote; my words had become her words.'" --Gawker, Oct. 11, 2007

A reference to Solomon's piece appeared in this journal in 2003.

See also yesterday's entry, today's 9 AM entry, and (for the Academy) an example of knight's move thinking.

Thursday, October 8, 2009  9:00 AM

Aesthetics continued...

In memory
of Irving Penn:


Chessboard (Detail)

Christopher Knight
on a current exhibit
of Penn's work:

"In American Vogue,
strict grids of nine pictures
establish an egalitarian
framework; the design
anticipates Minimalist art
by a decade."

Wednesday, October 7, 2009  7:59 AM

A Passage:

Finucane's Wake

Terence McKenna,
"Surfing on Finnegans Wake"--

"Shall I try and find a passage?....

But I am a worker, a tombstone mason, anxious to pleace averyburies and jully glad when Christmas comes his once a year. You are a poorjoist, unctuous to polise nopebobbies...."

The Finucane of the title
was a Holy Ghost Father.

Related material:
"Haunting Time,"
June 3, 2007.

Tuesday, October 6, 2009  11:07 AM

Annals of Aesthetics, continued--

A Halmos
for Gelfand:

Black monolith

See also
The Unity
 of Mathematics

Monday, October 5, 2009  4:00 AM

Annals of Aesthetics:

Continued from Saturday-- 

Pieces missing from Wechsler block design test and from IZZI puzzle

for the 16:

Block Designs
and Art

for the 70:

and Counting
  "Kunst ist nicht einfach."
-- Sondheim in translation

Saturday, October 3, 2009  3:31 AM

Annals of Aesthetics:

Missing Pieces:  Conceptual art by Cullinane and Bochner

Related material:

Frame Tales, as well as
The Sacred Day of Kali,
this morning's
 New York Times obituaries,
Mental Health Month, 2003:

Wechsler blocks

WAIS blocks

IZZI puzzle
IZZI puzzle

Michael Douglas in 'The Game'

Sondheim: 'Putting It Together'

Friday, October 2, 2009  6:00 AM

Some Finite Geometry:

Edge on Heptads

Part I: Dye on Edge

....we obtain various orbits of partitions of quadrics over GF(2a) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell's heptagons for the Klein quadric in PG(5,2)."

In 1910 Conwell... produced his heptagons in PG(5,2) associated with the Klein quadric K whose points represent the lines of PG(3,2).... Edge... constructed the 8 heptads of complexes in PG(3,2) directly. Both he and Conwell used their 8 objects to establish geometrically the isomorphisms SL(4,2)=A8 and O6(2)=S8 where O6(2) is the group of K...."

-- "Partitions and Their Stabilizers for Line Complexes and Quadrics," by R.H. Dye, Annali di Matematica Pura ed Applicata, Volume 114, Number 1, December 1977, pp. 173-194

Part II: Edge on Heptads

"The Geometry of the Linear Fractional Group LF(4,2)," by W.L. Edge, Proc. London Math Soc., Volume s3-4, No. 1, 1954, pp. 317-342. See the historical remarks on the first page.

Note added by Edge in proof:
"Since this paper was finished I have found one by G. M. Conwell: Annals of Mathematics (2) 11 (1910), 60-76...."

Some context:

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

Wednesday, September 30, 2009  9:48 AM

Art and Religion:

Midnight in the Garden,
 Autumn 2009


The New York Times Magazine
on Sunday, Sept. 20, 2009:

'The Holy Grail of the Unconscious' at The New York Times

From this journal on the
following day, Sept. 21:

Pearl Jam 'Backspacer' album released Sept. 20, 2009

Happy birthday,
 Stephen King.

Today's previous entry is based on
a song, "Unthought Known," from
 the above album; the cover of the
album uses the 3x3 grid shown in
  Sept. 20's midnight review.

For related material on
the unconscious, see
June 13-15, 2005.

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

-- Tom O'Bedlam's Song

Wednesday, September 30, 2009  2:02 AM

Synchronistic Recognition:

Not So Second-Rate

From the above link in this journal on Thursday, Sept. 24, 2009:

"... the unthought-known dispatches a steady stream of 'notes' from underground. If it cannot gain a full return to consciousness, it at least would like some recognition-- even if disguised."

-- Randall Hoedeman, Sunnyhill Church, Pittsburgh
Randall Hoedeman

Randall Hoedeman

Another discovery reportedly also made last Thursday, Sept. 24:

"The Cullinan mine has again given the world a spectacularly beautiful and important diamond."

-- Petra Diamonds Ltd. CEO Johan Dippenaar
Picture from Fox News:
507-carat diamond from the Cullinan mine reportedly found on Thursday, Sept. 24, 2009

Tuesday, September 29, 2009  3:00 AM

Annals of Religion:

From Harper's Magazine
for the Feast of
St. Michael and All Angels:

Note on a poem by Rilke

Monday, September 28, 2009  3:00 AM

Annals of Aesthetics, continued:

for Germany

See Annals of Aesthetics,
 January 13, 2009,
which features the following
example of modernism:

Modernist chess set, Lanier Graham, 1966

... and for readers of
the Sunday New York Times ...

Highgate Cemetery, London, on cover of NY Times Book Review Sept. 27, 2009

The funereal heart illustrates a review of a book titled Her Fearful Symmetry. The book is set, partly, in London's Highgate Cemetery.

The book's author, Audrey Niffenegger, has stated that her title refers to "the doubling and twinning and opposites" that are "essential to the theme and structure of the book." For examples of doubling, twinning, and opposites that I prefer to Niffenegger's, see this journal's Saturday and Sunday entries.

Fans of the New York Times's cultural coverage may prefer Niffenegger's own art work. They may also enjoy images from the weekend's London Art Book Fair that suggested the rather different sort of book in Saturday's entry.

Sunday, September 27, 2009  3:00 AM

Today's Sermon:

A Pleasantly
Discursive Treatment

In memory of Unitarian
minister Forrest Church,
 dead at 61 on Thursday:
NY Times Sept. 27, 2009, obituaries, featuring Unitarian minister Forrest Church
Unitarian Universalist Origins: Our Historic Faith--
"In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history."
Gravity's Rainbow--
"For every kind of vampire, there is a kind of cross."
Unitarian minister Richard Trudeau--
"... I called the belief that

(1) Diamonds-- informative, certain truths about the world-- exist

the 'Diamond Theory' of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that

(2) The theorems of Euclidean geometry are diamonds....

As the news about non-Euclidean geometry spread-- first among mathematicians, then among scientists and philosophers-- the Diamond Theory began a long decline that continues today.

Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory's only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.

Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*-- first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain." --The Non-Euclidean Revolution, p. 255
H. S. M. Coxeter, 1987, introduction to Trudeau's book--
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'."
As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase "diamond theory" that is closely related to...

A kind
 of cross:

Diamond formed by four diagonally-divided two-color squares

See both
Theme and
and some more
poetic remarks,
 of the Fourfold.

* As recent Log24 entries have pointed out, diamond theory (in the original 1976 sense) is a type of non-Euclidean geometry, since finite geometry is not Euclidean geometry-- and is, therefore, non-Euclidean, in the strictest sense (though not according to popular usage).

Saturday, September 26, 2009  8:28 AM

ART WARS continued--

Suggested by today's
London Art Book Fair:

at Amazon

'Four Quartets' paperback cover

Amazon image upload: 'Four Quartets' cover, 2/27/06

Related material:


Friday, September 25, 2009  3:09 AM

Verbum Sat...

Gian-Carlo Rota


Shift Lock key from manual typewriter, linking to book 'The Philosopher's Gaze'

Thursday, September 24, 2009  2:01 AM

Aesthetics for King, continued:

Who Knows
What Evil Lurks...

The brain-in-a-jar on the cover of the new Pearl Jam album "Backspacer" (previous two entries) is apparently there because of a song on the album, "Unthought Known"--
"All the thoughts you never see
You are always thinking
Brain is wide, the brain is deep
Oh, are you sinking?"
The song title is from a book, The Shadow of the Object (Columbia U. Press, 1987), by psychoanalyst Christopher Bollas.

The "unthought known" phrase has been quoted widely by second-rate psychologizers and by some not so second-rate. Their lucubrations suggest that sinking brain-worshippers should seek a...

Amy Adams and Meryl Streep ('Doubt') as Catholic psychoanalysts, with their couch

The couch is from a 2002 exhibit
at London's White Cube gallery.

Wednesday, September 23, 2009  9:26 AM

Aesthetics for King, continued:


Some background for the images in the previous entry's album cover:
Pearl Jam's 'Backspacer' album cover
See, Aug. 3, 2009, and Aug. 6, 2009.

The brain image is apparently based on a photo at Flickr.

Monday, September 21, 2009  2:56 AM

Aesthetics for King:


A Google search for "Das Scheinen," a very rough translation into Heidegger's German of "The Shining," leads to a song. A search for the English version of the song leads to a site with a sidebar advertising Pearl Jam's new (Sept. 20) album "Backspacer."


Pearl Jam 'Backspacer' album released Sept. 20, 2009

Happy birthday,
 Stephen King.


Yesterday's entries
and the plot of
L'Engle's classic
A Wrinkle in Time.
(See this journal's entries
for March 2008.)

The Pearl Jam album cover art
is of particular interest in light
of King's story "Apt Pupil" and
of Katherine Neville's remark
"Nine is a very powerful
Nordic number.

Those who prefer more sophisticated
aesthetic theory may click on the
following keys:

Back Space key from manual typewriter, linking to Babich on Music, Nietzsche, and Heidegger
Shift Lock key from manual typewriter, linking to Levin's 'The Philosopher's Gaze'

Sunday, September 20, 2009  9:00 AM

Today's Sermon:

The Appearances

German verb:
  1. to shine; to gleam     
  2. to seem; to appear....
Quine, Pursuit of Truth,
second edition,
Harvard U. Press, 1990,

Google search:

Owen Barfield,
Saving the Appearances:

George S. Lensing,
Wallace Stevens and the Seasons:

"Poetry is often a revelation 
of the elements of appearance.

Sunday, September 20, 2009  12:00 AM


Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The 3x3 grid

Nichts ist wie es scheint.

Saturday, September 19, 2009  4:23 PM

Block that Metaphor, continued:

Old Year, Raus!

Also in today's New York Times obituaries index:

 John T. Elson, Editor Who Asked
"Is God Dead?" at Time, Dies at 78

John T. Elson and Budd Schulberg

Wikipedia article on George Polya:

From the date of Elson's death:

A Crystal Block --

Cube, 4x4x4

Four coloring pencils, of four different colors

Related material:
"A Four-Color Theorem."

Saturday, September 19, 2009  2:22 AM

A New Year's Prayer...

Towards Kristen

Kristen Wiig as Michele Bachmann, SNL Thursday update, Sept. 17, 2009

Jerusalem Post Interview
with Charles Krauthammer

by Hilary Leilea Krieger, JPost Correspondent, Washington

Krauthammer, a columnist for The Washington Post, is a winner of the Irving Kristol award.

Jerusalem Post, June 10, 2009:

Can you talk a little bit about your own Jewish upbringing and sense of Jewishness, and how that influences you? I assume it's a factor in this particular project.

I grew up in a Modern Orthodox home [in Montreal]. I went to Jewish day school right through high school, so half of my day was spent speaking Hebrew from age six to 16. I studied thousands of hours of Talmud. My father thought I didn't get enough Talmud at school, so I took the extra Talmud class at school and he had a rabbi come to the house three nights a week. One of those nights was Saturday night, so in synagogue Saturday morning my brother and I would pray very hard for snow so he wouldn't be able to come on Saturday night and we could watch hockey night in Canada. That's where I learned about prayer.

That didn't seem to you to be a prayer that was likely to go unanswered?

Yeah, I was giving it a shot to see what side God was on.

And what did you determine?

It rarely snowed.


More on Krauthammer's Canadian childhood:

"His parents were Orthodox and sent him to
 Hebrew day school. He also took
 private Gomorrah lessons twice a week."

-- "Charles Krauthammer: Prize Writer,"
     by Mitchell Bard


Also in the Jerusalem Post interview:

.... What, then, did you mean by a Jewish sensibility?

".... In literature it's an interesting question, what's a Jewish novel?"

My Prayer:

Private Gomorrah lessons
with Kristen.


"Heaven Can Wait"

Happy Rosh Hashanah
(and Gemara).

Update, 5:01 AM Sept. 19

Before becoming a writer,
Krauthammer was, his
Washington Post biography says,
a resident and then chief resident
in psychiatry at
Massachusetts General Hospital.

Related Metaphors

This morning's New York Times:

NY Times obituary for Irving Kristol, with squirrel-and-nuts ad this morning:

Squirrel with acorns at Michele Bachmann home page, Sept. 19, 2009

See also:

James Hillman's "acorn theory"
of personality development
(yesterday's entry).

Friday, September 18, 2009  2:22 PM

Annals of Aesthetics

An Alternative

to the worldview
of Dan Brown:

'The Soul's Code,' by James Hillman

Thursday, September 17, 2009  8:00 PM

Instant Review Department:

Jennifer's Body

The following remark this evening by Ann Hornaday of The Washington Post serves as an instant review of today's previous cinematic Log24 offering starring the late Patrick Swayze:
"Watch it, forget it, move on."
A perhaps more enduring tribute:

Patrick Swayze in 'King Solomon's Mines'

Related material:

Solomon's Cube,
Solomon and Sheba,
Raiders of the Lost Stone.

"Ready when you are, C.B."

Thursday, September 17, 2009  11:07 AM

Words and Music:

Symbologist Robert Langdon and a corner of Solomon's Cube

Patrick Swayze and Jennifer Grey in  'Dirty Dancing'

"Nobody puts Baby in a corner."

Thursday, September 17, 2009  2:45 AM

On Wings of Song:

Thomas Disch
welcomes Mary Travers

NY Times obituaries, 2:30 AM Sept. 17, 2009: Mary Travers and others

Click to enlarge

Wednesday, September 16, 2009  11:07 AM

For Dan Brown, continued:

The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Tuesday, September 15, 2009  2:02 PM

Johnson's Wake:

In memory of
Harvard literature professor
Barbara Ellen Johnson
(Oct. 4, 1947 -
 Aug. 27, 2009)

" has to be willing
to tolerate ambiguity,
even to be crazy."

"Bohr's words?"

"The party line...."

-- Quotation from
Secret Passages linked to on
 the date of Johnson's death

"Yes and no (what else?)."
-- Barbara Johnson in
The Wake of Deconstruction

Related material:

Harvard Crimson obituary
and a
Funeral Service obituary
with comments.

For more on ambiguity,
see this journal's entries of
 March 7, 8, and 9, 2007.

For more on craziness,
see this journal's entries of
March 10, 2007.

Tuesday, September 15, 2009  12:00 AM

For Dan Brown:

Table of Unicode dingbats

My personal favorite:

Dingbat 275A, 
"heavy vertical bar"--

Unicode symbol 'heavy vertical bar'

Cf. March 7, 2003.

Monday, September 14, 2009  3:09 PM

Back-to-School Special, continued:


Generating permutations for the Klein simple group of order 168 acting on the eightfold cube

The Sept. 8 entry on non-Euclidean* blocks ended with the phrase "Go figure." This suggested a MAGMA calculation that demonstrates how Klein's simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.

* i.e., other than Euclidean. The phrase "non-Euclidean" is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not "non-Euclidean."

Friday, September 11, 2009  1:00 PM

In Memoriam:

For 9/11

Cover of 'Underworld,' by Don DeLillo, First Edition, Advance Reader's Copy, 1997

Cover of Underworld,
 by Don DeLillo, First Edition,
 Advance Reader's Copy, 1997

"Time and chance
happeneth to them all."
-- Ecclesiastes 9:11  

Related material:

1. The previous entry, on
  Copenhagen physicist
Aage Bohr, and      
2. Notes from this journal
 from Bohr's birthday,
 June 19th, through  
        Midsummer Night, 2007...
 including notes on   
  Faust in Copenhagen
   3. Walpurgisnacht 2008 and
 Walpurgisnacht 2009

Friday, September 11, 2009  2:56 AM

Annals of Aesthetics:

 in memory of
physicist Aage Bohr,
who died at 87 on
Tuesday, Sept. 8, 2009

Swarthmore physics honors thesis, 136 pp., 2007--


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

Footnote 55, page 114:

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

Interpolated figure
from Log24:

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

Cf. August 5, 2009.

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

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

Tuesday, September 8, 2009  12:25 PM

Back-to-School Special:

Magic Box  
Box containing Froebel's Third Gift-- The Eightfold Cube
 Continued from Dec. 7, 2008,
and from yesterday.


Passages from a classic story:

... he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads....


"Your mind has been conditioned to Euclid," Holloway said. "So this-- thing-- bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."

"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child-- especially a baby-- might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

"Hardening of the thought-arteries," Jane interjected.

Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only--"

"Well, look. Let's suppose there are two kinds of geometry-- we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid--"

"Poor kid," Jane said.

Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra-- they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic--"

"Blocks? What kind?"

Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."

-- "Mimsy Were the Borogoves," Lewis Padgett, 1943

Padgett (pseudonym of a husband-and-wife writing team) says that alphabet blocks are the intuitive "basis of Euclid." Au contraire; they are the basis of Gutenberg.

For the intuitive basis of one type of non-Euclidean* geometry-- finite geometry over the two-element Galois field-- see the work of...

Friedrich Froebel
 (1782-1852), who
 invented kindergarten.

His "third gift" --
Froebel's Third Gift-- The Eightfold Cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring

Go figure.

* i.e., other than Euclidean

Monday, September 7, 2009  12:00 AM

Midnight in the Garden, continued:

Magic Boxes

"Somehow it seems to fill my head with ideas-- only I don't exactly know what they are!.... Let's have a look at the garden first!"

-- A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part-- but not the "ideas" part-- was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

 on the passage:

Part I:  "The Magic Box," shown on Turner Classic Movies earlier tonight

Part II: "Mimsy Were the Borogoves," a classic science fiction story:
"... he lifted a square, transparent crystal block, small enough to cup in his palm-- much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example-- They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."
Part III:  A Crystal Block --

Cube, 4x4x4

Four coloring pencils, of four different colors

Image of pencils is by
Diane Robertson Design.

Related material:
"A Four-Color Theorem."

Part IV:

David Carradine displays a yellow book-- the Princeton I Ching.

"Click on the Yellow Book."

Sunday, September 6, 2009  11:07 AM


Back at  
Harvard Yard...

(Click images to enlarge.)

H-Bomb Pioneer Dies

Harvard's New Line

Related material:
Aug. 31 and Sept. 1

Sunday, September 6, 2009  11:00 AM

Today's Sermon:

Oscar Speech

(continued from
February 25th, 2007)

Nicolas Cage as Ghost Rider

 "From wrong to wrong the exasperated spirit   
     Proceeds, unless restored by that refining fire     
     Where you must move in measure, like a dancer."

-- Four Quartets

Saturday, September 5, 2009  10:31 PM

Annals of Aesthetics:

For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

Background --

Part I:

Sophists (August 20th)

Part II:


Escher's 'Verbum'

Escher's Verbum

Solomon's Cube

Part III:

From August 25th --

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.

Friday, September 4, 2009  2:02 PM


Closing the Circle

Continued from Monday

"This is a chapel 
 of mischance;
ill luck betide it, 'tis
the cursedest kirk
  that ever I came in!"

Philip Kennicott on
Kirk Varnedoe in
The Washington Post:

"Varnedoe's lectures were
ultimately about faith,
about his faith in
the power of abstraction,
 and abstraction as a kind of
    anti-religious faith in itself...."

Kennicott's remarks were
 on Sunday, May 18, 2003.
They were subtitled
"Closing the Circle
on Abstract Art."

Also on Sunday, May 18, 2003:

"Will the circle be unbroken?
  As if some southern congregation
  is praying we will come to understand."

Princeton University Press

Empty canvas on cover of Varnedoe's 'Pictures of Nothing'

See also

  Giordano Bruno --

Parmiggiani's Bruno: empty canvas with sculpture of Durer's solid

Dürer's Melencolia I --

Durer, Melencolia I

and Log24 entries
of May 19-22, 2009,
ending with
    "Steiner System" --

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design

George Steiner on chess
(see yesterday morning):

"There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything-- marriage, mortgages, careers, the Russian Revolution-- in order to spend their days and nights moving little carved objects up and down a quadrate board."

Steiner continues...

"Allegoric associations of death with chess are perennial...."

Yes, they are.

Log24, April 7, 2003:

April is Math Awareness Month.
This year's theme is "mathematics and art."

Mathematics and Art: Totentanz from Seventh Seal

Cf. both of yesterday's entries.

Thursday, September 3, 2009  2:02 PM

Painting the Mystical:

White Space

"White space should not be considered merely 'blank' space-- it is an important element of design which enables the objects in it to exist at all." --Wikipedia

Related material (or non-material)--

White space resulting from a recent lack of ad sales in the New York Times obituaries section leads to the following composition--

White Space
with Voices

Click on images to enlarge.

Thursday, September 3, 2009  11:07 AM

Annals of Aesthetics:

Autistic Enchantment
"Music and mathematics are among the pre-eminent wonders of the race. Levi-Strauss sees in the invention of melody 'a key to the supreme mystery' of man-- a clue, could we but follow it, to the singular structure and genius of the species. The power of mathematics to devise actions for reasons as subtle, witty, manifold as any offered by sensory experience and to move forward in an endless unfolding of self-creating life is one of the strange, deep marks man leaves on the world. Chess, on the other hand, is a game in which thirty-two bits of ivory, horn, wood, metal, or (in stalags) sawdust stuck together with shoe polish, are pushed around on sixty-four alternately coloured squares. To the addict, such a description is blasphemy. The origins of chess are shrouded in mists of controversy, but unquestionably this very ancient, trivial pastime has seemed to many exceptionally intelligent human beings of many races and centuries to constitute a reality, a focus for the emotions, as substantial as, often more substantial than, reality itself. Cards can come to mean the same absolute. But their magnetism is impure. A mania for whist or poker hooks into the obvious, universal magic of money. The financial element in chess, where it exists at all, has always been small or accidental.

To a true chess player, the pushing about of thirty-two counters on 8x8 squares is an end in itself, a whole world next to which that of a mere biological or political or social life seems messy, stale, and contingent. Even the patzer, the wretched amateur who charges out with his knight pawn when the opponent’s bishop decamps to R4, feels this daemonic spell. There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything-- marriage, mortgages, careers, the Russian Revolution-- in order to spend their days and nights moving little carved objects up and down a quadrate board. At the sight of a set, even the tawdriest of plastic pocket sets, one’s fingers arch and a coldness as in a light sleep steals over one’s spine. Not for gain, not for knowledge or reknown, but in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra."

-- George Steiner in "A Death of Kings," The New Yorker, issue dated September 7, 1968, page 133

"Examples are the stained-glass windows of knowledge." --Nabokov
Quaternion rotations in a finite geometry
Click above images for some context.

See also:

Log24 entries of May 30, 2006, as well as "For John Cramer's daughter Kathryn"-- August 27, 2009-- and related material at Wikipedia (where Kathryn is known as "Pleasantville").

Wednesday, September 2, 2009  9:48 PM

Thanks, Janet:

Outside the Box

Living Outside the Box-- Janet Maslin on Tim Page

Makes sense to me.

Wednesday, September 2, 2009  6:00 PM

Chronicles of Boston:

Zoo Story

Boston Herald
 this afternoon:

Christopher, The Lion of Boston
Photo by Lisa Hornak (file)

Christopher the lion
 was 'secured'
 at Franklin Park Zoo
when a teen toppled
 into the lion's den.

You can't make this stuff up.

Wednesday, September 2, 2009  11:09 AM

Soul of the Party, continued:

Back to School

Canto I:

NY Times-- 'Soul' of a Party Is Memorialized

Canto II:

Friday, August 28, 2009,
in this journal

Annals of Religion:

Rites of Passage

"Things fall apart;
   the centre cannot hold...."

Part I:

"Inside the church,  
    the grief was real...."

Canto III:

Sunday, August 30, 2009,
in The New York Times

"Things fall apart; the center cannot hold. Facebook, the online social grid, could not command loyalty forever."

-- Virginia Heffernan, "Facebook Exodus," NY Times Magazine, Sunday, August 30, 2009

Canto IV:

A Season in Purgatory, by Dominick Dunne

Click for details.
Canto V:
Dorm Room Feng Shui: 'YOU ARE HERE'

Tuesday, September 1, 2009  11:30 PM

Soul of the Party, continued:

Back to the Garden

The previous entry
dealt with an artist who died last Wednesday (August 26).

Dominick Dunne, producer of the film version of "The Boys in the Band," also died last Wednesday.

In his memory, four readings:

1. "Pilot Fish," by Hemingway

2.  Self-profile by Stephen Vider, author of "American Mystic" (see previous entry)

3. "Party Animal," Vider's essay on "The Boys in the Band" published on Sinatra's birthday, 2008

4.  Back to the Garden of Forking Paths (also on Sinatra's birthday, 2008)

Related material
from last Sunday morning:

"'Soul' of a Party Is Memorialized"
--New York Times online front page
"In the Details."

The following illustration from
August 16th may also be relevant:

The Expulsion from Eden

Click cover to enlarge