Log24

Friday, August 30, 2019

The Coxeter Aleph

Filed under: General — Tags: — m759 @ 4:21 AM

(Continued)

The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.

A meditation, with apologies to Coleridge:

In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.

A related video —

Beware, beware, her flashing eyes, her floating hair:

Set design —

As opposed to block design

Monday, February 20, 2012

Coxeter and the Relativity Problem

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

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"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

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of  the primitive root λ or 10 are

λ = 10,  λ2 = 21,  λ3 = 22,  λ4 = 02,
λ5 = 20,  λ6 = 12,  λ7 = 11,  λ8 = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"

http://www.log24.com/log/pix12/120220-CoxeterFig10.jpg

Coxeter's Figure 10 yields...

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The Aleph

The details:

(Click to enlarge)

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

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

Friday, January 7, 2011

Coxeter and the Aleph

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

In a nutshell —

Epigraph to "The Aleph," a 1945 story by Borges:

O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2

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

The story in book form, 1949

A 2006 biography of geometer H.S.M. Coxeter:

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

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The details:

(Click to enlarge)

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

Related material: Group Actions, 1984-2009.

Friday, July 5, 2019

The Motive for Metaphor

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

"János Bolyai was a nineteenth-century mathematician who
set the stage for the field  of non-Euclidean geometry."

Transylvania Now , October 26, 2018

 

From  Coxeter and the Relativity Problem

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

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

Wallace Stevens, "The Motive for Metaphor"

Wednesday, August 9, 2017

Implosion

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

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

See also Coxeter's Aleph.

Monday, July 17, 2017

Athens Meets Jerusalem . . .

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

At the Googleplex .

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

Plato's Diamond and the Hebrew letter Aleph —

          

and some related (if only graphically) mathematics —

Click the above image for some related purely mathematical  remarks.

Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That  cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

Sunday, October 18, 2015

Coordinatization Problem

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

There are various ways to coordinatize a 3×3 array
(the Chinese "Holy Field'). Here are some —

See  Cullinane,  Coxeter,  and  Knight tour.

Saturday, November 16, 2013

Raiders of the Lost Theorem

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

IMAGE- The 'atomic square' in Lee Sallows's article 'The Lost Theorem'

Yes. See

The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =

(1,1) (1,0) (1,2)
(0,1) (0,0) (0,2)
(2,1) (2,0) (2,2)

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

Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and  
all four broken diagonals (with arithmetic modulo 3).

For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.

Tuesday, August 13, 2013

The Story of N

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

(Continued from this morning)

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The above stylized "N," based on
an 8-cycle in the 9-element Galois field
GF(9), may also be read as
an Aleph.

Graphic designers may prefer a simpler,
bolder version:

Tuesday, May 14, 2013

Raiders of the Lost Aleph

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

See Coxeter + Aleph in this journal.

Epigraph to "The Aleph," a 1945 story by Borges:

"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2

Sunday, December 9, 2012

Eve’s Menorah

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

"Now the serpent was more subtle
than any beast of the field…."
Genesis 3:1

"“The serpent’s eyes shine
As he wraps around the vine….”
Don Henley

"Nine is a vine."
Folk rhyme

Part I

Part II

Part III

Halloween 2005

The image “http://log24.com/log/pix03/030109-gridsmall.gif” cannot be displayed, because it contains errors.

Click images for some background.

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.

Counterexample—

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

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics
http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

Monday, October 10, 2011

10/10

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

See last year's Day of the Tetraktys.

Those who prefer Hebrew to Greek may consult Coxeter and the Aleph.

See also last midnight's The Aleph as well as Saturday morning's
An Ordinary Evening in Hartford and Saturday evening's
For Whom the Bell (with material from March 20, 2011).

For connoisseurs of synchronicity, there is …

THE LAST CONCERT

Cached from http://mrpianotoday.com/tourdates.htm —
The last concert of Roger Williams — March 20, 2011 —

   March 20

"Roger Williams" In Concert,
The Legendary Piano Man!!
Roger Williams & his Band
(Sierra Ballroom)
7:30-9:00pm

Palm Desert, CA    

Background music… Theme from "Somewhere in Time"

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cube” part of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Thursday, May 19, 2011

The Aleph, the Lottery, and the Eightfold Way

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

Three links with a Borges flavor—

Related material

The 236 in yesterday evening's NY lottery may be
 viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.

A new illustration for that topic—

http://www.log24.com/log/pix11A/110519-8-3-Configuration.jpg

This shows a reconcilation of the triples described by Sloane
 in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.

Note that  the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of  Sloane's 83 configuration—

http://www.log24.com/log/pix11A/110519-SloaneDesign.jpg

The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.

Wednesday, May 18, 2011

Minimalist Icon

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

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

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

— has been identified.

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

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

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

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

Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube

http://www.log24.com/log/pix11A/110509-SpideySuperStories39Sm.jpg

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

http://www.log24.com/log/pix11A/110510-GaloisAleph.GIF

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Friday, January 7, 2011

Ayn Sof

Filed under: General — Tags: , — m759 @ 7:26 PM

(A continuation of this morning's Coxeter and the Aleph)

"You've got to pick up every stitch… Must be the season of the witch."
Donovan song at the end of Nicole Kidman's "To Die For"

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics

http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

"As is well known, the Aleph is the first letter of the Hebrew alphabet.
Its use for the strange sphere in my story may not be accidental.
For the Kabbala, the letter stands for the En Soph ,
the pure and boundless godhead; it is also said that it takes
the shape of a man pointing to both heaven and earth, in order to show
that the lower world is the map and mirror of the higher; for Cantor's
Mengenlehre , it is the symbol of transfinite numbers,
of which any part is as great as the whole."

— Borges, "The Aleph"

From WorldLingo.com

Ein Sof

Ein Soph or Ayn Sof (Hebrew  אין סוף, literally "without end", denoting "boundlessness" and/or "nothingness"), is a Kabbalistic term that usually refers to an abstract state of existence preceding God's Creation of the limited universe. This Ein Sof , typically referred to figuratively as the "light of Ein Sof " ("Or Ein Sof "), is the most fundamental emanation manifested by God. The Ein Sof  is the material basis of Creation that, when focused, restricted, and filtered through the sefirot , results in the created, dynamic universe.
….

Cultural impact

Mathematician Georg Cantor labeled different sizes of infinity using the Aleph. The smallest size of infinity is aleph-null (0), the second size is aleph-one (1), etc. One theory about why Cantor chose to use the aleph is because it is the first letter of Ein-Sof. (See Aleph number)

"Infinite Jest… now stands as the principal contender
for what serious literature can aspire to
in the late twentieth and early twenty-first centuries."

All Things Shining, a work of pop philosophy published January 4th

http://www.log24.com/log/pix10B/101231-AllThingsShining-Cover.jpg

"You're gonna need a bigger boat." — Roy Scheider in "Jaws"

"We're gonna need more holy water." — "Season of the Witch," a film opening tonight

See also, with respect to David Foster Wallace, infinity, nihilism,
and the above reading of "Ayn Sof" as "nothingness,"
the quotations compiled as "Is Nothing Sacred?"

Tuesday, March 16, 2010

Variations on a Theme

Filed under: General — Tags: — m759 @ 2:29 PM

Today's previous entry was "Gameplayers of the Academy."

More on this theme–

David Corfield in the March 2010
European Mathematical Society newsletter

    "Staying on the theme of games, the mathematician
Alexandre Borovik* once told me he thinks of mathematics
as a Massively-Multiplayer Online Role-Playing Game. If
so, it would show up very clearly the difference between
internal and external viewpoints. Inside the game people
are asking each other whether they were right about
something they encountered in it– 'When you entered
the dungeon did you see that dragon in the fireplace or
did I imagine it?' But someone observing them from the
outside wants to shout: 'You’re not dealing with anything
real. You’ve just got a silly virtual reality helmet on.' External
nominalists say the same thing, if more politely, to
mathematical practitioners. But in an important way the
analogy breaks down. Even if the players interact with
the game to change its functioning in unforeseen ways,
there were the original programmers who set the bounds
for what is possible by the choices they made. When they
release the next version of the game they will have made
changes to allow new things to happen. In the case of
mathematics, it’s the players themselves who make these
choices. There’s no further layer outside.
    What can we do then instead to pin down internal reality?"

*See previous references to Borovik in this journal.

Related material:

The Diamond Theory vs. the Story Theory of Truth,

Infantilizing the Audience, and

It's Still the Same Old Story…God of War III

Thursday, February 18, 2010

Theories: An Outline

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)

Sunday, February 14, 2010

Sunday School

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

"Simplify, simplify." — Henry David Thoreau

"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."

Alexandre Borovik, 2010 (See previous entry.)

Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

A Simple Reflection Group
of Order 168
.

Background:

Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups

15.1 (p. 114): Every finite reflection group is a Coxeter group.

15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.

Consider in this context the above simple reflection group of order 168.

(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General,Geometry — m759 @ 6:30 PM
Hall of Mirrors

Epigraph to
Deploying the Glass Bead Game, Part II,”
by Robert de Marrais:

“For a complete logical argument,”
Arthur began
with admirable solemnity,
“we need two prim Misses –”
“Of course!” she interrupted.
“I remember that word now.
And they produce — ?”
“A Delusion,” said Arthur.

— Lewis Carroll,
Sylvie and Bruno

Prim Miss 1:

Erin O’Connor’s weblog
“Critical Mass” on May 24:

Roger Rosenblatt’s Beet [Ecco hardcover, Jan. 29, 2008] is the latest addition to the noble sub-genre of campus fiction….

Curricular questions and the behavior of committees are at once dry as dust subjects and areas ripe for sarcastic send-up– not least because, as dull as they are, they are really both quite vital to the credibility and viability of higher education.

Here’s an excerpt from the first meeting, in which committee members propose their personal plans for a new, improved curriculum:

“… Once the students really got into playing with toy soldiers, they would understand history with hands-on excitement.”

To demonstrate his idea, he’d brought along a shoe box full of toy doughboys and grenadiers, and was about to reenact the Battle of Verdun on the committee table when Heilbrun stayed his hand. “We get it,” he said.

“That’s quite interesting, Molton,” said Booth [a chemist]. “But is it rigorous enough?”

At the mention of the word, everyone, save Peace, sat up straight.

“Rigor is so important,” said Kettlegorf.

“We must have rigor,” said Booth.

“You may be sure,” said the offended Kramer. “I never would propose anything lacking rigor.”

Smythe inhaled and looked at the ceiling. “I think I may have something of interest,” he said, as if he were at a poker game and was about to disclose a royal flush. “My proposal is called ‘Icons of Taste.’ It would consist of a galaxy of courses affixed to several departments consisting of lectures on examples of music, art, architecture, literature, and other cultural areas a student needed to indicate that he or she was sophisticated.”

“Why would a student want to do that?” asked Booth.

“Perhaps sophistication is not a problem for chemists,” said Smythe. Lipman tittered.

“What’s the subject matter?” asked Heilbrun. “Would it have rigor?”

“Of course it would have rigor. Yet it would also attract those additional students Bollovate is talking about.” Smythe inhaled again. “The material would be carefully selected,” he said. “One would need to pick out cultural icons the students were likely to bring up in conversation for the rest of their lives, so that when they spoke, others would recognize their taste as being exquisite yet eclectic and unpredictable.”

“You mean Rembrandt?” said Kramer.

Smythe smiled with weary contempt. “No, I do not mean Rembrandt. I don’t mean Beethoven or Shakespeare, either, unless something iconic has emerged about them to justify their more general appeal.”

“You mean, if they appeared on posters,” said Lipman.

“That’s it, precisely.”

Lipman blushed with pride.

“The subject matter would be fairly easy to amass,” Smythe said. “We could all make up a list off the top of our heads. Einstein–who does have a poster.” He nodded to the ecstatic Lipman. “Auden, for the same reason. Students would need to be able to quote ‘September 1939[ or at least the last lines. And it would be good to teach ‘Musee des Beaux Arts’ as well, which is off the beaten path, but not garishly. Mahler certainly. But Cole Porter too. And Sondheim, I think. Goya. Warhol, it goes without saying, Stephen Hawking, Kurosawa, Bergman, Bette Davis. They’d have to come up with some lines from Dark Victory, or better still, Jezebel. La Dolce Vita. Casablanca. King of Hearts. And Orson, naturally. Citizen Kane, I suppose, though personally I prefer F for Fake.”

“Judy!” cried Heilbrun.

“Yes, Judy too. But not ‘Over the Rainbow.’ It would be more impressive for them to do ‘The Trolley Song,’ don’t you think?” Kettlegorf hummed the intro.

Guernica,” said Kramer. “Robert Capa.” Eight-limbed asterisk

“Edward R. Murrow,” said Lipman.

“No! Don’t be ridiculous!” said Smythe, ending Lipman’s brief foray into the world of respectable thought.

“Marilyn Monroe!” said Kettlegorf.

“Absolutely!” said Smythe, clapping to indicate his approval.

“And the Brooklyn Bridge,” said Booth, catching on. “And the Chrysler Building.”

“Maybe,” said Smythe. “But I wonder if the Chrysler Building isn’t becoming something of a cliche.”

Peace had had enough. “And you want students to nail this stuff so they’ll do well at cocktail parties?”

Smythe sniffed criticism, always a tetchy moment for him. “You make it sound so superficial,” he said.

Prim Miss 2:

Siri Hustvedt speaks at Adelaide Writers’ Week– a story dated March 24, 2008

“I have come to think of my books as echo chambers or halls of mirrors in which themes, ideas, associations continually reflect and reverberate inside a text. There is always point and counterpoint, to use a musical illustration. There is always repetition with difference.”

A Delusion:

Exercise — Identify in the following article the sentence that one might (by unfairly taking it out of context) argue is a delusion.

(Hint: See Reflection Groups in Finite Geometry.)

A. V. Borovik, 'Maroids and Coxeter Groups'

Why Borovik’s Figure 4
is included above:

Euclid, Peirce, L’Engle:
No Royal Roads.

For more on Prim Miss 2
and deploying
the Glass Bead Game,
see the previous entry.

The image “http://www.log24.com/log/images/asterisk8.gif” cannot be displayed, because it contains errors. And now, perhaps, his brother Cornell Capa, who died Friday.

 Related material: Log24 on March 24– Death and the Apple Tree— with an excerpt from
George MacDonald, and an essay by David L. Neuhouser mentioning the influence of MacDonald on Lewis Carroll– Lewis Carroll: Author, Mathematician, and Christian (pdf).

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