Thursday, March 31, 2011

Women’s History Month

Filed under: General — m759 @ 3:48 PM

The Center


Anonymous Woman

Adapted from Harvard Crimson
March 2008 photo by Isabella N. Lai
at the Harvard Women's Center

See also …

Yeats –
the centre cannot hold,”

Stevens –
the center of resemblance,”

and Zelazny –
center loosens,
forms again elsewhere

Wednesday, March 30, 2011


Filed under: General — m759 @ 11:11 PM

This way, not that.

Tuesday, March 29, 2011

Diamond Star

Filed under: General,Geometry — m759 @ 4:03 PM

From last night's note on finite geometry—

"The (83, 83) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (94, 123) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point." An illustration—

This suggests a search for "diamond+star."

Design Theory

Filed under: General,Geometry — m759 @ 2:12 AM

See the new note Configurations and Squares at finitegeometry.org/sc/.

Monday, March 28, 2011

Music for Zenna

Filed under: General — m759 @ 1:00 PM

Part I:   Naturalized Epistemology

Part II:  Peter Cameron

Part III: Linda Ronstadt, Dolly Parton, and Emmylou Harris


Filed under: General — m759 @ 12:00 PM

"Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter…."

Related material: Appalachian Spring at Harvard and Reba.com.

Sunday, March 27, 2011

A Many-Sided Theory

Filed under: General,Geometry — m759 @ 5:48 PM

On this date 106 years ago…

Prefatory note from Hudson's classic Kummer's Quartic Surface ,
Cambridge University Press, 1905—

been twenty-nine years old in July of this year; educated at
St Paul's School, London, and at St John's College, Cambridge,
he obtained the highest honours in the public examinations of the
University, in 1898, 1899, 1900; was elected a Fellow of St John's
College in 1900; became a Lecturer in Mathematics at University
College, Liverpool, in 1902; was D.Sc. in the University of London
in 1903; and died, as the result of a fall while climbing in Wales,
in the early autumn of 1904….

A many-sided theory such as that of this volume is
generally to be won only by the work of many lives;
one who held so firmly the faith that the time is well spent
could ill be spared.

— H. F. Baker, 27 March 1905

For some more recent remarks related to the theory, see
Defining Configurations and its updates, March 20-27, 2011.

Sunday Dinner

Filed under: General — m759 @ 2:00 PM


Elizabeth Taylor: Gerard Manley Hopkins Fan


Filed under: General — m759 @ 3:33 AM
Crazy Arms
Ray Price
Chart position: No. 1, 1956
(written by Ralph E. Mooney)

Now blue ain't the word for the way that I feel
And a storm is brewing in this heart of mine
This ain't no crazy dream I know that it's real
You're someone else's love now, you're not mine

Crazy arms that reach to hold somebody new
But my yearning heart keeps saying you're not mine
My troubled mind knows soon to another you'll be wed
That's why I'm lonely all the time

Please take these treasured dreams I had for you and me
And take all the love I thought was mine
Someday my crazy arms will hold somebody new
But right now I'm so lonesome I could die

Crazy arms that reach to hold somebody new
But my yearning heart keeps saying you're not mine
My troubled mind knows soon to another you'll be wed
You're someone else's love now, you're not mine
Well you're someone else's love now, you're not mine


Portions of this work contain the intellectual property
of third party authors and contributors. As such,
third party authors and contributors retain all copyrights
in the individual works and reserve all rights
not specifically granted herein.

Tonight's dream

                                                                 … All her nubied
companions were asleeping with the squirrels. Their mivver,
Mrs Moonan, was off in the Fuerst quarter scrubbing the back-
steps of Number 28. Fuvver, that Skand, he was up in Norwood's
sokaparlour, eating oceans of Voking's Blemish. Nuvoletta lis-
tened as she reflected herself, though the heavenly one with his
constellatria and his emanations stood between….

* See The Black Queen.


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

"I just seemed to have more frames per second than other kids."

— Mary Karr, "Facing Altars: Poetry and Prayer"


 See also "Signs and Symbols."

Art based on a cover of Salinger's 'Nine Stories'

Friday, March 25, 2011

Combinatorial Delight

Filed under: General — m759 @ 12:00 PM

See Margaret Atwood in this journal.

This link was suggested by the phrase "combinatorial* delight" in last night's quote from Nabokov, which also appears in Douglas Glover's review essay, "Her Life Entire," in Books in Canada , Volume 17, Number 7, October 1988—

Cat's Eye  is Atwood's seventh novel. It is dense, intricate, and superb, as thematically diverse and complex as anything she has written. It is what you might expect from a writer at mid-career, mid-life: a portrait of the artist, a summation of what she knows about art and people. It is also an Atwoodian Under the Volcano , a vision of Toronto as Hell.

See also Under the Volcano  and Toronto in this journal.

"Right through hell there is a path." –Under the Volcano

* Update: Corrected on Dec. 13, 2014, to "combinational  delight."

Thursday, March 24, 2011

Signs and Symbols

Filed under: General — m759 @ 10:07 PM

                      " … I feel I understand
Existence, or at least a minute part
Of my existence, only through my art,
In terms of combinational delight;
And if my private universe scans right,
So does the verse of galaxies divine
Which I suspect is an iambic line.
I'm reasonably sure that we survive
And that my darling somewhere is alive…."

— Vladimir Nabokov, Pale Fire

The Call

Filed under: General — m759 @ 3:00 PM

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

An image from a post linked to in Tuesday's Koan for Larsson

The image “http://www.log24.com/log/pix06A/060916-Art.jpg” cannot be displayed, because it contains errors.

An image from today's New York Times  obituaries—


Wikipedia  on singer Loleatta Holloway, who died Monday—

[Her] "Like a Prayer," a Madonna cover, was a track on the Madonna tribute album Virgin Voices.

New York Times  on actress Helen Stenborg, who died Tuesday—

A Minnesotan of Swedish descent, she naturally brought to all her roles the kind of reserve that reflected her upbringing.


Wednesday, March 23, 2011

The Heroic Finger

Filed under: General — m759 @ 4:09 PM

From The New York Times  on February 27th, 1949— the day Elizabeth Taylor turned 17

"Tragedy enlightens-and it must, in that it points the heroic finger at the enemy of man's freedom."

— Arthur Miller, "Tragedy and the Common Man"

For an illustration, see the link in last Saturday's Night of the Supermoon to July 2005.

Well, she was just 17…*

Filed under: General — m759 @ 10:48 AM


* For the title, see The Dance.

Tuesday, March 22, 2011

Koan for Larsson

Filed under: General — m759 @ 4:09 PM

"On the one-ton temple bell
 a moon-moth, folded into sleep,
 sits still." — Haiku by Buson

From the day author Stieg Larsson died—

The Nine (November 9th, 2004).

See also Pandora's Box (September 16th, 2006).

Sunday, March 20, 2011


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

"a sort of… Dr. Strangelove" —Review of Point Omega





Context— From March 13— The Counter and Twenty-Four.

Saturday, March 19, 2011

Night of the Supermoon

Filed under: General — m759 @ 2:02 PM

Readings for tonight, the Night of the Supermoon

  1. Configuration
  2. Moonrise over Pyramid
  3. July 2005

Knecht Moves

Filed under: General — m759 @ 2:45 AM

In memory of David Rumelhart, who "explored the possibility of formulating a formal grammar to capture the structure of stories." (Wikipedia)

Rumelhart died on Sunday, March 13, 2011.*

So set 'em up, Joe

I've got a little schema you oughta know….

So make it one for my baby and one more for the road.

* From that date: The Counter and Twenty-Four.

Friday, March 18, 2011

Defining Configurations*

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

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.


* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.


Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—


Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Gone with the Time

Filed under: General — m759 @ 8:00 AM

A phrase from yesterday evening's post "Time Travel Poem"—

"what Joe Public is thinking."

An expert on Joe Public delivers his  opinion on time travel—


"The time is now."
Source: NY Times obituaries—


Thursday, March 17, 2011

Time Travel Poem

Filed under: General — Tags: — m759 @ 7:11 PM

From “This Week’s Hype II,” a post at Peter Woit’s physics weblog this afternoon, a comment—

TedUnger says:
March 17, 2011 at 5:34 pm

“… there’s been nothing from these CERN scientists
except some lousy boring data on physics!
They better at least give us some time travel or else!

You know that is what Joe Public is thinking.”

The commenter’s identity is not clear. Even less clear is the identity of his subject, Joe Public.

For some remarks on time travel from literature rather than science, see “Damnation Morning” in this journal.

Erin O’Connor’s St. Patrick’s Day post this morning says,

“[Roddy] Doyle’s take on the Irish struggle for independence,
A Star Called Henry , has a lovely touch of magical realism.”

Note that the remarks by Henry Baker in this morning’s post here  were dated Thursday, 11 September 1913.

Related material—

Yet they were of a different kind
The names that stilled your childish play,
They have gone about the world like wind,
But little time had they to pray
For whom the hangman’s rope was spun,
And what, God help us, could they save:
Romantic Ireland’s dead and gone,
It’s with O’Leary in the grave.

William Butler Yeats, “September 1913

Remarks on Reality

Filed under: General,Geometry — Tags: — m759 @ 9:12 AM

Conclusion of “The Place of Pure Mathematics” —

“Dogmas and philosophies, it would seem, rise and fall. But gradually accumulating throughout the ages, from the earliest dawn of history, there is a body of doctrine, a reasoned insight into the relations of exact ideas, painfully won and often tested. And this remains the main heritage of man; his little beacon of light amidst the solitudes and darknesses of infinite space; or, if you prefer, like the shout of children at play together in the cultivated valleys, which continues from generation to generation.

Yes, and continues for ever! A universe which has the potentiality of becoming thus conscious of itself is not without something of which that which we call memory is but an image. Somewhere, somehow, in ways we dream not of, when you and I have merged again into the illimitable whole, when all that is material has ceased, the faculty in which we now have some share, shall surely endure; the conceptions we now dimly struggle to grasp, the joy we have in the effort, these are but part of a greater whole. Some may fear, and some may hope, that they and theirs shall not endure for ever. But he must have studied Nature in vain who does not see that our spiritual activities are inherent in the mighty process of which we are part; who can doubt of their persistence.

And, on the intellectual side, of all that is best ascertained, and surest, and most definite, of these; of all that is oldest and most universal; of all that is most fundamental and far-reaching, of these activities, Pure Mathematics is the symbol and the sum.”

— From a 1913 address by geometry saint Henry Frederick Baker, who died on this date in 1956

The feast of another saint, Patrick, also falls on 3/17.  The date itself is related, if only by chance, to the following remark—

“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”

— From a 1940 book by the somewhat less saintly number theorist G. H. Hardy

Wednesday, March 16, 2011

Time and Chance (continued)

Filed under: General,Geometry — Tags: — m759 @ 8:25 PM

Accidental Time and Space

New York Lottery today— midday 987, evening 522.


The midday 987 may be interpreted as "…nine, eight, seven, …."—

"The countdown as we know it, 10-9-8-u.s.w.,
was invented by Fritz Lang in 1929 for
the Ufa film Die Frau im Mond . He put it into
the launch scene to heighten the suspense.
'It is another of my damned "touches,"' Fritz Lang said."

Gravity's Rainbow


The evening 522 suggests the date 5/22. From that date last year

Art Space (2:02 AM EDT)

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Space: what you damn well have to see.
– James Joyce, Ulysses

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM


Related material:
See a search for the author of
Venus on the Half-Shell .

"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow

Manhattan Project

Filed under: General — m759 @ 11:32 AM

"Hang a shining star upon the highest bough."

Hugh Martin. Martin died on March 11th, 2011.

"The shaping of a work of art means, paradoxically, preserving some space for ambiguity."

— Rebecca Newberger Goldstein in The Wall Street Journal , December 11th, 2010

From January 16th


Here are links to the source posts: December 3, 2010 and December 4, 2010.

Tuesday, March 15, 2011

Cramer’s Bridge

Filed under: General — m759 @ 8:28 PM

A comment yesterday at Peter Woit's weblog—

Glenn says:
March 14, 2011 at 8:49 pm

Perhaps John G. Cramer’s prediction will come true after all?


Of course, in that case, the proof would exist on a world-line
inaccessible to any living observer.

New York Lottery—

IMAGE- NY Lottery March 15, 2011- Midday 016, Evening 928

Related material:

From the weblog of Cramer's daughter Kathryn on Feb. 28—


For 928. see the two posts from last year's 9/28 in this journal—

Midnight at the Still Point and Brightness at Noon.

In Hoc Signo

Filed under: General — m759 @ 1:06 PM

From "Mathematicians and Poets," by Cai Tianxin, in the April 2011 AMS Notices

Gauss, “the prince of mathematics”, wrote to tell a
friend, after solving a problem (symbols of Gaussian
summation) that had been bothering him for
years, “Finally, two days ago, I succeeded—not on
account of my hard efforts, but by the grace of the
Lord. Like a sudden flash of lightning, the riddle
was solved. I am unable to say what the conducting
thread was that connected what I previously knew
with what made my success possible.”

Gauss appears also in the results of the search for the phrase "only connect"
in this morning's Ides of Art post.

See as well the Grateful Dead logo in that post, and the following ad
shown today at Secret Blogging Seminar's Oct. 11, 2008, post
"The Sign of the Gauss Sum"—

IMAGE- Google ad for 'Quality Church Signs'

The Die is Cast

Filed under: General — m759 @ 10:31 AM

Meditation for the Ides of March—

IMAGE- Grateful Dead logo based on a concept by Owsley Stanley

   Only connect.

Party On, Kid Charlemagne

Filed under: General — m759 @ 1:06 AM

Part I:    A search in this journal for Cuernavaca+garden
Part II:   This life can be very strange –Steely Dan, 1976
Part III:  Owsley Stanley Dies at 76

Midnight in the Garden (continued)

Filed under: General — m759 @ 12:00 AM

From Rough Magic , starring Russell Crowe and Bridget Fonda (1995)—

Bridget Fonda in 'Rough Magic'

Bridget brought her rabbits,
There was magic in the air…

— Adapted from "Garden Party"—

Can't please everyone, so you
Got to please yourself.

From The Chronicles of Narnia, Voyage of the Dawn Treader

"… Sometimes, perhaps, I am a little impatient, waiting for the day
when they can be governed by wisdom instead of this rough magic."

"All in good time, Coriakin," said Aslan.

"Yes, all in very good time, Sir," was the answer.

From Another Manic Monday (Feb. 21)—

We are now at the Year of the Rabbit

(Click images for sources.)

http://www.log24.com/log/pix11/110221-DarkoRabbitSm.jpg IMAGE- Scene from a film based on the old SF story 'Mimsy Were the Borogoves'

Monday, March 14, 2011

A Dante for Our Times

Filed under: General — m759 @ 7:59 PM

(Continued from this date two years ago)

"Hell is other people." —Sartre
"With a laugh track." —Cullinane

A sequel to Good Will Hunting and Hereafter

The Emory Board

Sunday, March 13, 2011


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

"Poetry never left me stranded, and as an atheist most of my life, I presumed its mojo was a highbrow, intellectual version of what religion did for those more gullible believers in my midst— dumb bunnies to a one, the faithful seemed to me, till I became one.

In the Texas oil town where I grew up, fierceness won fights, but I was thin-skinned— an unfashionably bookish kid whose brain wattage was sapped by a consuming inner life others didn’t seem to bear the burden of. I just seemed to have more frames per second than other kids."

— "Facing Altars: Poetry and Prayer," by Mary Karr

"The original movie had been slowed to a running time of twenty-four hours.
What he was watching seemed pure film, pure time.
The broad horror of the old gothic movie was subsumed in time."

Point Omega , by Don DeLillo

The Counter

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

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

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

Cover Illustration Number 1 (1976):


Cover Illustration Number 2 (1991):


   The Shrikhande Graph



This post was prompted by two remarks…

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

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

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

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

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

     –Martin Gardner in 1976 on the diamond theorem

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

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

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

Saturday, March 12, 2011

A Songwriter’s Apology

Filed under: General — m759 @ 7:59 PM

From this evening's online New York Times , a death from yesterday

Mr. Hardy’s song “St. Clare” was covered by Ms. Vega
and appears on her 2001 album “Songs in Red and Gray.”

[Lyrics here.]

See also "red and gray" and "The Eye" in this journal.

Friday, March 11, 2011

Table Talk

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

The following was suggested by a link within this evening's earlier Kane site link.

Peter J. Cameron's weblog on August 26, 2010

A Latin square  of order n  is a n × n  array with entries from the symbol set {1, 2, …, n }, such that each symbol occurs once in each row and once in each column. Now it is not hard to show that, up to permutations of the rows, columns and symbols, there are only two Latin squares of order 4:

1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3

 Some related literary remarks—

Proginoskes and Latin Squares.

See also "It was a perfectly ordinary night at Christ's high table…."

Evening Star

Filed under: General — m759 @ 7:11 PM

The following link was suggested by
this morning's Now Lens and noon's Citizen Julie

The Citizen Kane of Horror.

Related material— Damnation Morning and Punch Line.

Citizen Julie

Filed under: General — m759 @ 12:00 PM

yesterday on Julie Taymor and "Spider-Man"—

"This isn't a time for schadenfreude. Jobs are on the line, careers hang in the balance and the Fed isn't going to ride to the rescue of megamusicals as it did for Wall Street banks. But you'll forgive me for being a pessimist about the chances of an 11th hour redemption. The only way I can see this train wreck turning into an artistic success is if the investors were somehow able to resurrect Orson Welles to adapt the whole unfortunate episode into a 'Citizen Kane' sequel, the tale of an avant-garde idealist who loses her way after being enabled by heedless businessmen determined to duplicate the multibillion-dollar bonanza of 'The Lion King.'"

See also this morning's post and…


 — Errol Morris in The New York Times , March 9th

Now Lens

Filed under: General — m759 @ 10:00 AM

A Story in Pictures

Errol Morris in The New York Times  on March 9

"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]"

Now Lens


Uncertified Copy


Del Toro


Plato's Diamond


Portrait of an Artist


Meanwhile, back at the Times


Thursday, March 10, 2011

Paradigms Lost

Filed under: General,Geometry — Tags: — m759 @ 5:48 PM

(Continued from February 19)

The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—


This journal on January 19, 2011

IMAGE- A Galois cube: model of the 27-point affine 3-space

If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean  3x3x3 cube on the Kuhn cover do not  form an affine plane
in the coordinate system of the Galois  cube in the second image, but they
at least suggest  such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik  cube, are not Galois  transformations,
but they at least suggest  such transformations.

See also today's online Harvard Crimson  illustration of problems of translation
not unrelated to the problems of commensurability  discussed by Kuhn.


Wednesday, March 9, 2011

Ash Wednesday Surprise

Filed under: General — m759 @ 11:07 AM

IMAGE- Errol Morris- 'The Ashtray'- at The New York Times

 Part 3 of 5  (See also Part 1 and Part 2) begins as follows…

"Incommensurable. It is a strange word. I wondered, why  did Kuhn choose it? What was the attraction? 

Here’s one clue. At the very end of 'The Road Since Structure,' a compendium of essays on Kuhn’s work, there is an interview with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a brief account of the historical origins of his idea. Here is the relevant segment of the interview.

T. KUHN: Look, 'incommensurability' is easy.

V. KINDI: You mean in mathematics?

T. KUHN: …When I was a bright high school mathematician and beginning to learn Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was sort of a big two-volume Calculus book by, I can’t remember whom. And then I never really read it. I read the early parts of it. And early on it gives the proof of the irrationality of the square root of 2. And I thought it was beautiful. That was terribly exciting, and I learned what incommensurability was then and there. So, it was all ready for me, I mean, it was a metaphor but it got at nicely what I was after. So, that’s where I got it.

'It was all ready for me.' I thought, 'Wow.' The language was suggestive. I imagined √2 provocatively dressed, its lips rouged. But there was an unexpected surprise. The idea didn’t come from the physical sciences or philosophy or linguistics, but from mathematics ."

A footnote from Morris (no. 29)—

"Those who are familiar with the proof [of irrationality] certainly don’t want me to explain it here; likewise, those who are unfamiliar with it don’t want me to explain it here, either. There are many simple proofs in many histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline, etc., etc. Barry Mazur offers a proof in his book, 'Imagining Numbers (particularly the square root of minus fifteen),' New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, 'How Did Theaetetus Prove His Theorem?', available on Mazur’s Harvard Web site."

There may, actually, be a few who do  want the proof. They may consult the sources Morris gives, or the excellent  description by G.H. Hardy in A Mathematician's Apology , or, perhaps best of all for present purposes, the proof as described in a "sort of a big two-volume Calculus book" (perhaps the one Kuhn mentioned)…  See page 6 and page 7 of  Volume One  of Richard Courant's classic Differential and Integral Calculus  (second edition, 1937, reprinted many times through 1970, and again in a Wiley Classics Library Edition in 1988).

The Uploading

Filed under: General — m759 @ 9:00 AM

Philosophy versus Stories


The above uploading was done on December 10th, 2006.
For some context, see the Log24 posts for December 2006.

See also the German version of a nursery rhyme
that one commenter has called "morbid and horrifying"—

"Dein Vater sitzt auf der Schwelle:
  Flieg in Himmel aus der Hölle."

The rhyme suggests characters in the novel The Quest for the 36
related to the Triangle Shirtwaist Factory fire often recalled during
women's history month. It also suggests the oeuvre  of Stephen King.

“History, Stephen said, is a nightmare
from which I am trying to awake.”

Tuesday, March 8, 2011

Country Strong

Filed under: General — m759 @ 11:32 AM

For Women’s History Month —

IMAGE- Awakenings: Kate Beckinsale at Pearl Harbor and Gwyneth Paltrow on the cover of W on 9/11, 2001

See also related remarks by Admiral Yamamoto.

Monday, March 7, 2011

Point Taken

Filed under: General,Geometry — m759 @ 4:00 PM

Recommended— An essay (part 1 of 5 parts) in today's New York TImes—


I don’t want to die in
a language I can’t understand.
— Jorge Luis Borges

Comment 71

"I agree with one of the earlier commenters that this is a piece of fine literary work. And in response to some of those who have wondered 'WHAT IS THE POINT?!' of this essay, I would like to say: Must literature always answer that question for us (and as quickly and efficiently as possible)?"

For an excellent survey of the essay's historical context, see The Stanford Encyclopedia of Philosophy article

"The Incommensurability of Scientific Theories,"
First published Wed., Feb. 25, 2009,
by Eric Oberheim and Paul Hoyningen-Huene.

Related material from this journal—

Paradigms, Paradigms Lost, and a search for "mere geometry." This last includes remarks contrasting Euclid's definition of a point ("that which has no parts") with a later notion useful in finite geometry.

See also (in the spirit of The Abacus Conundrum )…

The Monolith Epiphany


(Note the Borges epigraph above.)


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

"Time it goes so fast
When you're having fun"

— "Another Manic Monday"

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

– Dante, Paradiso , XVII, 17-18

See mirando  in this journal.
       See also Time Fold.

Sunday, March 6, 2011

The Sound of Music*

Filed under: General — m759 @ 5:24 PM

For the cast and writers of Saturday Night Live ,
a word from Charlie Sheen's sponsor—

"Gracia," as in…
"una poca de gracia."


Google Search on March 6, 2011—

THE HYPE: New stuff…….pay attention!

Feb 12, 2007 You're not going very far….
after Mason's Mobile City  you want to make a right
on the next road which is Dutch Hill.

Related rather personal material—

Memories of Mason's Mobile City  in 1959,
  the music of the late Johnny Preston,
and Kristen Wiig as last night's SNL Rock-A-Billy Lady.



eBay image of Johnny Preston 45

* Update of 10:22 PM EST March 6—
A video of the SNL sketch from which the title was taken
is now available.




Filed under: General — m759 @ 11:00 AM

The Year of Magical Realism

"The non-linear story is narrated via different time frames,
a technique derived from the Argentine writer Jorge Luis Borges
(as in The Garden of Forking Paths )."

—Wikipedia on One Hundred Years of Solitude

One year ago today, in "Deconstructing Alice"—

"When you come to a fork in the road, take it." –Yogi Berra

Happy birthday to Gabriel García Márquez.

The Eye

Filed under: General — m759 @ 7:59 AM

In memory of a Brazilian writer who died at 73 on Oscar day— three symbols and a link.

http://www.log24.com/log/pix11/110301-Diamond-RedOnGray.bmp http://www.log24.com/log/pix11/110301-Diamond-RedOnWhite.bmp

Related material—

"I really have nothing to add," "Ojo de Dios," and "Red on Gray."

Saturday, March 5, 2011

Personal Link

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

A search for some background on Dmitri Tymoczko, the subject of yesterday's evening entry on music theory, shows that his name and mine once both appeared in the same web page— "This Week's Finds in Mathematical Physics (Week 234)," by John Baez, June 12, 2006 (linked to by the Wikipedia article on transformational music theory).

In that page, Baez speculates on the possibility of a connection between music theory and Mathieu groups and says—

"For a pretty explanation of M24, also try this:

Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html."

I know of no connection* between the groups I discussed there and music theory. For some background on Tymoczko's work, see the helpful survey "Exploring Musical Space," by Julian Hook (Science  magazine, 7 July 2006).

* Apart, that is, from the tesseract (see Geometry of the 4 × 4 Square) shown by Tymoczko in a 2010 lecture


This is perhaps "Chopin's tesseract" from section 8.5 of Tymoczko's new book
A Geometry of Music  (Oxford University Press, 2011).

Friday, March 4, 2011

Ageometretos Medeis Eisito*

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

Your mission, should you choose to accept it…

IMAGE- Future Bead Game Master Joseph Knecht's mission to a Benedictine monastery

See also “Mapping Music” from Harvard Magazine , Jan.-Feb. 2007—

“Life inside an orbifold is a non-Euclidean world”

— as well as the cover story “The Shape of Music” from Princeton Alumni Weekly ,
Feb. 9, 2011, and “Bead Game” + music in this  journal (click, then scroll down).
Those impressed by the phrase “non-Euclidean” may also enjoy
Non-Euclidean Blocks and Pilate Goes to Kindergarten.

The “Bead Game” + music search above includes, notably, a passage describing a
sort of non-Euclidean abacus in the classic 1943 story “Mimsy Were the Borogoves.”
For a visually related experience, see the video “Chord Geometries Demo: Chopin
on a Mobius Strip” at a music.princeton.edu web page.

* Motto of the American Mathematical Society, said to be also the motto of Plato’s Academy.

Infinite Jest

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

Two items from the August 5, 2005, anniversary
of the day Marilyn Monroe was found dead—

1.  New Chapter in the Mystery

2.  Literary Symbol —

IMAGE- Lemniscate

See also related material on Hollywood.

Oh, When the Saints…

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

In memory of John Miner

March First

NY Times obits index: Jane Russell and Peter J. Gomes

See also
Venus at St. Anne's.

Related symbols:



AMS logo—Note resemblance
to Harvard's Memorial Church.

Click on pictures for details.

This morning's LA Times —


Related remarks —

Wednesday, March 2, 2011

Labyrinth of the Line

Filed under: General,Geometry — Tags: — m759 @ 11:24 AM

“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—

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

Related material —

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.

A search in the latter for miracle octad is updated below.


This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

Tuesday, March 1, 2011

Ariadne’s Clue

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

IMAGE- The final page of 'Deathly Hallows' is 759.

Related symbolism from Plato’s Cave—

Recall that Ariadne in “Inception” is played by Ellen Page .


Show me all  the blueprints.”
— Howard Hughes, according to Hollywood

Supper at Eight (continued)

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

Review— “It was a perfectly ordinary night at Christ’s high table….”

See also this afternoon’s Other Literary Symbolism.

Other Literary Symbolism

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

NY Times obits index: Jane Russell and Peter J. Gomes

See also Venus at St. Anne’s.

Women’s History Month

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

Susanne for Suzanne

From pages 7-8 of William York Tindall’s Literary Symbolism  (Columbia U. Press, 1955)—

                                     ... According to Cassirer's Essay 
on Man, as we have seen, art is a symbolic form, parallel in respect 
of this to religion or science. Each of these forms builds up a universe 
that enables man to interpret and organize his experience; and each 
is a discovery, because a creation, of reality. Although similar in func- 
tion, the forms differ in the kind of reality built. Whereas science
builds it of facts, art builds it of feelings, intuitions of quality, and 
the other distractions of our inner life— and in their degrees so do 
myth and religion. What art, myth, and religion are, Cassirer con- 
fesses, cannot be expressed by a logical definition. 

Nevertheless, let us see what Clive Bell says about art. He calls 
it "significant form," but what that is he is unable to say. Having 
no quarrel with art as form, we may, however, question its signifi- 
cance. By significant he cannot mean important in the sense of 
having import, nor can he mean having the function of a sign; 
for to him art, lacking reference to nature, is insignificant. Since, 
however, he tells us that a work of art "expresses" the emotion of 
its creator and "provokes" an emotion in its contemplator,he seems 
to imply that his significant means expressive and provocative. The 
emotion expressed and provoked is an "aesthetic emotion," contem- 
plative, detached from all concerns of utility and from all reference. 

Attempting to explain Bell's significant form, Roger Fry, equally 
devoted to Whistler and art for art's sake, says that Flaubert's "ex- 
pression of the idea" is as near as he can get to it, but neither Flaubert 
nor Fry tells what is meant by idea. To "evoke" it, however, the artist 
creates an "expressive design" or "symbolic form," by which the 
spirit "communicates its most secret and indefinable impulses." 

Susanne Langer,who occupies a place somewhere between Fry 
and Cassirer, though nearer the latter, once said in a seminar that a 
work of art is an "unassigned syntactical symbol." Since this defini- 

End of page 7 

tion does not appear in her latest book, she may have rejected it, but 
it seems far more precise than Fry's attempt. By unassigned she prob- 
ably intends insignificant in the sense of lacking sign value or fixed 
reference; syntactical implies a form composed of parts in relation- 
ship to one another; and a symbol, according to Feeling and Form, 
is "any device whereby we are enabled to make an abstraction." Too 
austere for my taste, this account of symbol seems to need elaboration, 
which, to be sure, her book provides. For the present, however, taking 
symbol to mean an outward device for presenting an inward state, 
and taking unassigned and syntactical as I think she uses them, let 
us tentatively admire her definition of the work of art.



Oh, the red leaf looks to the hard gray stone
To each other, they know what they mean

— Suzanne Vega, “Song in Red and Gray

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