Thursday, November 19, 2020


Filed under: General — m759 @ 10:28 AM

Hexagram 15:

Click on the hexagram
for some related posts.

Tuesday, August 15, 2017

Every Picture Tells a Story

Filed under: General,Geometry — m759 @ 11:48 AM

Hexagram 15:

See also remarks today by David Brooks at The New York Times .

Saturday, March 12, 2016


Filed under: General — Tags: , — m759 @ 1:00 AM

Patrick Hodgkinson, a British architect, reportedly died at 85 on 
February 21, 2016. From his March 4 obituary in the Telegraph

Before Brunswick, came Harvey Court for Gonville and Caius College, Cambridge. Colin St John Wilson, exLCC, his senior in the Martin studio, had done a scheme with four freestanding ranges in concrete. Hodgkinson radically transformed this at short notice into the final version presented to the College, a tight, connected square finished in local brick with a stepped section and impressive close-spaced brick columns on the exterior faces where the section overhung.

Never afflicted by modesty, Hodgkinson called it “designed to a brick-perfect, three-dimensional grid clear of ugly moments: the builders enjoyed making it”. It was attributed to Martin, Wilson and Hodgkinson jointly, but Hodgkinson felt that his contribution was under-appreciated, and again with the Law Library at Oxford, normally credited to Martin and Wilson. The theory of compact medium-rise courtyard forms derived from the Harvey Court design became central to Martin’s research programme at Cambridge in the 1960s; Hodgkinson felt that he deserved more credit for this too.

Saturday, August 2, 2008

Saturday August 2, 2008

Filed under: General,Geometry — m759 @ 6:23 AM

There is an article in today’s Telegraph on mathematician Simon Phillips Norton– co-author, with John Horton Conway, of the rather famous paper “Monstrous Moonshine” (Bull. London Math. Soc. 11, 308–339, 1979).
“Simon studies one of the most complicated groups of all: the Monster. He is, still, the world expert on it ….

Simon tells me he has a quasi-religious faith in the Monster. One day, he says, … the Monster will expose the structure of the universe.

… although Simon says he is keen for me to write a book about him and his work on the Monster and his obsession with buses, he doesn’t like talking, has no sense of anecdotes or extended conversation, and can’t remember (or never paid any attention to) 90 per cent of the things I want him to tell me about in his past. It is not modesty. Simon is not modest or immodest: he just has no self-curiosity. To Simon, Simon is a collection of disparate facts and no interpretative glue. He is a man without adjectives. His speech is made up almost entirely of short bursts of grunts and nouns.

This is the main reason why we spent three weeks together …. I needed to find a way to make him prattle.”

Those in search of prattle and interpretive glue should consult Anthony Judge’s essay “Potential Psychosocial Significance of Monstrous Moonshine: An Exceptional Form of Symmetry as a Rosetta Stone for Cognitive Frameworks.”  This was cited here in Thursday’s entry “Symmetry in Review.”  (That entry is just a list of items related in part by synchronicity, in part by mathematical content. The list, while meaningful to me and perhaps a few others, is also lacking in prattle and interpretive glue.)

Those in search of knowledge, rather than glue and prattle, should consult Symmetry and the Monster, by Mark Ronan.  If they have a good undergraduate education in mathematics, Terry Gannon‘s survey paper “Monstrous Moonshine: The First Twenty-Five Years” (pdf) and book– Moonshine Beyond the Monster— may also be of interest.

Monday, June 19, 2006

Monday June 19, 2006

Filed under: General — m759 @ 4:00 PM

A Reply to John Updike

See Updike on digitized snippets.

The following four snippets were pirated from the end of MathPages Quotations, compiled by Kevin Brown.

They are of synchronistic interest in view of the previous two Log24 entries, which referred (implicitly) to a Poe story and (explicitly) to Pascal.

"That is another of your odd notions,"
said the Prefect, who had the fashion
of calling everything 'odd' that was
beyond his comprehension, and thus
lived amid an absolute legion of 'oddities.'
Edgar Allan Poe

I knew when seven justices could not
take up a quarrel, but when the parties
were met themselves, one of them
thought but of an If, as, 'If you said so,
then I said so'; and they shook hands
and swore brothers. Your If is the only
peacemaker; much virtue in If.

I have made this letter longer than usual
because I lack the time to make it shorter.
Blaise Pascal

S'io credessi che mia risposta fosse
a persona che mai tornasse al mondo,
questa fiamma staria senza piu scosse.
Ma per cio che giammai di questo fondo
non torno vivo alcun, s'i'odo il vero,
senza tema d'infamia ti rispondo.
Dante, 1302

For translations of the Dante (including one by Dorothy Sayers), see everything2.com.

An anonymous author there notes that Dante describes a flame in which is encased a damned soul. The flame vibrates as the soul speaks:

If I thought that I were making
Answer to one that might return to view
The world, this flame should evermore
cease shaking.

But since from this abyss, if I hear true,
None ever came alive, I have no fear
Of infamy, but give thee answer due.

-- Dante, Inferno, Canto 27, lines 61-66,
translated by Dorothy Sayers

Updike says,

“Yes, there is a ton of information on the web but much of it is grievously inaccurate, unedited, unattributed and juvenile. The electronic marvels that abound around us serve, I have the impression, to inflame what is most informally and non-critically human about us. Our computer screens stare back at us with a kind of giant, instant aw-shucks, disarming in its modesty.”

Note Updike’s use of “inflame.”

For an aw-shucks version of “what is most informally and non-critically human about us,” as well as a theological flame, see both the previous entry and the above report from Hell.

Note that the web serves also to correct material that is inaccurate, unedited, unattributed, and juvenile. For examples, see Mathematics and Narrative. The combination of today’s entry for Pascal’s birthday with that web page serves both to light one candle and to curse the darkness.

Monday, December 12, 2005

Monday December 12, 2005

Filed under: General — m759 @ 12:06 PM
Modestly Yours

The image “http://www.log24.com/log/pix05B/051212-Cash.jpg” cannot be displayed, because it contains errors.
American IV:
The Man Comes Around

The virgins are all
trimming their wicks.

Johnny Cash   

From a Dec. 9 Mona Charen column promoting modesty:

Modestlyyours.net is an antidote to the vulgarity that is shoved in our faces from magazine covers….”

Related material
(click on covers
for details):

“For Jennifer Connelly
on Her Birthday,”

The image “http://www.log24.com/log/pix05B/051212-ConnellyThumb.jpg” cannot be displayed, because it contains errors.

and “For Rita Moreno
on Her Birthday.”

The image “http://www.log24.com/log/pix05B/051211-Rita21Thumb.jpg” cannot be displayed, because it contains errors.

For Frank Sinatra
on His Birthday:

The image “http://www.log24.com/log/pix05B/051212-Muse.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05B/051212-Crop.jpg” cannot be displayed, because it contains errors.

Modesty, my ass!

Monday, September 20, 2004

Monday September 20, 2004

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

Pi continued:

(see 9/15/04)


Renegade mathematician Max Cohen (Sean Gullette, left) and the leader of the Kabbalah sect, Lenny Meyer (Ben Shenkman) have a chance encounter on a Chinatown street corner.

The Magic Schmuck

"Confucius is said to have received only one inappropriate answer, i.e., hexagram 22, GRACE — a thoroughly aesthetic hexagram. This is reminiscent of the advice given to Socrates by his daemon — 'You ought to make more music' — whereupon Socrates took to playing the flute. Confucius and Socrates compete for first place as far as reasonableness and a pedagogic attitude to life are concerned; but it is unlikely that either of them occupied himself with 'lending grace to the beard on his chin,' as the second line of this hexagram advises. Unfortunately, reason and pedagogy often lack charm and grace, and so the oracle may not have been wrong after all."

— Carl Jung, Foreword to the I Ching 

Yesterday, class, in keeping with our morning German lesson, our evening (5:01:22 PM ET) entry was Hexagram 22, Pi (pronounced "bee"). The Chinese term pi may be translated in various ways… As ornament, as adornment, or as in a German web page:

I-Ching 22 Pi Der Schmuck

The Wilhelm translation of pi is "grace."  This suggests we examine yesterday's evening lottery number in the State of Grace, Pennsylvania:


As kabbalists know, there are many ways of interpreting numbers.  In keeping with the viewpoint of Ecclesiastes — "time and chance happeneth" — let us interpret this instance of chance as an instance of time… namely, 4/08.  Striving for consistency in our meditations, let us examine the lessons for…

4/08 2003 — Death's Dream Kingdom

and 4/08 2004 — Triple Crown

From the former:

"When smashing monuments, save the pedestals; they always come in handy."

Stanislaw J. Lec

From the latter: 

"The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated…. Philosophical ambition and formal modesty still constitute Minimalism's bottom line."

Michael Kimmelman

In keeping with the above, from
this year's Log24.net
Rosh Hashanah service

A Minimalist


For a poetic interpretation
of this symbol, see
Hexagram 20,
Contemplation (View).

Thursday, April 8, 2004

Thursday April 8, 2004

Filed under: General — m759 @ 12:00 AM

Triple Crown

“The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated…. Philosophical ambition and formal modesty still constitute Minimalism’s bottom line.”

Michael Kimmelman, April 2, 2004 


From Hans Reichenbach‘s

The Rise of Scientific Philosophy:

Ch. 18 – The Old and the New Philosophy

“The speculative philosophers allotted to art a dignified position by putting art on a par with science and morality: truth, beauty and the good were for them the triple crown of human searching and longing.”

Ch. 15 – Interlude: Hamlet’s Soliloquy

“I have good evidence.  The ghost was very conclusive in his arguments.  But he is only a ghost.  Does he exist?  I could not very well ask him.  Maybe I dreamed him.  But there is other evidence….

It is really a good idea: that show I shall put on.  It will be a crucial experiment.  If they murdered him they will be unable to hide their emotions.  That is good psychology.  If the test is positive I shall know the whole story for certain.  See what I mean?  There are more things in heaven and earth than are dreamt of in your philosophy, my dear logician.
    I shall know it for certain?  I see your ironical smile.  There is no certainty….
   There I am, the eternal Hamlet.  What does it help me to ask the logician….?  His advice confirms my doubt rather than giving me the courage I need for my action.  One has to have more courage than Hamlet to be always guided by logic.”


On this Holy Thursday, the day of Christ’s Last Supper, let us reflect on Quine’s very pertinent question in Quiddities (under “Communication”):

“What transubstantiation?”

“It is easiest to tell what transubstantiation is by saying this: little children should be taught about it as early as possible. Not of course using the word…because it is not a little child’s word. But the thing can be taught… by whispering…”Look! Look what the priest is doing…He’s saying Jesus’ words that change the bread into Jesus’ body. Now he’s lifting it up. Look!”

From “On Transubstantiation” by Gertrude Elizabeth Margaret Anscombe, Collected Philosophical Papers, V.III: Ethics, Religion, and Politics, 1981, Univ. of Minnesota Press, as quoted in the weblog of William Luse, Sept, 28, 2003

A perhaps more credible instance of transubstantiation may be found in this account of Anscombe on the Feast of Corpus Christi:

“In her first year at Oxford, she converted to Catholicism. In 1938, after mass at Blackfriars on the Feast of Corpus Christi, she met Peter Geach, a young man three years her senior who was also a recent convert to Catholicism. Like her, Geach was destined to achieve eminence in philosophy, but philosophy played no role in bringing about the romance that blossomed. Smitten by Miss Anscombe’s beauty and voice, Geach immediately inquired of mutual friends whether she was ‘reliably Catholic.’ Upon learning that she was, he pursued her and, swiftly, their hearts were entangled.”

— John M. Dolan, Living the Truth

Christ plays in ten thousand places,
Lovely in limbs, and
    lovely in eyes not his
To the Father through
    the features of men’s faces.

— Gerard Manley Hopkins

Concluding reflections for Holy Thursday:

Truth, Beauty, and The Good

Art is magic delivered from
the lie of being truth.
 — Theodor Adorno, Minima moralia,
London, New Left Books, 1974, p. 222
(First published in German in 1951.)

The director, Carol Reed, makes…
 impeccable use of the beauty of black….
V. B. Daniel on The Third Man 

I see your ironical smile.
Hans Reichenbach (see above)

Adorno, The Third Man, and Reichenbach
are illustrated below (l. to r.) above the names of cities with which they are associated. 


In keeping with our transubstantiation theme, these three cities may be regarded as illustrating the remarks of Jimmy Buffett

on culinary theology.

Monday, April 5, 2004

Monday April 5, 2004

Filed under: General,Geometry — Tags: , — m759 @ 4:03 AM

Ideas and Art

Motto of
Plato's Academy


From Minimalist Fantasies,
by Roger Kimball, May 2003:

All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
—Frank Stella, 1966

Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
— Clement Greenberg, 1967

The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
— Michael Kimmelman, 2003

There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics.
— Roger Kimball, 2003



From How Not Much Is a Whole World,
by Michael Kimmelman, April 2, 2004

Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it

"William Clark,"
by Patricia Johanson, 1967

should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it.


The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line.

If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence….

That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one.


I maintain that of course
we can see ideas.

Example: the idea of
invariant structure.

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235.
    Pergamon Press Ltd., 1978

For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.

Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)…  That is to say, they are not lost in translation.  (See previous entry.)

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