Thursday, November 5, 2015

The Monster

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

In memory of Princeton mathematician John Nash

"For the past six years all over the world 
experts in the branch of abstract algebra
called group theory have been struggling
to capture a group known as the monster."

—Martin Gardner, Scientific American ,  June 1980

"When the Hawkline Monster moved to get a better view
of what was happening, the shadow, after having checked
all the possibilities of light, had discovered a way that it
could shift itself in front of the monster, so that the monster
at this crucial time would be blinded by darkness for a few
seconds, did so, causing confusion to befall the monster.

This was all that the shadow could do and it hoped that this
would give Greer and Cameron the edge they would need
to destroy the Hawkline Monster using whatever plan they
had come up with, for it seemed that they must have a plan
if they were to have any chance at all with the monster and
they did not seem like fools.

When Cameron yelled at Greer, the shadow interpreted this
as the time to move and did so. It obscured the vision of the
Hawkline Monster for a few seconds, knowing full well that if
the monster were destroyed it would be destroyed, too, but
death was better than going on living like this, being a part of
this evil."

— Richard Brautigan, The Hawkline Monster , 1974

From the post For Scientific Witch Hunters of October 30,
an illustration from The Boston Globe —

From the post Colorful Story (All Souls' Day),  
an Illustration from Google Book Search —

Earlier in Brautigan's tale

" Everybody started to leave the parlor to go downstairs
and pour out the Hawkline Monster but just as
they reached the door and one of the Hawkline women
had her hand on the knob, Cameron said, 'Hold it for a
second. I want to get myself a little whiskey.' "

Friday, July 25, 2014

Magic for Jews

Filed under: General — Tags: — m759 @ 8:00 PM

(Continued from April 8, 2013.)

See Two Blocks Short of a Design (May 5, 2011).

Monday, April 8, 2013

Magic for Jews

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

A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.

Here is such a look (click to enlarge):

I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram  is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)

The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not  be encouraged to waste their time on this topic.

For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems

Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .

Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.

Related material from today's Daily Princetonian :

See also Nash + Princeton in this journal.

Saturday, March 16, 2013


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

For readers of The Daily Princetonian :

IMAGE- 4x4 array in 'Ancient Jewels' puzzle

(From a site advertised in the
Princetonian  on March 11, 2013)

For readers of The Harvard Crimson :

IMAGE- Harvard Crimson ad, Easter Sunday, 2008: 'Finite projective geometry as a graphic grammar of abstract design'

For some background, see Crimson Easter Egg and the Diamond 16 Puzzle.

For some (very loosely) related narrative, see Crosswicks in this journal
and the Crosswicks Curse  in a new novel by Joyce Carol Oates.

"There is  such a thing as a tesseract."
— Crosswicks author Madeleine L'Engle

Friday, March 15, 2013

Moran and Molloy

Filed under: General — m759 @ 11:00 AM


IMAGE- Story on jazz pianist Jason Moran in The New Yorker of March 11, 2013


"I lived in the garden. I have spoken of a voice
telling me things. I was getting to know it better
now, to understand what it wanted. It did not
use the words that Moran had been taught
when he was little and that he in his turn had
taught to his little one. So that at first I did not
know what it wanted. But in the end I understood
this language. I understood it, I understood it,
all wrong perhaps. That is not what matters.
It told me to write the report. Does this mean
I am freer now than I was? I do not know.
I shall learn. Then I went back into the house
and wrote, It is midnight. The rain is beating on
the windows. It was not midnight. It was not

Molloy , by Samuel Beckett

The above excerpts are in memory of some wordplay
in this journal on March 2, of a sneering joke in the 
Daily Princetonian  on March 11, and of a possible saint
who reportedly died around midnight on the night of
March 13-14. 

See also the morning of March 13.

Note, at the end of the Princetonian  piece, a comment
worthy of Beckett—

"These words. They've been played on."

Thursday, January 3, 2008

Thursday January 3, 2008

Filed under: General — m759 @ 1:01 PM
Context-Sensitive Theology

The Revelation Game 
New Year’s reading for
the tigers of Princeton

Two reviews from the February 2008 Notices of the American Mathematical Society:

From a review of

A Certain Ambiguity
(A Mathematical Novel)

by Gaurav Suri and Hartosh Singh Bal
Princeton University Press
Hardcover, US$27.95, 281 pages —

“From the Habermas-Lyotard debate (see [1] for an introduction) to the Sokal hoax ([4]), to recent atheist manifestos on the bestseller lists (e.g., [2]) the question of foundations for intellectual thought and especially for intellectual debate has never been more critical or urgent.”

[1] M. Bérubé, What’s Liberal about the Liberal Arts? Classroom Politics and “Bias” in Higher Education, W. W. Norton, 2006.
[2] S. Harris, Letter to a Christian Nation, Knopf, 2006.
[4] A. Sokal and P. Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, 1999.

Danny Calegari of Caltech

Also in the February Notices– a review of a book, Superior Beings: If They Exist, How Would We Know?, in which the author

“.. uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….

In the ‘Revelation Game,’ for example,
the Person (P) has two options:
1) P can believe in SB’s existence
2) P can not believe in SB’s existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself….

… [and] goals allow us to rank all the outcomes for each player from best… to worst…. The question we must answer is: what is the Nash equilibrium in this case?”

The answer is what one might expect from the American Mathematical Society:

“… the dominant strategy for both is when SB does not reveal Himself and P does not believe in His existence.”

Other strategies are, of course, possible. See last year’s entries.

See also
the life of John Nash,


for whom the above
equilibrium is named.

Friday, February 2, 2007

Friday February 2, 2007

Filed under: General — m759 @ 7:11 AM

The Night Watch

For Catholic Schools Week
(continued from last year)–

Last night’s Log24 Xanga
footprints from Poland:

Poland 2/2/07 1:29 AM
2/20/06: The Past Revisited
(with link to online text of
Many Dimensions, by Charles Williams)

Poland 2/2/07 2:38 AM
1/15/06 Inscape
(the mathematical concept, with
square and “star” diagrams)

Poland 2/2/07 3:30 AM
2/8/05 The Equation
(Russell Crowe as John Nash
with “star” diagram from a
Princeton lecture by Langlands)

Poland 2/2/07 4:31 AM
8/29/06 Hollywood Birthday
(with link to online text of
Plato on the Human Paradox,
by a Fordham Jesuit)

Poland 2/2/07 4:43 AM
8/30/06 Seven
(Harvard, the etymology of the
word “experience,” and the
Catholic funeral of a professor’s
23-year-old daughter)

Poland 2/2/07 4:56 AM
12/19/05 Quarter to Three (cont.)

(remarks on permutation groups
for the birthday of Helmut Wielandt)

Poland 2/2/07 5:03 AM
5/29/06 For JFK’s Birthday
(The Call Girls revisited)

Poland 2/2/07 5:32 AM
8/24/06 Beginnings
(Nasar in The New Yorker and
T. S. Eliot in Log24, both on the 2006
Beijing String Theory conference)

Poland 2/2/07 5:46 AM
2/22/06 In the Details
(Harvard’s president resigns,
with accompanying “rosebud”)

Sunday, August 7, 2005

Sunday August 7, 2005

Filed under: General — m759 @ 7:20 AM
Presbyterian Justice

News from today’s New York Times:

The Rev. Dr. Theodore Alexander Gill Sr., a Presbyterian theologian, a philosophy teacher, and an influential provost emeritus of John Jay College of Criminal Justice in Manhattan, died at 85 on June 10 in Princeton.  In retirement from John Jay, The Rev. Dr. Gill was theologian in residence at Nassau Presbyterian Church in Princeton.

In memory of The Rev. Dr. Gill:

Religious Symbolism at Princeton
    (on Nassau Presbyterian Church),
    (on number theory at Princeton),
For the Mad Musicians of Princeton,
     (on Schroeder and Bernstein),
Movie Date and its preceding entries
   (on Princeton’s St. John von Neumann),
Why Me?
   (for Princeton theologian Elaine Pagels),
Notes on Literary and Philosophical Puzzles
   (Princeton’s John Nash as Ya Ya Fontana), and
Go Tigers!
   (for the Princeton Evangelical Fellowship).

For a more conventional memorial, see

The image “http://www.log24.com/log/pix05B/050807-SFTS-Logo.gif” cannot be displayed, because it contains errors.

the obituary from

San Francisco Theological Seminary.

Monday, January 24, 2005

Monday January 24, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:45 PM

Old School Tie

From a review of A Beautiful Mind:

"We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "

The image “http://www.log24.com/log/pix05/050124-Tie.gif” cannot be displayed, because it contains errors.

"Three readings of diamond and box
have been extremely influential."

Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke

"Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections….

… diamonds and boxes are upper and lower adjoints of Galois connections…."

— "Modal Kleene Algebra
and Applications: A Survey"
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
See also
Galois Correspondence

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

and Log24.net, May 20, 2004:

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

Monday, November 25, 2002

Monday November 25, 2002

Filed under: General — m759 @ 11:32 AM

The Artist’s Signature

This title is taken from the final chapter of Carl Sagan’s novel Contact.

“There might be a game in which paper figures were put together to form a story, or at any rate were somehow assembled. The materials might be collected and stored in a scrap-book, full of pictures and anecdotes. The child might then take various bits from the scrap-book to put into the construction; and he might take a considerable picture because it had something in it which he wanted and he might just include the rest because it was there.”

— Ludwig Wittgenstein, Lectures and Conversations on Aesthetics, Psychology and Religious Belief

“Not games. Puzzles. Big difference. That’s a whole other matter. All art — symphonies, architecture, novels — it’s all puzzles. The fitting together of notes, the fitting together of words have by their very nature a puzzle aspect. It’s the creation of form out of chaos. And I believe in form.”

— Stephen Sondheim, in Stephen Schiff, Deconstructing Sondheim,” The New Yorker, March 8, 1993, p. 76

Vesica Piscis

Arch at
Glastonbury Tor

“All goods in this world, all beauties, all truths, are diverse and partial aspects of one unique good. Therefore they are goods which need to be ranged in order. Puzzle games are an image of this operation. Taken all together, viewed from the right point and rightly related, they make an architecture. Through this architecture the unique good, which cannot be grasped, becomes apprehensible. All architecture is a symbol of this, an image of this. The entire universe is nothing but a great metaphor.”

Simone Weil, sister of Princeton mathematician André Weil, First and Last Notebooks, p. 98

This passage from Weil is quoted in
Gateway to God,
p. 42, paperback, fourth impression,
printed in Glasgow in 1982 by
Fontana Books

“He would leave enigmatic messages on blackboards,
signed Ya Ya Fontana.”

Brian Hayes on John Nash,
The Sciences magazine, Sept.-Oct., 1998

“I have a friend who is a Chief of the Aniunkwia (Cherokee) people and I asked him the name of the Creator in which
he replied… Ya Ho Wah. This is also how it is spoken in Hebrew. In my native language it is spoken
Ya Ya*,
which is also what Moses was told
at the ‘Burning Bush’ incident.”

“Tank” (of Taino ancestry), Bronx, NY, Wednesday, April 17, 2002

From a website reviewing books published by

Master and Commander (Patrick O’Brian)”

1/17/02: NEW YORK (Variety) – Russell Crowe is negotiating to star in 20th Century Fox’s “Master and Commander,” the Peter Weir-directed adaptation of the Patrick O’Brian book series.


*For another religious interpretation of this phrase, see my note of October 4, 2002, “The Agony and the Ya-Ya.”

Wednesday, November 6, 2002

Wednesday November 6, 2002

Filed under: General — m759 @ 2:22 PM

Today's birthdays: Mike Nichols and Sally Field.

Who is Sylvia?
What is she? 


From A Beautiful Mind, by Sylvia Nasar:


Where the statue stood
Of Newton with his prism and silent face,
The marble index of a mind for ever
Voyaging through strange seas of Thought, alone.

John Forbes Nash, Jr. — mathematical genius, inventor of a theory of rational behavior, visionary of the thinking machine — had been sitting with his visitor, also a mathematician, for nearly half an hour. It was late on a weekday afternoon in the spring of 1959, and, though it was only May, uncomfortably warm. Nash was slumped in an armchair in one corner of the hospital lounge, carelessly dressed in a nylon shirt that hung limply over his unbelted trousers. His powerful frame was slack as a rag doll's, his finely molded features expressionless. He had been staring dully at a spot immediately in front of the left foot of Harvard professor George Mackey, hardly moving except to brush his long dark hair away from his forehead in a fitful, repetitive motion. His visitor sat upright, oppressed by the silence, acutely conscious that the doors to the room were locked. Mackey finally could contain himself no longer. His voice was slightly querulous, but he strained to be gentle. "How could you," began Mackey, "how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?"

Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. "Because," Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, "the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously."

What I  take seriously:

Introduction to Topology and Modern Analysis, by George F. Simmons, McGraw-Hill, New York, 1963 

An Introduction to Abstract Harmonic Analysis, by Lynn H. Loomis, Van Nostrand, Princeton, 1953

"Harmonic Analysis as the Exploitation of Symmetry — A Historical Survey," by George W. Mackey, pp. 543-698, Bulletin of the American Mathematical Society, July 1980

Walsh Functions and Their Applications, by K. G. Beauchamp, Academic Press, New York, 1975

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp, P. Simon, W. R. Wade, and J. Pal, Adam Hilger Ltd., 1990

The review, by W. R. Wade, of Walsh Series and Transforms (Golubov, Efimov, and Skvortsov, publ. by Kluwer, Netherlands, 1991) in the Bulletin of the American Mathematical Society, April 1992, pp. 348-359

Music courtesy of Franz Schubert.

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