Wednesday, June 30, 2010

Field Dream

Filed under: General,Geometry — Tags: — m759 @ 10:23 AM

In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday

Image-- The Dream of the Expanded Field

“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead and try,’ he exclaimed.  ‘You’ll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”

— Hermann Hesse, The Glass Bead Game, translated by Richard and Clara Winston

“The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever.”

Jürgen Teichmann. Teichmann notes that “the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his ‘Kurzgefasster Lebenslauf,’ 1925.”

Who Wants to Be a Mathematician?

Filed under: General — Tags: — m759 @ 9:48 AM

— A followup to yesterday's note on mathematics as a post-Communist activity

From Log24 on May 20, 2008

The Dictatorship of Talent, by David Brooks in The New York Times of December 4, 2007:

“When you talk to Americans, you find that they have all these weird notions about Chinese communism. You try to tell them that China isn’t a communist country anymore. It’s got a different system: meritocratic paternalism. You joke: Imagine the Ivy League taking over the shell of the Communist Party and deciding not to change the name. Imagine the Harvard Alumni Association with an army.”

The New York Times this morning

BEIJING (AP) — China threw open the gates of its secretive Central Party School on Wednesday, offering foreign journalists a rare but carefully scripted peek at the leafy campus where the country's Communist elite are trained.

The tour is part of a drive by the Communist government to show it's becoming more open and transparent…

The tour was also part of activities marking the 89th anniversary this week of the founding of China's Communist Party.

The American Mathematical Society's top news item today

"Data collected this March by the AMS from approximately sixty mathematics departments in the U.S. shows that the number of open full-time academic positions requiring a Ph.D. in 2010 is down 57% in two years."

Party on.

Tuesday, June 29, 2010

Kind of Bleu

Filed under: General — Tags: — m759 @ 8:28 AM

Image-- Kind of Bleu-- Global Partners of Cambridge, Mass.

Click for a Global Partners story.

Related material:

"We want to promote the vitality of mathematics
by playing an increasingly active role in political affairs."

Princeton Class Notes, Jan. 27, 1999, remark by Felix E. Browder,
then president-elect of the American Mathematical Society (AMS).

See also web pages on Browder's brothers
William (also an AMS president, 1989-1990)
and Andrew and their father Earl.

Earl was General Secretary of
the Communist Party USA from 1930 to 1944.

Princeton Class Notes on the Browders— "The senior Browder 'discouraged me and my two brothers from taking an active part in politics, but strongly encouraged our intellectual interests.' That all three brothers became mathematicians– the others are Princeton professor William Browder '58 (a former president of the AMS) and Brown professor Andrew Browder– is an outcome for which Felix Browder 'can offer no rational explanation.'"

"As a trusted partner, we do more than consult and train.
We add a new dimension to our client’s thinking…."

Global Partners, Inc., of Cambridge, Mass.

Happy St. Peter’s Day…

Filed under: General — m759 @ 12:00 AM

to Antoine de Saint-Exupéry (born on this date in 1900).

Monday, June 28, 2010

Star Wars

Filed under: General — m759 @ 11:30 PM

(Continued from 8 PM Sunday)

Sunday night
    in this journal

Six-rayed asterisk from Tahoma font

The above asterisk, from the Tahoma font, suggests
a figure from "Diablo Ballet" (Jan. 21, 2003).

Shown tonight on AMC—

Star of Life on helipad at St. Peter's Charity Hospital in 'The Client'

The helipad at John Grisham's fictional
St. Peter's Charity Hospital
with the Star of Life symbol.

Byrd Song

Filed under: General — m759 @ 1:00 PM

Robert Byrd, longest-serving member of Congress, dead at 92

CNN – Dana Bash, Mike Roselli – ‎7 minutes ago‎ at Google News

By the CNN Wire Staff NEW:
Byrd: "When I am dead …
they will find West Virginia written on my heart"
Washington (CNN) — West Virginia Sen. Robert Byrd,
the self-educated son of a coal miner who became
the longest-serving member of Congress
See also Sunday at the Apollo
and the Apollo Civic Theatre
in Martinsburg, West Virginia.

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM

See David Corfield,
"The Robustness of Mathematical Entities."

This is an abstract from a paper at a conference,
"From Practice to Results in Logic and Mathematics"
(June 21st-23rd, 2010, Archives Henri Poincaré,
University of Nancy (France)).

See also Corfield's post "Inevitability in Mathematics"
at the n-Category Café today. He links to an earlier
post, "Mathematical Robustness." From that post—

…let’s see what Michiel Hazewinkel has to say
 in his paper Niceness theorems:

It appears that many important mathematical objects
(including counterexamples) are unreasonably
nice, beautiful and elegant. They tend to have
(many) more (nice) properties and extra bits
of structure than one would a priori expect….

Shall I Compare Thee

Filed under: General,Geometry — Tags: — m759 @ 11:02 AM

Margaret Soltan on a summer's-day poem by D.A. Powell

first, a congregated light, the brilliance of a meadowland in bloom
and then the image must fail, as we must fail, as we

graceless creatures that we are, unmake and befoul our beds
don’t tell me deluge.     don’t tell me heat, too damned much heat

"Specifically, your trope is the trope of every life:
 the organizing of the disparate parts of a personality
 into a self (a congregated light), blazing youth
 (a meadowland in bloom), and then the failure
 of that image, the failure of that self to sustain itself."

Alternate title for Soltan's commentary, suggested by yesterday's Portrait:

Smart Jewish Girl Fwows Up.

Midrash on Soltan—

Congregated Light

The 13 symmetry axes 
of the cube


Appalachian meadow


Wert thou my enemy, O thou my friend,
How wouldst thou worse, I wonder, than thou dost
Defeat, thwart me?


"…meadow-down is not distressed
For a rainbow footing…."

Sunday, June 27, 2010

Sunday at the Apollo

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



The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the 27-part (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

Star Wars

Filed under: General — m759 @ 8:00 PM


The above asterisk, from the Tahoma font, suggests
a figure from "Diablo Ballet" (Jan. 21, 2003)—

“At the still point,
there the dance is.”
— T. S. Eliot

Another asterisk figure,  
from Twelfth Night 2010—

Kenneth Nolad, 'Play,' 1960. Noland died on January 5, 2010.

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

Rubén Darío


Filed under: General — m759 @ 12:00 PM

Smart* Jewish Girl


Rivka Galchen

See Galchen's essay on Stevenson and Borges
on the last page of today's New York Times Book Review.

With Typewriter


Photo from Flickr.com

See also Borges's "Lottery in Babylon."

* Galchen writes on quantum theory here and here.

Bright Star (continued)

Filed under: General — m759 @ 9:00 AM

From Epiphany 2010

The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato's cave dwellers watching the shadows, without knowing what had projected them, which was the Word."

– Tom Wolfe, The Painted Word

Pennsylvania Lottery yesterday—

Saturday, June 26, 2010: Midday 846, Evening 106


Yesterday's morning post, Plato's Logos
Yesterday's evening post, Bold and Brilliant Emergence

Poem 846, Oxford Book of English Verse, 1919:
"bird-song at morning and star-shine at night"

Poem 106, Oxford Book of English Verse, 1919:
" All labourers draw home at even"

The number 106 may also be read as 1/06, the date of Epiphany.

Posts on Epiphany 2010—

9:00 AM    Epiphany Revisited
12:00 PM  Brightness at Noon
9:00 PM    The Difference

Related material—


Star and Diamond: A Tombstone for Plato

Saturday, June 26, 2010

Bold and Brilliant Emergence

Filed under: General — m759 @ 7:20 PM

"Rosemary Desjardins argues boldly and brilliantly that the Theaetetus  contains not only an answer to the question of the character of knowledge, but considerably more besides — an outline of a Platonic ontology. That ontology is neither materialist nor idealist (it is not a theory of forms), but like the twentieth century theory known as generative emergence holds that beings are particular interactive combinations of material elements. On this view, while wholes (for example, words, to use a Platonic model) may be analyzed into their elemental parts (letters), each whole has a property or quality separate from the aggregated properties of its parts."

— Stephen G. Salkever, 1991 review of The Rational Enterprise : Logos in Plato's Theaetetus  (SUNY Press, 1990)

See also "strong emergence" in this journal.

Plato’s Logos

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

“The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was ‘The Idea of the Beautiful in Plato’s Dialogues’…. My investigation traces the historical destiny of the same concept….”

* See Cassirer’s Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.

— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924

On a figure from Plato’s Meno

IMAGE- Plato's diamond and finite geometry

The above figures illustrate Husserl’s phrase  “eidetic variation”
a phrase based on Plato’s use of eidos, a word
closely related to the word “idea” in Panofsky’s title.

For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his “The Concept of Group and the Theory of Perception.”

Sketch of some further remarks—


The Waterfield question in the sketch above
is from his edition of Plato’s Theaetetus
(Penguin Classics, 1987).

The “design theory” referred to in the sketch
is that of graphic  design, which includes the design
of commercial logos. The Greek  word logos
has more to do with mathematics and theology.

“If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity.”

— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato’s Theaetetus
, SUNY Press, 1990

Related material—

(Click to enlarge.)


Friday, June 25, 2010

ART WARS continued

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

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

See The Klein Correspondence.

How Deep the Darkness

Filed under: General — m759 @ 12:25 PM

Image-- Rosalind Krauss and The Ninefold Square

Art Theorist Rosalind Krauss and The Ninefold Square

Thursday, June 24, 2010

The Door to Stockholm

Filed under: General — m759 @ 2:02 PM

Today's midday lottery number in Pennsylvania
(State of Grace ) was 752.

Related material from this journal—
Page 752 of Gravity's Rainbow 
(Penguin Classics, 1995).

("With Pointsman it's only habit, retro-scientism:
  a last look back at the door to Stockholm,
  closing behind him forever.")

Bright Star

Filed under: General — m759 @ 12:31 PM

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

– Rubén Darío

Midsummer Noon

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

Geometry Simplified

Image-- The Three-Point Line: A Finite Projective Space
(a projective space)

The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
 (Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

(an affine space)

The (Galois) points of this affine plane are
  not the single and combined (Euclidean)
line segments that play the role of
  points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

For further details, see Galois Geometry.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —


Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

Wednesday, June 23, 2010

Group Theory and Philosophy

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM

Excerpts from "The Concept of Group and the Theory of Perception,"
by Ernst Cassirer, Philosophy and Phenomenological Research,
Volume V, Number 1, September, 1944.
(Published in French in the Journal de Psychologie, 1938, pp. 368-414.)

The group-theoretical interpretation of the fundaments of geometry is,
from the standpoint of pure logic, of great importance, since it enables us to
state the problem of the "universality" of mathematical concepts in simple
and precise form and thus to disentangle it from the difficulties and ambigui-
ties with which it is beset in its usual formulation. Since the times of the
great controversies about the status of universals in the Middle Ages, logic
and psychology have always been troubled with these ambiguities….

Our foregoing reflections on the concept of group  permit us to define more
precisely what is involved in, and meant by, that "rule" which renders both
geometrical and perceptual concepts universal. The rule may, in simple
and exact terms, be defined as that group of transformations  with regard to
which the variation of the particular image is considered. We have seen
above that this conception operates as the constitutive principle in the con-
struction of the universe of mathematical concepts….

                                                              …Within Euclidean geometry,
a "triangle" is conceived of as a pure geometrical "essence," and this
essence is regarded as invariant with respect to that "principal group" of
spatial transformations to which Euclidean geometry refers, viz., displace-
ments, transformations by similarity. But it must always be possible to
exhibit any particular figure, chosen from this infinite class, as a concrete
and intuitively representable object. Greek mathematics could not
dispense with this requirement which is rooted in a fundamental principle
of Greek philosophy, the principle of the correlatedness of "logos" and
"eidos." It is, however, characteristic of the modern development of
mathematics, that this bond between "logos" and "eidos," which was indis-
soluble for Greek thought, has been loosened more and more, to be, in the
end, completely broken….

                                                            …This process has come to its logical
conclusion and systematic completion in the development of modern group-
theory. Geometrical figures  are no longer regarded as fundamental, as
date of perception or immediate intuition. The "nature" or "essence" of a
figure is defined in terms of the operations  which may be said to
generate the figure.
The operations in question are, in turn, subject to
certain group conditions….

                                                                                                    …What we
find in both cases are invariances with respect to variations undergone by
the primitive elements out of which a form is constructed. The peculiar
kind of "identity" that is attributed to apparently altogether heterogen-
eous figures in virtue of their being transformable into one another by means
of certain operations defining a group, is thus seen to exist also in the
domain of perception. This identity permits us not only to single out ele-
ments but also to grasp "structures" in perception. To the mathematical
concept of "transformability" there corresponds, in the domain of per-
ception, the concept of "transposability." The theory  of the latter con-
cept has been worked out step by step and its development has gone through
various stages….
                                                                                 …By the acceptance of
"form" as a primitive concept, psychological theory has freed it from the
character of contingency  which it possessed for its first founders. The inter-
pretation of perception as a mere mosaic of sensations, a "bundle" of simple
sense-impressions has proved untenable…. 

                             …In the domain of mathematics this state of affairs mani-
fests itself in the impossibility of searching for invariant properties of a
figure except with reference to a group. As long as there existed but one
form of geometry, i.e., as long as Euclidean geometry was considered as the
geometry kat' exochen  this fact was somehow concealed. It was possible
to assume implicitly  the principal group of spatial transformations that lies
at the basis of Euclidean geometry. With the advent of non-Euclidean
geometries, however, it became indispensable to have a complete and sys-
tematic survey of the different "geometries," i.e., the different theories of
invariancy that result from the choice of certain groups of transformation.
This is the task which F. Klein set to himself and which he brought to a
certain logical fulfillment in his Vergleichende Untersuchungen ueber neuere
geometrische Forschungen

                                                          …Without discrimination between the
accidental and the substantial, the transitory and the permanent, there
would be no constitution of an objective reality.

This process, unceasingly operative in perception and, so to speak, ex-
pressing the inner dynamics of the latter, seems to have come to final per-
fection, when we go beyond perception to enter into the domain of pure
thought. For the logical advantage and peculiar privilege of the pure con –
cept seems to consist in the replacement of fluctuating perception by some-
thing precise and exactly determined. The pure concept does not lose
itself in the flux of appearances; it tends from "becoming" toward "being,"
from dynamics toward statics. In this achievement philosophers have
ever seen the genuine meaning and value of geometry. When Plato re-
gards geometry as the prerequisite to philosophical knowledge, it is because
geometry alone renders accessible the realm of things eternal; tou gar aei
ontos he geometrike gnosis estin
. Can there be degrees or levels of objec-
tive knowledge in this realm of eternal being, or does not rather knowledge
attain here an absolute maximum? Ancient geometry cannot but answer
in the affirmative to this question. For ancient geometry, in the classical
form it received from Euclid, there was such a maximum, a non plus ultra.
But modern group theory thinking has brought about a remarkable change
In this matter. Group theory is far from challenging the truth of Euclidean
metrical geometry, but it does challenge its claim to definitiveness. Each
geometry is considered as a theory of invariants of a certain group; the
groups themselves may be classified in the order of increasing generality.
The "principal group" of transformations which underlies Euclidean geome-
try permits us to establish a number of properties that are invariant with
respect to the transformations in question. But when we pass from this
"principal group" to another, by including, for example, affinitive and pro-
jective transformations, all that we had established thus far and which,
from the point of view of Euclidean geometry, looked like a definitive result
and a consolidated achievement, becomes fluctuating again. With every
extension of the principal group, some of the properties that we had taken
for invariant are lost. We come to other properties that may be hierar-
chically arranged. Many differences that are considered as essential
within ordinary metrical geometry, may now prove "accidental." With
reference to the new group-principle they appear as "unessential" modifica-

                 … From the point of view of modern geometrical systematization,
geometrical judgments, however "true" in themselves, are nevertheless not
all of them equally "essential" and necessary. Modern geometry
endeavors to attain progressively to more and more fundamental strata of
spatial determination. The depth of these strata depends upon the com-
prehensiveness of the concept of group; it is proportional to the strictness of
the conditions that must be satisfied by the invariance that is a universal
postulate with respect to geometrical entities. Thus the objective truth
and structure of space cannot be apprehended at a single glance, but have to
be progressively  discovered and established. If geometrical thought is to
achieve this discovery, the conceptual means that it employs must become
more and more universal….

Tuesday, June 22, 2010

Hermeneutics for Bernstein

Filed under: General — m759 @ 8:28 PM

J. M. Bernstein (previous post) has written of moving toward "a Marxist hermeneutic."

I prefer lottery hermeneutics.

Some background from Bernstein—


I would argue that at least sometimes, lottery numbers may be regarded, according to Bernstein's definition, as story statements. For instance—

Today's New York State Lottery— Midday 389, Evening 828.

For the significance of 389, see

A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.
 The Evolution of Concepts of Space and Symmetry
 by Pierre Cartier, Bulletin of the American Mathematical Society,
 Vol. 38 (2001) No. 4, beginning on page 389.

The philosophical import of page 389 is perhaps merely in Cartier's title (see previous post).

For the significance of 828, see 8/28, the feast of St. Augustine, in 2006.

See also Halloween 2007. (Happy birthday, Dan Brown.)

1984 Story (continued)

Filed under: General — m759 @ 8:00 PM


"There are many accounts of
moral and political anger in
the philosophical literature."

— J. M. Bernstein in today's NY Times

J.M. Bernstein is University Distinguished Professor
of Philosophy at the New School for Social Research.

He is the author of a work
that Google Books files under
"Communism and Literature"—

The Philosophy of the Novel:
Lukács, Marxism, and the Dialectics of Form

(University of Minnesota Press, 1984)

Mathematics and Narrative, continued

Filed under: General,Geometry — Tags: — m759 @ 2:14 PM

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

A 1973 review of Koestler's book—

"Koestler's 'call girls,' summoned here and there
 by this university and that foundation
 to perform their expert tricks, are the butts
 of some chilling satire."

Examples of Light—

Felix Christian Klein (1849- June 22, 1925) and Évariste Galois (1811-1832)

Klein on Galois—

"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."

— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.

"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."

— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.

Examples of Darkness—

Martin Gardner on Galois—

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Gardner was reviewing a recent book about Galois by one Amir Alexander.

Alexander himself has written some reviews relevant to the Koestler book above.

See Alexander on—

The 2005 Mykonos conference on Mathematics and Narrative

A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."

Monday, June 21, 2010


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

From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—

Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—


Peterson associates the four-diamond figure
with the Pythagorean theorem.

A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—

Image-- The Four-Diamond Tesseract

This figure is relevant because of a
tesseract sculpture by Peter Forakis—


This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.

The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).

For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).

"There is  such a thing
as a tesseract."
A Wrinkle in TIme   

1984 Story (continued)

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

This journal's 11 AM Sunday post was "Lovasz Wins Kyoto Prize." This is now the top item on the American Mathematical Society online home page—


Click to enlarge.

For more background on Lovasz, see today's
previous Log24 post, Cube Spaces, and also
Cube Space, 1984-2003.

"If the Party could thrust its hand into the past and
say of this or that event, it never happened…."

— George Orwell, 1984

Cube Spaces

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

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

Example 1— The 2×2×2 Cube—

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—


Version by Laszlo Lovasz et al., 2003—


Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—


Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
  3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 
note by Cullinane

Pegg gives no reference to the 1985 work on group actions.

Example 3— The 4×4×4 Cube

A note from 27 years ago today—


As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Sunday, June 20, 2010

Happy Father’s Day

Filed under: General — m759 @ 4:00 PM

Image-- Poster for 'The Good, The Bad, and the Ugly'

Lovasz Wins Kyoto Prize

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

From a June 18 press release

KYOTO, Japan, Jun 18, 2010 (BUSINESS WIRE) — The non-profit Inamori Foundation (President: Dr. Kazuo Inamori) today announced that Dr. Laszlo Lovasz will receive its 26th annual Kyoto Prize in Basic Sciences, which for 2010 focuses on the field of Mathematical Sciences. Dr. Lovasz, 62, a citizen of both Hungary and the United States, will receive the award for his outstanding contributions to the advancement of both the academic and technological possibilities of the mathematical sciences.

Dr. Lovasz currently serves as both director of the Mathematical Institute at Eotvos Lorand University in Budapest and as president of the International Mathematics Union. Among many positions held throughout his distinguished career, Dr. Lovasz also served as a senior research member at Microsoft Research Center and as a professor of computer science at Yale University.

Related material: Cube Space, 1984-2003.

See also "Kyoto Prize" in this journal—

The Kyoto Prize is "administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”

— – Montreal Gazette, June 20, 2008



Wittgenstein and Fly from Fly-Bottle

Fly from Fly Bottle

Sunday School

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

Limited— Good   
Évariste Galois  


Unlimited— Bad
H.S.M. Coxeter


Jamie James in The Music of the Spheres

"The Pythagorean philosophy, like Zoroastrianism, Taoism, and every early system of higher thought, is based upon the concept of dualism. Pythagoras constructed a table of opposites from which he was able to derive every concept needed for a philosophy of the phenomenal world. As reconstructed by Aristotle in his Metaphysics, the table contains ten dualities (ten being a particularly important number in the Pythagorean system, as we shall see):


Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited (man, finite time, and so forth) and the unlimited (the cosmos, eternity, etc.) is not only the aim of Pythagoras's system but the central aim of all Western philosophy."

Saturday, June 19, 2010

Imago Creationis

Filed under: General,Geometry — Tags: , , , , — m759 @ 6:00 PM

Image-- The Four-Diamond Tesseract

In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.

Four-Part Tesseract Divisions


The above figure shows how four-part partitions
of the 16 vertices  of a tesseract in an infinite
Euclidean  space are related to four-part partitions
of the 16 points  in a finite Galois  space

Euclidean spaces versus Galois spaces
in a larger context—



Infinite versus Finite

The central aim of Western religion —

"Each of us has something to offer the Creator...
the bridging of
                 masculine and feminine,
                      life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist  (1998)

The central aim of Western philosophy —

              Dualities of Pythagoras
              as reconstructed by Aristotle:
                 Limited     Unlimited
                     Odd     Even
                    Male     Female
                   Light      Dark
                Straight    Curved
                  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres  (1993)

Another picture related to philosophy and religion—

Jung's Four-Diamond Figure from Aion


This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—

Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—



Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science…  reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).


  Paul Valéry, Oeuvres  (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—

… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.

* That is, uses multi-dimensional symbols beyond our grasp.

Related material:

Imago Creationis

A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).


Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—

Frame of Reference


The Diamond Theorem


Some context by a British mathematician —



by Wallace Stevens

Who can pick up the weight of Britain, 
Who can move the German load 
Or say to the French here is France again? 
Imago. Imago. Imago. 

It is nothing, no great thing, nor man 
Of ten brilliancies of battered gold 
And fortunate stone. It moves its parade 
Of motions in the mind and heart, 

A gorgeous fortitude. Medium man 
In February hears the imagination's hymns 
And sees its images, its motions 
And multitudes of motions 

And feels the imagination's mercies, 
In a season more than sun and south wind, 
Something returning from a deeper quarter, 
A glacier running through delirium, 

Making this heavy rock a place, 
Which is not of our lives composed . . . 
Lightly and lightly, O my land, 
Move lightly through the air again.

Friday, June 18, 2010

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM




Image-- Leibniz medal

Leibniz Medal

Search this journal  
for "Leibniz medal."

   Related material— Street of the Fathers and Game Over.

Blue Ribbon

Filed under: General — m759 @ 10:00 AM

"I think there's a lot of meritocracy, a lot of blue-ribbon talk here."

– Chris Matthews on President Obama's Tuesday night speech

And here…

Image-- Detail of New Yorker cover 'Finish Line,' double fiction issue of June 14 & 21, 2010
Detail from cover of current New Yorker
in Thursday afternoon's Log24 post

Related material:

Image-- Nobel Prize-Winning Writer Saramago Dies at 87

See also "Saramago" in this journal
as well as his Nobel Prize lecture.

Word of the Day

Filed under: General — m759 @ 8:28 AM

… and week, year, decade, century…

"I think there's a lot of meritocracy, a lot of blue-ribbon talk here."

— Chris Matthews, at 5:00 of 6:44 minutes in a YouTube video of an MSNBC discussion at the news blog ArlingtonCardinal.com Wednesday morning. The post containing the video was headlined "Word of the Day: Meritocracy."

"There is a growing meme that Mr. Obama is too impressed by credentialism, by the meritocracy, by those who hold forth in the faculty lounge, and too strongly identifies with them."

— Peggy Noonan in today's Wall Street Journal

Some background—

Lost in the Meritocracy:
The Undereducation of an Overachiever

by Walter Kirn, 224 pp. Doubleday, 2009

and the review by Laura Miller in the Sunday,
May 24, 2009, New York Times Book Review.

See also Log24 on this date three years ago
a post on Harvard, Bunker Hill Community College,
and the Mystic River area in between.

Thursday, June 17, 2010


Filed under: General — Tags: — m759 @ 2:02 PM

Continued from yesterday evening's "Long Day's Journey into Nighttown"—

A detail from that post—

Image-- Detail of New Yorker cover 'Finish Line,' double fiction issue of June 14 & 21, 2010

Related material from Nighttown—
The Sebastian Horsley Guide to Whoring

Image-- YouTube video, 'The Sebastian Horsley Guide to Whoring'

Horsley, the author of Dandy in the Underworld, was
found dead this morning of a suspected heroin overdose.

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

Wednesday, June 16, 2010

Field of Dreams

Filed under: General — Tags: , — m759 @ 8:28 PM

Long Day's Journey into Nighttown


Click for larger, clearer image.

New Yorker  cover, fiction issue of June 14 and 21, 2010.
"Finish Line," by Chris Ware.

See also Shakespeare's Birthday, 2009.

Brightness at Noon

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

David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —

Image-- Arthur Koestler by David Levine, NY Review of Books, Dec. 17, 1964, review of 'The Act of Creation'

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Image-- The 64 I Ching hexagrams in the 4 layers of the Cullinane cube

Geometry of the I Ching

See also this morning's post as well as
Monday's post quoting George David Birkhoff

"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."

Geometry of Language

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

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: , , , — m759 @ 11:07 AM

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).


  Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion  of the Self as a four-diamond figure:


and Cullinane’s purely geometric four-diamond figure:


For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Monday, June 14, 2010

Birkhoff on the Galois “Theory of Ambiguity”

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

The Principle of Sufficient Reason

by George David Birkhoff

from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940


My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2

In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is

2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought

3 That is, uses multi-dimensional symbols beyond our grasp.

the appropriate instrument of thought to remedy our deficiency
in this respect.

The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were

Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.

Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D;  B, C, D, A;  C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a

4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.

null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:

This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."


Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:

in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).

HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.

The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.

In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.

It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:

Image-- Birkhoff diagram relating Galois's theory of ambiguity to metaphysics

While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the


A more recent lecture on the same subject —

"From Leibniz to Quantum World:
Symmetries, Principle of Sufficient Reason
and Ambiguity in the Sense of Galois

by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)

Theory of Ambiguity

Filed under: General,Geometry — m759 @ 5:01 PM

Théorie de l'Ambiguité

According to a 2008 paper by Yves André of the École Normale Supérieure  of Paris—

"Ambiguity theory was the name which Galois used
 when he referred to his own theory and its future developments."

The phrase "the theory of ambiguity" occurs in the testamentary letter Galois wrote to a friend, Auguste Chevalier, on the night before Galois was shot in a duel.

Hermann Weyl in Symmetry, Princeton University Press, 1952—

"This letter, if judged by the novelty and profundity of ideas it contains, is perhaps
  the most substantial piece of writing in the whole literature of mankind."

Conclusion of the Galois testamentary letter, according to
the 1897 Paris edition of Galois's collected works—

Image-- Galois on his theory of ambiguity, from Collected Works, Paris, 1897

The original—

Image-- Concluding paragraphs, Galois's 'last testament' letter to Chevalier, May 29, 1832

A transcription—

Évariste GALOIS, Lettre-testament, adressée à Auguste Chevalier—

Tu sais mon cher Auguste, que ces sujets ne sont pas les seuls que j'aie
explorés. Mes principales méditations, depuis quelques temps,
étaient dirigées sur l'application à l'analyse transcendante de la théorie de
l'ambiguité. Il s'agissait de voir a priori, dans une relation entre des quantités
ou fonctions transcendantes, quels échanges on pouvait faire, quelles
quantités on pouvait substituer aux quantités données, sans que la relation
put cesser d'avoir lieu. Cela fait reconnaitre de suite l'impossibilité de beaucoup
d'expressions que l'on pourrait chercher. Mais je n'ai pas le temps, et mes idées
ne sont pas encore bien développées sur ce terrain, qui est

Tu feras imprimer cette lettre dans la Revue encyclopédique.

Je me suis souvent hasardé dans ma vie à avancer des propositions dont je n'étais
pas sûr. Mais tout ce que j'ai écrit là est depuis bientôt un an dans ma
tête, et il est trop de mon intérêt de ne pas me tromper pour qu'on
me soupconne d'avoir énoncé des théorèmes dont je n'aurais pas la démonstration

Tu prieras publiquement Jacobi et Gauss de donner leur avis,
non sur la vérité, mais sur l'importance des théorèmes.

Après cela, il y aura, j'espère, des gens qui trouveront leur profit
à déchiffrer tout ce gachis.

Je t'embrasse avec effusion.

                                               E. Galois   Le 29 Mai 1832

A translation by Dr. Louis Weisner, Hunter College of the City of New York, from A Source Book in Mathematics, by David Eugene Smith, Dover Publications, 1959–

You know, my dear Auguste, that these subjects are not the only ones I have explored. My reflections, for some time, have been directed principally to the application of the theory of ambiguity to transcendental analysis. It is desired see a priori  in a relation among quantities or transcendental functions, what transformations one may make, what quantities one may substitute for the given quantities, without the relation ceasing to be valid. This enables us to recognize at once the impossibility of many expressions which we might seek. But I have no time, and my ideas are not developed in this field, which is immense.

Print this letter in the Revue Encyclopédique.

I have often in my life ventured to advance propositions of which I was uncertain; but all that I have written here has been in my head nearly a year, and it is too much to my interest not to deceive myself that I have been suspected of announcing theorems of which I had not the complete demonstration.

Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of the theorems.

Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess.

J t'embrasse avec effusion.
                                                     E. Galois.   May 29, 1832.

Translation, in part, in The Unravelers: Mathematical Snapshots, by Jean Francois Dars, Annick Lesne, and Anne Papillaut (A.K. Peters, 2008)–

"You know, dear Auguste, that these subjects are not the only ones I have explored. For some time my main meditations have been directed on the application to transcendental analysis of the theory of ambiguity. The aim was to see in a relation between quantities or transcendental functions, what exchanges we could make, what quantities could be substituted to the given quantities without the relation ceasing to take place. In that way we see immediately that many expressions that we might look for are impossible. But I don't have the time and my ideas are not yet developed enough in this vast field."

Another translation, by James Dolan at the n-Category Café

"My principal meditations for some time have been directed towards the application of the theory of ambiguity to transcendental analysis. It was a question of seeing a priori in a relation between quantities or transcendent functions, what exchanges one could make, which quantities one could substitute for the given quantities without the original relation ceasing to hold. That immediately made clear the impossibility of finding many expressions that one could look for. But I do not have time and my ideas are not yet well developed on this ground which is immense."

Related material

"Renormalisation et Ambiguité Galoisienne," by Alain Connes, 2004

"La Théorie de l’Ambiguïté : De Galois aux Systèmes Dynamiques," by Jean-Pierre Ramis, 2006

"Ambiguity Theory, Old and New," preprint by Yves André, May 16, 2008,

"Ambiguity Theory," post by David Corfield at the n-Category Café, May 19, 2008

"Measuring Ambiguity," inaugural lecture at Utrecht University by Gunther Cornelissen, Jan. 16, 2009


Filed under: General — m759 @ 7:20 AM

Continued from Sunday morning.

Image-- Album by Jimmy Dean-- '20 Great Story Songs'

Breakfast at Tiffany's  (Vintage reprint), page 73—

"Doc really loves me, you know. And I love him. He may have looked old and tacky to you. But you don't know the sweetness of him, the confidence he can give to birds and brats and fragile things like that. Anyone who ever gave you confidence, you owe them a lot. I've always remembered Doc in my prayers. Please stop smirking!" she demanded, stabbing out a cigarette. "I do  say my prayers."

… Page 74 …

She glanced at the clock. "He must be in the Blue Mountains by now."

The image 
“http://www.log24.com/log/pix05A/050703-Cold.jpg” cannot be displayed, 
because it contains errors.

Adapted from cover of
German edition of Cold Mountain

Sunday, June 13, 2010

Today’s Sermon

Filed under: General — m759 @ 10:30 AM



Sunday School

Filed under: General — m759 @ 9:00 AM

What on earth is a 'concrete universal'?"
Said to be an annotation (undated)
by Robert M. Pirsig of A History of Philosophy,
by Frederick Copleston, Society of Jesus.

From Aaron Urbanczyk's 2005 review of Christ and Apollo  by William Lynch, S.J., a book first published in 1960—

"Lynch's use of analogy vis-a-vis literature provides, in a sense, a philosophical basis to the theoretical paradox popularized by W. K. Wimsatt (1907-1975), which contends that literature is a sort of 'concrete universal.'"

The following figure has often been
offered in this journal as a symbol of Apollo

Image-- 3x3 array of white squares

Arguments that it is, rather, a symbol of Christ
may be left to the Society of Jesus.

One possible approach—
Urbanczyk's review says that
"Christianity offers the critic
   a privileged ontological window…."

"The world was warm and white when I was born:
Beyond the windowpane the world was white,
A glaring whiteness in a leaded frame,
Yet warm as in the hearth and heart of light."

Delmore Schwartz

Saturday, June 12, 2010

Holy Geometry

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

The late mathematician V.I. Arnold was born on this date in 1937.

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy


Image-- AMS site screenshot of V.I. Arnold obituary, June 12, 2010


Image-- AMS site screenshot of Martin Gardner tribute, May 25, 2010

Choosing light rather than darkness, we observe Arnold's birthday with a quotation from his 1997 Paris talk 'On Teaching Mathematics.'

"The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact…."

The "experimental fact" part, perhaps offered with tongue in cheek, is of less interest than the assertion that the Jacobi identity forces the altitude-intersection theorem.

Albert Einstein on that theorem in the "holy geometry book" he read at the age of 12—

"Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

Arnold's much less  evident assertion about altitudes and the Jacobi identity is discussed in "Arnol'd, Jacobi identity, and orthocenters" (pdf) by Nikolai V. Ivanov.

Ivanov says, without giving a source,  that the altitudes theorem "was known to Euclid." Alexander Bogomolny, on the other hand, says it is "a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements  or subsequent writings of the Greek scholars. The timing of the first proof is still an open question."

For other remarks on geometry, search this journal for the year of Arnold's birth.

Friday, June 11, 2010

The Raven

Filed under: General — m759 @ 12:00 PM

"Why is a raven like a writing-desk?" — Alice's Adventures in Wonderland

Arthur Koestler, The Roots of Coincidence

"In his The Nature of the Physical World  (1928) Sir Arthur Eddington introduced his famous 'parable of the two writing desks.' One is the antique piece of furniture on which his elbows solidly rest while writing; the other is the desk as the physicist conceives it, consisting almost entirely of empty space, sheer nothingness…. Eddington concluded:

In the world of physics we watch a shadowgraph performance of familiar life. The shadow of my elbow rests on the shadow-table as the shadow-ink flows over the shadow-paper….

Though the constituents of matter could be described with great mathematical accuracy as patterns of vibrations, the question remained—  what was it that vibrated? On the one hand, these matter-waves produced physically real phenomena, such as interference patterns on a screen, or the currents in a transistor radio. On the other hand, the whole conception of matter-waves excludes by definition any medium with physical attributes as a carrier of the waves. A wave is movement; but what is that something that moves, producing the shadows on Eddington's shadow-desk? Short of calling it the grin of the Cheshire Cat, it was named the 'psi field' or 'psi function.'"

What is it that moves? Perhaps not the Cheshire Cat, but rather The Raven

Closeup, he’s blue—streaked iris blue, india-ink blue—and
black—an oily, fiery set of blacks—none of them
true—as where hate and order touch—something that cannot
become known. Stages of black but without
graduation. So there is no direction.
All of this happened, yes.

 — Jorie Graham, "The Dream of the Unified Field"

See also notes on darkness in this journal.

Toward the Light

Filed under: General — m759 @ 11:01 AM

The title is a reference to yesterday's noon post.

For the late Vladimir Igorevich Arnold

"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical Mathematicks of the City of Heaven."

— Sir Thomas Browne, The Garden of Cyrus, Chapter V

Arnold's own mystical mathematics may be found in his paper

"Polymathematics: Is Mathematics a Single Science or a Set of Arts?"

Page 13–
"In mathematics we always encounter mysterious analogies, and our trinities [page 8] represent only a small part of these miracles."

Also from that paper—

Page 5, footnote 2–
"The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization."

Arnold died in Paris on June 3. A farewell gathering was held there on June 8—

"Celles et ceux qui le souhaitent pourront donner un dernier adieu à Vladimir Igorevitch
mardi 8 juin, de 14h a 16h, chambre mortuaire de l'hopital Saint Antoine…."

An International Blue Diamond

In Arnold's memory—  Here, in the Russian style, is a link to a "first nontrivial case" of a blue diamond— from this journal on June 8 (feast of St. Gerard Manley Hopkins). For those who prefer French style, here is a link to a blue diamond from May 18

From French cinema—


"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa"

Thursday, June 10, 2010

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

Continued from Christmas 2009 and from last Sunday

The serious reflection is composed
Neither of comic nor tragic but of commonplace."

Wallace Stevens

The Celestine Dream

Filed under: General — m759 @ 9:00 AM

(Pace  James Redfield)

"…after all, the word 'bolshoi' means big, great or grand."

Celestine Bohlen, 1999

"How much for a dream?" — This journal, January 1, 2010

Star of Bolshoi Ballet Dies at 102

                                                                  I was eight—
I saw the different weights of things,
saw the vivid performance of the present,
saw the light rippling almost shuddering where her body finally
the image, the silver film between them like something that would have
                                                                           shed itself in nature now
but wouldn’t, couldn’t, here, on tight,
not thinning, not slipping off to let some
through, no signal in it, no information …

— Jorie Graham, "The Dream of the Unified Field"

Update of 10:25 AM June 10— Click to enlarge—


Wednesday, June 9, 2010

Academy Award

Filed under: General — m759 @ 2:29 PM

Continued from Saturday, June 5


"These fitful sayings are, also, of tragedy:
The serious reflection is composed
Neither of comic nor tragic but of commonplace."

Wallace Stevens

Shaggy Dance

Filed under: General — m759 @ 11:30 AM

David Brooks in The New York Times  yesterday:

The Big Shaggy

…deep down, people have passions and drives…  that reside in an inner beast you could call The Big Shaggy….

If you spend your life riding the links of the Internet, you probably won’t get too far into The Big Shaggy either, because the fast, effortless prose of blogging (and journalism) lacks the heft to get you deep below.

You have to know where to look.


Tuesday, June 8, 2010

From Plato to Finite Geometry

Filed under: General,Geometry — m759 @ 12:30 PM

A supplement to yesterday's post on variation of an eidos

Image-- Plato's diamond and a modern version from finite geometry


Monday, June 7, 2010

Inspirational Combinatorics

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

According to the Mathematical Association of America this morning, one purpose of the upcoming June/July issue of the Notices of the American Mathematical Society  is

"…to stress the inspirational role of combinatorics…."

Here is another contribution along those lines—

Eidetic Variation

from page 244 of
From Combinatorics to Philosophy: The Legacy of  G.-C. Rota,
hardcover, published by Springer on August 4, 2009

(Edited by Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, and Fabrizio Palombi)

"Rota's Philosophical Insights," by Massimo Mugnai—

"… In other words, 'objectivism' is the attitude [that tries] to render a particular aspect absolute and dominant over the others; it is a kind of narrow-mindedness attempting to reduce to only one the multiple layers which constitute what we call 'reality.' According to Rota, this narrow-mindedness limits in an essential way even of [sic ] the most basic facts of our cognitive activity, as, for example, the understanding of a simple declarative sentence: 'So objectivism is the error we [make when we] persist in believing that we can understand what a declarative sentence means without a possible thematization of this declarative sentence in one of [an] endless variety of possible contexts' (Rota, 1991*, p. 155). Rota here implicitly refers to what, amongst phenomenologists is known as eidetic variation, i.e. the change of perspective, imposed by experience or performed voluntarily, from which to look at things, facts or sentences of the world. A typical example, proposed by Heidegger, in Sein und Zeit  (1927) and repeated many times by Rota, is that of the hammer."

* Rota, G.-C. (1991), The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT, Cambridge, MA, MIT Mathematics Department

The example of the hammer appears also on yesterday's online New York Times  front page—


Related material:

From The Blackwell Dictionary of Western Philosophy

Eidetic variation — an alternative expression for eidetic reduction

Eidetic reduction

Husserl's term for an intuitive act toward an essence or universal, in contrast to an empirical intuition or perception. He also called this act an essential intuition, eidetic intuition, or eidetic variation. In Greek, eideo  means “to see” and what is seen is an eidos  (Platonic Form), that is, the common characteristic of a number of entities or regularities in experience. For Plato, eidos  means what is seen by the eye of the soul and is identical with essence. Husserl also called this act “ideation,” for ideo  is synonymous with eideo  and also means “to see” in Greek. Correspondingly, idea  is identical to eidos.

An example of eidos— Plato's diamond (from the Meno )—


For examples of variation of this eidos, see the diamond theorem.
See also Blockheads (8/22/08).

Related poetic remarks— The Trials of Device.

Combinatorics Issue

Filed under: General — m759 @ 8:00 AM

From the MAA today

AMS Notices Combinatorics

June 7, 2010 

"The June/July 2010 issue of the Notices of the AMS  [not yet online] highlights the subject of combinatorics.

Over the last 30 years, combinatorics and finite mathematics have become central to the creation of collaboration graphs, Google search algorithms, random graphs, and internet routing. They also play a vital part in queueing theory and the solutions of coloring problems.  

To showcase these and other related topics and to stress the inspirational role of combinatorics, the June/July issue contains these key features: 'The Giant Component: The Golden Anniversary,' by Joel Spencer; 'The Mathematical Side of M. C. Escher,' by Doris Schattschneider; 'Graph Theory in the Information Age,' by Fan Chung, and 'Hadwiger's Conjecture,' by Maria Chudnovsky."

Sunday, June 6, 2010

The Trials of Device

Filed under: General — m759 @ 9:00 AM

Image-- 'The Trials of Device -- Excerpts from Two Poems by Wallace Stevens'

Saturday, June 5, 2010

Academy Award

Filed under: General — m759 @ 9:00 AM

The history of mathematics continues…

Image-- Academy of Athens announces a Jan. 26, 2010, speech by Professor Nicolaos Artemiadis


The Academician Professor Nicolaos Artemiadis will give a speech entitled "The Exploration of the Universe through the Mathematical Science" during a public session of the Academy of Athens (the speech will be in Greek).

The public session will be held on Tuesday, January 26th, 2010, at 19:00 at the Academy of Athens.

For some background on Professor Artemiadis, see two notes of July 2005 (the month an international conference on "Mathematics and Narrative" was held in Greece).

A post related by synchronicity to Artemiadis's Jan. 26 speech— Symbology.

Other philosophical remarks— "The Pediment of Appearance."

Friday, June 4, 2010

ART WARS continued

Filed under: General,Geometry — m759 @ 8:00 PM

Today's New York Times

Art Review

Painting Thin Air, Sometimes in Bright Blue

(“Yves Klein: With the Void, Full Powers”
  runs through Sept. 12 at the Hirshhorn.)

Related material—

Search this journal for klein + paris.

See also Art Space (May 22, 2010)—

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Brightness at Noon

Filed under: General — m759 @ 12:00 PM

(Continued from Epiphany 2010)

For a Languid Janitor

Image-- Matt Damon as an MIT janitor

For the Mothers of Invention

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

Today is Commencement Day at MIT.

A song by Joni—

Image-- 'You Turn Me On, I'm a Radio' lyrics by Joni Mitchell

"Who needs the static?" Well might you ask, Joni.

"The static boxes were an invention of Grandfather…."

In Memory of Mother*

Filed under: General — m759 @ 8:28 AM

"We are not saints."

* Click for name.

A Better Story

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

Continued from May 8
(Feast of Saint Robert Heinlein)

“Wells and trees were dedicated to saints.  But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden.  Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy.”

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Why "Saint" Robert? See his accurate depiction of evil– the Eater of Souls in Glory Road.

For more on Williams's "other capacities," see Heinlein's story "Lost Legacy."

A related story– Fritz Leiber's "The Mind Spider." An excerpt:

The conference—it was much more a hyper-intimate

"My static box bugged out for a few ticks this morning,"
Evelyn remarked in the course of talking over the
trivia of the past twenty-four hours.

The static boxes were an invention of Grandfather
Horn. They generated a tiny cloud of meaningless brain
waves. Without such individual thought-screens, there was
too much danger of complete loss of individual personality

—once Grandfather Horn had "become" his infant daughter
as well as himself for several hours and the unfledged
mind had come close to being permanently lost in its own
subconscious. The static boxes provided a mental wall be-
– hind which a mind could safely grow and function, similar
to the wall by which ordinary minds are apparently
always enclosed.

In spite of the boxes, the Horns shared thoughts and
emotions to an amazing degree. Their mental togetherness
was as real and as mysterious—and as incredible—as
thought itself . . . and thought is the original angel-cloud
dancing on the head of a pin. Their present conference
was as warm and intimate and tart as any actual family
gathering in one actual room around one actual table.
Five minds, joined together in the vast mental darkness
that shrouds all minds. Five minds hugged together for
comfort and safety in the infinite mental loneliness that
pervades the cosmos.

Evelyn continued, "Your boxes were all working, of
course, so I couldn't get your thoughts—just the blurs of
your boxes like little old dark grey stars. But this time
if gave me a funny uncomfortable feeling, like a spider
Crawling down my—Grayl! Don't feel so wildly! What
Is it?”

Then… just as Grayl started to think her answer…
something crept from the vast mental darkness and infinite
cosmic loneliness surrounding the five minds of the

Grayl was the first to notice. Her panicky thought had
ttie curling too-keen edge of hysteria. "There are six of
us now! There should only be five, but there are six.
Count! Count, I tell you! Six!"

To Mort it seemed that a gigantic spider was racing
across the web of their thoughts….

See also this journal on May 30– "720 in the Book"– and on May 31– "Memorial for Galois."

("Obnoxious nerds"— a phrase Martin Gardner recently applied to Galois— will note that 720 (= 6!) is one possible result of obeying Leiber's command "Count! Count, I tell you! Six!")

Thursday, June 3, 2010


Filed under: General — m759 @ 9:00 AM

Margaret Atwood (pdf) on Lewis Hyde’s
Trickster Makes This World: Mischief, Myth, and Art

“Trickster,” says Hyde, “feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation.” (71) As Hyde says, “…  almost everything that can be said about psychopaths can also be said about tricksters,” (158), although the reverse is not the case. “Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists.” (159)

What is “the next world”? It might be the Underworld….

The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.

See also George P. Hansen on Martin Gardner, Trickster.

Wednesday, June 2, 2010

The Harvard Style

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM

"I wonder if there's just been a critical mass
of creepy stories about Harvard
in the last couple of years…
A kind of piling on of
    nastiness and creepiness…"

Margaret Soltan, October 23, 2006

Harvard University Press
  on Facebook

Harvard University Press Harvard University Press
Martin Gardner on demythologizing mathematicians:
"Galois was a thoroughly obnoxious nerd"
  May 26 at 6:28 pm via Ping.f

The book that the late Gardner was reviewing
was published in April by Harvard University Press.

If Gardner's remark were true,
Galois would fit right in at Harvard. Example—
  The Harvard math department's pie-eating contest

Harvard Math Department Pi Day event

Rite of Passage

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


"On June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown."

Évariste Galois, Lettre de Galois à M. Auguste Chevalier

Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Martin Gardner on the above letter—

"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."

The Last Recreations, by Martin Gardner, published by Springer in 2007, page 156.

Leonard E. Dickson

Image-- Leonard E. Dickson on the posthumous fundamental memoir of Galois

Tuesday, June 1, 2010

The Gardner Tribute

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

"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue."

Roger Kimball of The New Criterion, May 23, 2010.

The Gardner piece is now online.  It contains…

Gardner's tribute to Galois—

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Annals of Art History

Filed under: General — m759 @ 11:00 AM

On Misplaced Concreteness

An excerpt from China and Vietnam: The Politics of Asymmetry, by Brantly Womack (Cambridge U. Press, 2006)—

The book is intended to be a contribution to the general theory of international relations as well as to the understanding of China and Vietnam, but I give greater priority to “the case” rather than to the theory. This is a deliberate methodological decision. As John Gerring has argued, case studies are especially appropriate when exploring new causal mechanisms.2  I would argue more broadly that the “case” is the reality to which the theory is secondary. In international relations theory, “realism” is often contrasted to “idealism,” but surely a more basic and appropriate meaning of “realism” is to give priority to reality rather than to theory. The philosopher Alfred North Whitehead defined the Fallacy of Misplaced Concreteness as “neglecting the degree of abstraction involved when an actual entity is considered merely so far as it exemplifies certain categories of thought.”3 In effect, the concept is taken as the concrete reality, and actual reality is reduced to a mere appendage of data. Misplaced Concreteness may well be the cardinal sin of modern social science. It is certainly pandemic in international relations theory, where a serious consideration of the complexities of real political situations is often dismissed as mere “area studies.” Like the Greek god Anteus who was sustained by touching his Mother Earth, theory is challenged and rejuvenated by planting its feet in thick reality.

2 John Gerring, "What Is a Case Study and What Is It Good For?"
   American Political Science Review  98:2 (May 2004), pp. 341-54
3 Alfred North Whitehead, Process and Reality
   (New York: Harper, 1929), p. 11


"Whitehead defined the Fallacy of Misplaced Concreteness…."

The phrase "misplaced concreteness" occurs in the title of a part of an exhibition, "Theme and Variations," by artist Josefine Lyche (Oslo, 2009). I do not know what Lyche had in mind when she used the phrase. A search for possible meanings yielded the above passage.

"In international relations theory, “realism” is often contrasted to “idealism….”

For a more poetic look at "realism" and "idealism" and international relations theory, see Midsummer Eve's Dream.

Contra Harvard

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

Today is commencement day at Princeton.

Sunday's A Post for Galois was suggested, in part, by the fact that the founder and CEO of Amazon.com was that day's Princeton baccalaureate speaker. The Galois post linked to the Amazon reviews of one Christopher G. Robinson, a resident of Cambridge, Mass., whose Amazon book list titled "Step Right Up!" reflects a continuing libertine tradition at Harvard.

For Princeton's commencement day, it seems fitting to cite another Amazon document that reflects the more conservative values of that university.

I recommend the review Postmodern Pythagoras, by Matthew Milliner. Milliner is, in his own words, "an art history Ph.D. candidate at Princeton University."

See also Milliner's other reviews at Amazon.com.

"For every kind of libertine,
there is a kind of cross."

— Saying adapted from Pynchon

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