Log24

Sunday, May 30, 2010

A Post for Galois

Filed under: General,Geometry — m759 @ 2:00 pm

Evariste Galois, 1811-1832 (Vita Mathematica, V. 11)

  • Paperback: 168 pages
  • Publisher: Birkhäuser Basel; 1 edition (December 6, 1996)
  • Language: English
  • ISBN-10: 3764354100
  • ISBN-13: 978-3764354107
  • Product Dimensions: 9.1 x 6 x 0.4 inches
  • Shipping Weight: 9.1 ounces
  • Average Customer Review: 5.0 out of 5 stars  (1 customer review)
  • Amazon Bestsellers Rank: #933,939 in Books

Awarded 5 stars by Christopher G. Robinson (Cambridge, MA USA).
See also other reviews by Robinson.

Galois was shot in a duel on today's date, May 30, in 1832. Related material for those who prefer entertainment to scholarship—

"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.

Today is, incidentally, the feast day of St. Joan of Arc, Die Jungfrau von Orleans. (See "against stupidity" in this journal.)

Thursday, February 18, 2010

Theories: An Outline

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

Truth, Geometry, Algebra

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

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

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

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

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

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

Monday, December 14, 2009

Peer Review at Wikipedia

Filed under: General,Geometry — m759 @ 5:40 pm

Recent Wikipedia activity in the area of finite geometry–

A list, complete up to now, of all Wikipedia changes made by anonymous user Marconet:

Note that all these items are related to changes in links that lead to my own web pages– with one exception, rather technical pages on finite geometry.

A list, complete up to now, of all Wikipedia changes made by anonymous user Greenfernglade:

Again, all these items are related to changes (in this case, deletions) in links that lead to my own web pages. Greenfernglade may or may not be the same person as Marconet. Neither one has a user home page at Wikipedia, but use of the pseudonyms has apparently served to cover up the IP address(es?) of the changes’ originator(s?).

For similar changes in the past, see my “user talk” page at Wikipedia. As I noted there on May 31, 2007, “There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.”

Sunday, November 15, 2009

The Dead Shepherd and…

Filed under: General — m759 @ 8:48 am

Chinese Boxes

Continued from “The  Dead Shepherd,” Jan. 24, 2007

Yesterday’s Washington Post:

“James R. Lilley, 81, a longtime CIA operative in Asia who served as ambassador to China during the Tiananmen Square crackdown… died Nov. 12.”

http://www.log24.com/log/pix09A/091115-JamesRLilley.jpg

James R. Lilley

From a page linked to here on the date of Lilley’s death:

“… the extraordinary set of nested Chinese boxes that we introduced earlier in this series….”

A seemingly unrelated item in today’s New York Times obituaries index:

http://www.log24.com/log/pix09A/091115-Pnueli.jpg

This suggests an article on temporal logic at IBM Developer Works, which contains a link to Time-Rover.com.

This in turn leads to…

http://www.log24.com/log/pix09A/091115-ManTakShing.jpg

Man-Tak Shing

Shing’s CV at the Naval Postgraduate School affirms the usefulness of temporal logic.

Those who prefer metaphysics may consult the novel Many Dimensions referred to in yesterday’s entries and in “Relativity Blues” (Feb. 20, 2005)–

From Many Dimensions:

“Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?”

Good question.

Saturday, November 14, 2009

Mathematics and Narrative, continued:

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

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Wednesday, August 19, 2009

Wednesday August 19, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 10:30 am

Group Actions, 1984-2009

From a 1984 book review:

"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

— Jonathan L. Alperin,
   review of books on group theory,
   Bulletin (New Series) of the American
   Mathematical Society
10 (1984) 121, doi:
   10.1090/S0273-0979-1984-15210-8
 

A more specific example:


Actions of GL(2,3) on a 3x3 coordinate-array

The same example
at Wolfram.com:

Ed Pegg Jr.'s program at Wolfram.com to display a large number of actions of small linear groups over finite fields

Caption from Wolfram.com:
 
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."

Citation data from Wolfram.com:

"GL(2,p) and GL(3,3) Acting on Points"
 from The Wolfram Demonstrations Project,
 http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
 Contributed by: Ed Pegg Jr"

As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:

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

Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
 

Wednesday, May 6, 2009

Wednesday May 6, 2009

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

“My pursuits are a joke
in that the universe is a joke.
One has to reflect
the universe faithfully.”

John Frederick Michell
Feb. 9, 1933 –
April 24, 2009 

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

For Marisa Tomei
  (born Dec. 4, 1964) —
on the day that
   Bob Seger turns 64 —

A Joke:
Points All Her Own

Points All Her Own,
Part I:

(For the backstory, see
the Log24 entries and links
on Marisa Tomei’s birthday
last year.)

Ad for a movie of the book 'Flatland'


Points All Her Own,

Part II:

(For the backstory, see
Galois Geometry:
The Simplest Examples
.)

Galois geometry: the simplest examples

Points All Her Own,

Part III:

(For the backstory, see
Geometry of the I Ching
and the history of
Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

In simpler terms:

Smackdown!

Garfield on May 6, 2009: Smackdown!

Saturday, February 28, 2009

Saturday February 28, 2009

Filed under: General,Geometry — Tags: — m759 @ 8:00 am
Mathematics
and Narrative

continued

Narrative:

xxx

Mathematics:

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference, (July 2000), Springer, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Mathematics:

"Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and O6 + (2)."

J. Sutherland Frame in "Orthogonal Groups over GF(2) and Related Graphs," Springer Lecture Notes in Mathematics vol. 642, Theory and Applications of Graphs (Proceedings, Michigan, May 11–15, 1976), edited by Y. Alavi and D. R. Lick, pp. 174-185

"One has O+(6) ≅ S8, the symmetric group of order 8!…."

— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint. This paper gives some context in superstring theory for the following work of Frame:

[F1] J.S. Frame, The classes and representations of the group of 27 lines and 28 bitangents, Annali
di Mathematica Pura ed Applicata, 32 (1951) 83–119.
[F2] J.S. Frame, Some characters of orthogonal groups over the field of two elements, In: Proc. of the
Second Inter. Conf. on the Theory of Groups, Lecture Notes in Math., Vol. 372, pp. 298–314,
Springer, 1974.
[F3] J. S. Frame, Degree polynomials for the orthogonal groups over GF(2), C. R. Math. Rep. Acad.
Sci. Canada 2 (1980) 253–258.

Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: General,Geometry — Tags: , , , , , — m759 @ 1:00 pm
 
Hollywood Nihilism
Meets
Pantheistic Solipsism

Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
 to hear about our religion
… that we made up."

Tina Fey and Steve Martin at the 2009 Oscars

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:

… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer

 A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.


Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."

As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.

Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.

Heinlein:

"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
    I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."

Stevens:

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:

B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":

"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'….  Its subject is its speaker's sense of nothingness and his need to be cured of it."

This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism.

More positively…

Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space
(or the corresponding
5-dimensional projective space)

The 4x4x4 cube

over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."

Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.

Cara:

Philippe Cara on the Klein correspondence
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.

Saturday, December 6, 2008

Saturday December 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 2:01 pm
Another Opening,
Another Show

"While feasts of Saint Nicholas are not observed nationally, cities with strong German influences like Milwaukee, Cincinnati, and St. Louis celebrate St. Nick's Day on a scale similar to the German custom." —Wikipedia

A footprint from Germany:

Germany
Python-urllib
/504856559/item.html 12/6/2008
1:21 PM

The link in the above footprint leads
to an entry of July 5, 2006.

The access method:

The urllib Module

"The Python urllib module implements a fairly high-level abstraction for making any web object with a URL act like a Python file: i.e., you open it, and get back an object…."


For more pictures and discussion
of the object fetched by Python,
see AntiChristmas 2007.

For a larger and more sophisticated
relative of that object,
 see Solomon's Cube and
the related three presents
from the German link's target:

Spellbound: A trinity of Christmas presents

1. Many Dimensions
2. Boggle
3. My Space

Monday, December 1, 2008

Monday December 1, 2008

Filed under: General — Tags: — m759 @ 12:00 pm
Pictures at
an Exhibition

Day Without Art:

Day Without Art logo: X'd-out frame

and therefore…

Art:

Art logo: frame not X'd out

From Braque's birthday, 2006:

"The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives."

— Georges Braque,
   Reflections on Painting, 1917

Those who wish to follow Braque's advice may try the following exercise from a book first published in 1937:

Carmichael on groups, exercise, p. 440
Hint: See the following
construction of a tesseract:
 
Point, line, square, cube, tesseract
From a page by Bryan Clair

For a different view
of the square and cube
see yesterday's entry
Abstraction and Faith.

Saturday, August 16, 2008

Saturday August 16, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:00 am

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Finite Structure
 on a Book Cover:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.


Walsh Series
states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

Saturday, May 10, 2008

Saturday May 10, 2008

MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in 'Paradise of Childhood,' 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Thursday, April 24, 2008

Thursday April 24, 2008

Filed under: General — m759 @ 9:00 am
Dimensions

George Steiner, interview in The Guardian of April 19:

“No culture has a pact with eternity,” he says. “The conditions which made possible the giants of the western poetic, aesthetic, philosophic tradition no longer really obtain.” Steiner doesn’t believe “there can be a Hamlet without a ghost, a Missa Solemnis without a missa,” and if you say that the questions addressed by religion are “nonsense or baby talk or trivial, I don’t believe that certain dimensions will be available to you. Particularly today, when the atheist case is being put, if I may say so, with such vulgarity of mind.”

Wednesday, April 23, 2008

Wednesday April 23, 2008

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

Upscale Realism

or, "Have some more
wine and cheese, Barack."

(See April 15, 5:01 AM)

  Allyn Jackson on Rebecca Goldstein
in the April 2006 AMS Notices (pdf)

"Rebecca Goldstein’s 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on everyday life. Her new book, Incompleteness: The Proof and Paradox of Kurt Gödel, is a volume in the 'Great Discoveries' series published by W. W. Norton….

In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work. The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped….

A member of the audience posed a question that has been on the minds of many of Goldstein’s readers: Is The Mind-Body Problem based on her own life? She did indeed study philosophy at Princeton, finishing her Ph.D. in 1976 with a thesis titled 'Reduction, Realism, and the Mind.' She said that while there are correlations between her life and the novel, the book is not autobiographical….

She… talked about the relationship between Gödel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, 'Einstein was a real mensch, and Gödel was very neurotic.' Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles– exiles from Europe and intellectual exiles. Gödel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as implying the sweeping view that 'everything is relative.' These misinterpretations irked both men, said Goldstein. 'Einstein and Gödel were realists and did not like it when their work was put to the opposite purpose.'"


Related material:

From Log24 on
March 22 (Tuesday of
Passion Week), 2005:

 
"'What is this Stone?' Chloe asked…. 'It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

Many Dimensions,
by Charles Williams, 1931

For more on this theme
appropriate to Passion Week
Jews playing God — see

The image “http://www.log24.com/log/pix05/050322-Trio.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences
Research Institute
at the
Commonwealth Club,
San Francisco,
Tuesday, March 22.

Wine and cheese
reception at 5:15 PM
(San Francisco time).

From
UPSCALE,
a website of the
physics department at
the University of Toronto:

Mirror Symmetry

 

Robert Fludd: Universe as mirror image of God

"The image [above]
is a depiction of
the universe as a
mirror image of God,
drawn by Robert Fludd
in the early 17th century.

The caption of the
upper triangle reads:

'That most divine and beautiful
counterpart visible below in the
flowing image of the universe.'

The caption of the
lower triangle is:

'A shadow, likeness, or
reflection of the insubstantial*
triangle visible in the image
of the universe.'"

* Sic. The original is incomprehensibilis, a technical theological term. See Dorothy Sayers on the Athanasian Creed and John 1:5.

For further iconology of the
above equilateral triangles,
see Star Wars (May 25, 2003),
Mani Padme (March 10, 2008),
Rite of Sping (March 14, 2008),
and
Art History: The Pope of Hope
(In honor of John Paul II
three days after his death
in April 2005).

Happy Shakespeare's Birthday.

Monday, April 7, 2008

Monday April 7, 2008

Filed under: General,Geometry — m759 @ 2:20 am

“Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?”

— Charles Williams, Many Dimensions

Saturday, February 23, 2008

Saturday February 23, 2008

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

"An acute study of the links
between word and fact"
Nina daVinci Nichols

 
Thanks to a Virginia reader for a reminder:
 
Virginia /391062427/item.html? 2/22/2008 7:37 PM
 
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.

The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure–

Change

The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Saturday, September 15, 2007

Saturday September 15, 2007

Filed under: General — Tags: — m759 @ 8:00 pm
The Crimson Passion
continues...
 
Professors: Post Your Syllabi
 
Professors should post their
course syllabi before move-in,
not after class has started

The Harvard Crimson

Published On Friday, September 14, 2007  12:54 AM

"Classes start in three days, and that means it’s time to… examine course syllabi– that is if you can find them…." More >>


Classics 101:
The Holy Spook
 

IMAGE- Anthony Hopkins in 'The Human Stain'

Prof. Coleman Silk introducing
 freshmen to academic values

The Course Begins:

Larry Summers, former president
of Harvard, was recently invited,
then disinvited, to speak at a
politically correct UC campus.

A Guest Lecturer Speaks:

"This is so pathetic. I used to write long disquisitions on the ethical dimensions of behavior like this, but years of it can make a girl get very tired. And that's because this stuff is tiresome, and boring, and wrong, and pathetic, and so very indicative of the derailed character of academic life. It's more important to keep punishing Summers for a comment he made years ago– and apologized for many times over, and essentially lost the presidency of Harvard over– than it is just to move on and let free exchange happen on campuses. I doubt Summers would have devoted his time before the Regents to theorizing gender (not that I would personally care much if he did– I was not so mortally wounded by his observations as others were), and he is a brilliant man with much of value to bring to a visit with the Regents. But what does that matter when the opportunity to mob a politically incorrect academic presents itself?" —Erin O'Connor on Sept. 15, 2007

Illustration of the Theme:

Clarinetist Ken Peplowski
plays "Cry Me a River"
as Nicole Kidman focuses
the students' attention.

A sample Holy Spook,
Kurt Vonnegut, was introduced
by Peplowski on the birthday
this year of Pope Benedict XVI.

"Deeply vulgar"
Academic characterization
of Harvard president Summers

"Do they still call it
 the licorice stick?"
Kurt Vonnegut

Related Material:

The image “http://www.log24.com/log/pix07A/070915-Summers.jpg” cannot be displayed, because it contains errors.

Midnight Drums for Larry

Monday, July 23, 2007

Monday July 23, 2007

Daniel Radcliffe
is 18 today.
Daniel Radcliffe as Harry Potter

Greetings.

“The greatest sorcerer (writes Novalis memorably)
would be the one who bewitched himself to the point of
taking his own phantasmagorias for autonomous apparitions.
Would not this be true of us?”

Jorge Luis Borges, “Avatars of the Tortoise”

El mayor hechicero (escribe memorablemente Novalis)
sería el que se hechizara hasta el punto de
tomar sus propias fantasmagorías por apariciones autónomas.
¿No sería este nuestro caso?”

Jorge Luis Borges, “Los Avatares de la Tortuga

Autonomous Apparition

At Midsummer Noon:

“In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew from
a brief description in Waite’s
The Holy Kabbalah (1929) of
a supernatural cubic stone
on which was inscribed
‘the Divine Name.’”
The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.
Related material:
It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure 

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

See also
as well as
Hofstadter on
his magnum opus:
“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”
Goedel Escher Bach coverHofstadter’s cover.

Here are three patterns,
“shadows” of a sort,
derived from a different
“central object”:
Faces of Solomon's Cube, related to Escher's 'Verbum'

Click on image for details.

Sunday, June 24, 2007

Sunday June 24, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm
Raiders of
the Lost Stone

(Continued from June 23)

Scott McLaren on
Charles Williams:
 
"In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew
from a brief description in Waite's
The Holy Kabbalah (1929)
of a supernatural cubic stone
on which was inscribed
'the Divine Name.'"

The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.

Related material:

The image “http://www.log24.com/log/pix07/070624-Cube.gif” cannot be displayed, because it contains errors.

Solomon's Cube,

Geometry of the 4x4x4 Cube,

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Sunday, March 18, 2007

Sunday March 18, 2007

Filed under: General,Geometry — m759 @ 2:20 pm

Update to
The Geometry of Logic:

A detailed description of a group of 16 “logical automorphisms” of the 16 binary connectives has been given in the paper “Simetria y Logica: La notacion de Peirce para los 16 conectivos binarios,” by Mireya Garcia, Jhon Fredy Gomez, and Arnold Oostra. (Published in the Memorias del XII Encuentro de Geometria y sus Aplicaciones, Universidad Pedagogica Nacional, Bogota, June 2001; on the Web at http://www.unav.es/gep/Articulos/SimetriaYLogica.pdf.) The authors do not identify this group as a subgroup of the affine group of A (the finite affine geometry of four dimensions over the two-element field); this can serve as an exercise.  Another exercise: determining whether the authors’ order-16 group includes all transformations that might reasonably be called “logical automorphisms” of the 16 binary connectives.

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
/446066083/item.html
2/20/06: The Past Revisited
(with link to online text of
Many Dimensions, by Charles Williams)

Poland 2/2/07 2:38 AM
/426273644/item.html
1/15/06 Inscape
(the mathematical concept, with
square and “star” diagrams)

Poland 2/2/07 3:30 AM
nextdate=2%252f8%252f20…
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
/524081776/item.html
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
/524459252/item.html
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
/409355167/item.html
12/19/05 Quarter to Three (cont.)

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

Poland 2/2/07 5:03 AM
/490604390/item.html
5/29/06 For JFK’s Birthday
(The Call Girls revisited)

Poland 2/2/07 5:32 AM
/522299668/item.html
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
/447354678/item.html
2/22/06 In the Details
(Harvard’s president resigns,
with accompanying “rosebud”)

Friday, November 24, 2006

Friday November 24, 2006

Filed under: General,Geometry — Tags: — m759 @ 1:06 pm
Galois’s Window:

Geometry
from Point
to Hyperspace


by Steven H. Cullinane

  Euclid is “the most famous
geometer ever known
and for good reason:
  for millennia it has been
his window
  that people first look through
when they view geometry.”

  Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace
,
by Leonard Mlodinow

“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
  concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”

— Paul Halmos in
I Want To Be a Mathematician

Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.

Each of these spaces
has infinitely many points.

Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:

+ 0 1
0 0 1
1 1 0
* 0 1
0 0 0
1 0 1
We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:
Galois affine spaces

From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call

Galois's Window

Galois’s Window.

The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.

For more on these cases, see

The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.

(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)

These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.

The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
  Elements of Finite Geometry.

Wednesday, November 22, 2006

Wednesday November 22, 2006

Filed under: General — m759 @ 9:00 pm
Rock of Ages

“Who knows where madness lies?”
— Rhetorical question
in “Man of La Mancha”
(See previous entry.)

Using madness to
seek out madness, let us
  consult today’s numbers…

Pennsylvania Lottery
Nov. 22, 2006:

Mid-day 487
Evening 814

The number 487 leads us to
page 487 in the
May 1977 PMLA,
The Form of Carnival
in Under the Volcano
“:

“The printing presses’ flywheel
marks the whirl of time*
    that will split La Despedida….”

The image “http://www.log24.com/log/pix06B/061122-Flywheel.gif” cannot be displayed, because it contains errors.

Flywheel

From Dana Grove,
A Rhetorical Analysis of
Under the Volcano
,
page 92:

“… In this way, mystical as well as psychological dimensions are established.  Later on, the two pass by a printer’s shop window and curiously stop to inspect, amidst wedding portraits and well in front of the revolving flywheel of the printing machines, ‘a photographic enlargement purporting to show the disintegration of a glacial deposit in the Sierra Madre, of a great rock split by forest fires.’  Significantly the picture is called ‘La Despedida,’ the Parting.  Yvonne cannot help but see the symbolic significance of the photograph and wishes with all of her might ‘to heal the cleft rock’ just as she wishes to heal the divorce….”

Some method in this madness
is revealed by the evening
lottery number, 814, which
leads to an entry of 8/14:

Cleavage Term

“… a point of common understanding
between the classic and romantic worlds.
Quality, the cleavage term between
hip and square, seemed to be it.”
Robert M. Pirsig 

The image “http://www.log24.com/log/pix06B/061122-Goldstein.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein

The 8/14 entry also deals with
Rebecca Goldstein, who
seems to understand
such cleavage
very well.

(See also today’s previous entry.)

* Cf. Shakespeare’s “whirligig of time
linked to in the previous entry.)

Monday, September 11, 2006

Monday September 11, 2006

Filed under: General — m759 @ 11:00 pm

A Sermon for Sartre

A sequel to
Les Mots:
Les Nombres

 “Words and numbers
are of equal value,
for, in the
cloak of knowledge,
one is warp
and the other woof.”
— The princesses
Rhyme and Reason
in The Phantom Tollbooth,
by Norton Juster, 1961

Lotteries
9/11/06

Midday

Evening
NY 394 628
PA 527 916

“Time and chance
happeneth to them all.”

— Ecclesiastes 9:11

Hermeneutics:

The numbers may be regarded
as coordinates in a map
of one spatial dimension
(a road dimension:
394 – Chautauqua, NY)
and of three
temporal dimensions
(birthday dimension 6/28,
Sartre dimension 5/27,
religious dimension 9/16).

This interpretation is of course
rather arbitrary, but so are most
interpretations.

Related material:
Sontag and Sartre this morning
and Sontag on Sunday.

Update of 1:29 AM 9/12:

 The image “http://www.log24.com/log/pix06A/060912-Doonesbury2.gif” cannot be displayed, because it contains errors.
“HASS-D”– Click here.

Tuesday, August 22, 2006

Tuesday August 22, 2006

Filed under: General,Geometry — m759 @ 12:00 am

Beginnings

“Nothing ever begins.

There is no first moment; no single word or place from which this or any other story springs.

The threads can always be traced back to some earlier tale, and to the tales that preceded that; though as the narrator’s voice recedes the connections will seem to grow more tenuous, for each age will want the tale told as if it were of its own making.”

— Clive Barker, Weaveworld

“No mathematical subject lies closer to intuition than the geometry of two and three dimensions.”

Robert E. Greene, beginning an April 1998 review of Three-Dimensional Geometry and Topology, by William P. Thurston

Thurston’s book provides some background for today’s opening lecture by Richard Hamilton, “The Poincare Conjecture,” at the beginning of the International Congress of Mathematicians in Madrid.

Hamilton is likely to discuss the Poincare conjecture in the wider context of Perelman‘s recent work on Thurston’s geometrization conjecture.

In “The Eight Model Geometries,” section 3.8 of his book, Thurston provides yet another beginning–

“What is a geometry?”

Saturday, July 29, 2006

Saturday July 29, 2006

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:02 pm

Big Rock

Thanks to Ars Mathematicaa link to everything2.com:

“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”

Another example:

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis.  See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:

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

The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts.  (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.

“The rock cannot be broken.
It is the truth.”

Wallace Stevens,
“Credences of Summer”

 

Thursday, July 13, 2006

Thursday July 13, 2006

Filed under: General — m759 @ 4:00 pm
Carpe Diem

From the new MySpace.com
weblog of Michio Kaku:

Thursday, July 13, 2006

Hyperspace and a Theory of Everything

What lies beyond our 4 dimensions?
By Michio Kaku

When I was a child, I used to visit the Japanese Tea Garden in San Francisco. I would spend hours fascinated by the carp, who lived in a very shallow pond just inches beneath the lily pads, just beneath my fingers, totally oblivious to the universe above them.

I would ask myself a question only a child could ask: what would it be like to be a carp?

 
A child, or Maurits Escher:

The image “http://www.log24.com/log/pix06A/060713-ThreeWorlds.jpg” cannot be displayed, because it contains errors.
Three Worlds,
1955

Wednesday, May 10, 2006

Wednesday May 10, 2006

Filed under: General,Geometry — Tags: , — m759 @ 4:29 pm
My Space

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

“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”

— Robert A. Heinlein,
The Number of the Beast

The image “http://www.log24.com/theory/images/Screenshot-64cube.gif” cannot be displayed, because it contains errors.

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

And so…

Compare and contrast:

Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.

Related material:

A more extensive excerpt from
The Number of the Beast, and

Story Theory and
the Number of the Beast.

Saturday, March 25, 2006

Saturday March 25, 2006

Filed under: General — m759 @ 4:23 pm
Built

In memory of Rolf Myller,
who died on Thursday,
March 23, 2006, at
Mount Sinai Hospital
in Manhattan:

Myller was,
according to the
New York Times,
an architect
whose eclectic pursuits
included writing
children’s books,
The Bible Puzzle Book, and
Fantasex: A Book of Erotic Games.

He also wrote, the Times says,
Symbols and Their Meaning
(1978), a graphic overview of
children’s nonverbal communication.”
This is of interest in view of the
Log24 reference to “symbol-mongers”
on the date of Myller’s death.

In honor of Women’s History Month
and of Myller’s interests in the erotic
and in architecture, we present
the following work from a British gallery.

The image “http://www.log24.com/log/pix06/060325-WhiteCube.jpg” cannot be displayed, because it contains errors.

This work might aptly be
  retitled “Brick Shithouse.”

Related material:

(1) the artist’s self-portrait

The image “http://www.log24.com/log/pix06/060325-LizaLouSelfPortrait.jpg” cannot be displayed, because it contains errors.

and, in view of the cover
illustration for Myller’s
The Bible Puzzle Book,

The image “http://www.log24.com/log/pix06/060325-Tower.jpg” cannot be displayed, because it contains errors.

(2) the monumental treatise
by Leonard Shlain

The Alphabet Versus
the Goddess: The Conflict
Between Word and Image
.

For devotees of women’s history
and of the Goddess,
here are further details from
the White Cube gallery:

Liza Lou

03.03.06 – 08.04.06

White Cube is pleased to present the first UK solo exhibition by Los Angeles-based artist Liza Lou.

Combining visionary, conceptual and craft approaches, Lou makes mixed-media sculptures and room-size installations that are suggestive of a transcendental reality. Lou’s work often employs familiar, domestic forms, crafted from a variety of materials such as steel, wood, papier-mâché and fibreglass, which is then covered with tiny glass beads that are painstakingly applied, one at a time, with tweezers. Dazzling and opulent and constantly glistening with refracted light, her sculptures bristle with what Peter Schjeldahl has aptly described as ‘surreal excrescence’.

This exhibition, a meditation on the vulnerability of the human body and the architecture of confinement, will include several new figurative sculptures as well as two major sculptural installations. Security Fence (2005) is a large scale cage made up of four steel, chain link walls, topped by rings of barbed wire and Cell (2004-2006), as its name suggests, is a room based on the approximate dimensions of a death row prison cell, a kind of externalized map of the prisoner’s mind. Both Security Fence and Cell, like Lou’s immense earlier installations Kitchen (1991-1995) and Back Yard (1995-1999) are characterized by the absence of their real human subject. But whereas the absent subject in Kitchen and Back Yard could be imagined through the details and accessories carefully laid out to view, in Lou’s two new installations the human body is implied simply through the empty volume created by the surrounding architecture. Both Cell and Security Fence are monochromatic and employ iconic forms that make direct reference to Minimalist art in its use of repetition, formal perfection and materiality. In contrast to this, the organic form of a gnarled tree trunk, Scaffold (2005-2006), its surface covered with shimmering golden beads, juts directly out from the wall.

Lou’s work has an immediate ‘shock’ content that works on different levels: first, an acknowledgement of the work’s sheer aesthetic impact and secondly the slower comprehension of the labour that underlies its construction. But whereas in Lou’s earlier works the startling clarity of the image is often a counterpoint to the lengthy process of its realization, for the execution of Cell, Lou further slowed down the process by using beads of the smallest variety with their holes all facing up in an exacting hour-by-hour approach in order to ‘use time as an art material’.

Concluding this body of work are three male figures in states of anguish. In The Seer (2005-2006), a man becomes the means of turning his body back in on himself. Bent over double, his body becomes an instrument of impending self-mutilation, the surface of his body covered with silver-lined beads, placed with the exactitude and precision of a surgeon. In Homeostasis (2005-2006) a naked man stands prostrate with his hands up against the wall in an act of surrender. In this work, the dissolution between inside and outside is explored as the ornate surface of Lou’s cell-like material ‘covers’ the form while exposing the systems of the body, both corporeal and esoteric. In The Vessel (2005-2006), Christ, the universal symbol of torture and agony holds up a broken log over his shoulders. This figure is beheaded, and bejewelled, with its neck carved out, becoming a vessel into which the world deposits its pain and suffering.

Lou has had numerous solo exhibitions internationally, including Museum Kunst Palast, Düsseldorf, Henie Onstad Kunstsenter, Oslo and Fondació Joan Miró, Barcelona. She was a 2002 recipient of the MacArthur Foundation Fellowship.

Liza Lou’s film Born Again (2004), in which the artist tells the compelling and traumatic story* of her Pentecostal upbringing in Minnesota, will be screened at 52 Hoxton Square from 3 – 25 March courtesy of Penny Govett and Mick Kerr.

Liza Lou will be discussing her work following a screening of her film at the ICA, The Mall, London on Friday 3 March at 7pm. Tickets are available from the ICA box office (+ 44 (0) 20 7930 3647).

A fully illustrated catalogue, with a text by Jeanette Winterson and an interview with Tim Marlow, will accompany the exhibition.

White Cube is open Tuesday to Saturday, 10.00 am to 6.00 pm.

For further information please contact Honey Luard or Susannah Hyman on + 44 (0) 20 7930 5373

* Warning note from Adrian Searle
    in The Guardian of March 21:
   “How much of her story is
    gospel truth we’ll never know.”

For deeper background on
art, patriarchal religion,
and feminism, see
The Agony and the Ya-Ya.

Monday, February 20, 2006

Monday February 20, 2006

Filed under: General — m759 @ 12:00 am

The Past Revisited

From Log24 a year ago on this date, a quote from Many Dimensions (1931), by Charles Williams:

“Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?”

For the rest of the story, see the downloadable version at Project Gutenberg of Australia.

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: General,Geometry — m759 @ 3:09 pm
Analogical
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

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

Another representation of
the 24-cell
:

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
,”
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:

[…]

“I’d like to wrap up with a few small comments about last Week.  There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’.  The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”

[…]

Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.

NDE


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics
:


Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London

Disclaimer:

The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Monday, August 22, 2005

Monday August 22, 2005

Filed under: General — Tags: , , — m759 @ 4:07 pm
The Hole

Part I: Mathematics and Narrative

The image “http://www.log24.com/log/pix05B/050822-Narr.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis on last month's conference on "mathematics and narrative"–

Doxiadis is describing how talks by two noted mathematicians were related to

    "… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "

Doxiadis says both talks had "a big hole in the middle."  

    "Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts.  [In one part] I’ll tell you a few things about proofs.  [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
  "And the hole?"
  "This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’  But the similarity was not clearly explained: the hole, you see, the ‘miracle.’  Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
 

Part II: Possible Worlds

"At times, bullshit can only be countered with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole
 

Part III: Modal Theology

 
  "'What is this Stone?' Chloe asked….
  '…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

  — Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)


"The lapis was thought of as a unity and therefore often stands for the prima materia in general."

  — Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)

 

"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."

  — The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)
 
"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

 

 

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: , — m759 @ 9:00 am
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Thursday, August 4, 2005

Thursday August 4, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:00 pm
Visible Mathematics, continued

 

Today's mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

 

Related material: Visualizing GL(2,p)

"The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

 

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

 

Sunday, May 8, 2005

Sunday May 8, 2005

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

Geometry and Theology

See

the science fiction writer mentioned in a Friday entry.

Mark Olson’s article is at the website of the New England Science Fiction Association, publisher of Ingathering: The Complete People Stories of Zenna Henderson.  This book, by one of my favorite science-fiction authors, was apparently edited by the same Mark Olson.

The following remarks seem relevant to the recurring telepathy theme in Henderson:

From the first article cited above,
David L. Neuhouser,
Higher Dimensions in the Writings of C. S. Lewis (pdf):

“If we are three-dimensional cross-sections of four-dimensional reality, perhaps we are parts of the same body. In fact, we know we are parts of the same body in some way, this four-dimensional idea just may help us to see it more clearly. Remember the preceding comments are mine, not Lewis’s. He puts it this way, ‘That we can die “in” Adam and live “in” Christ seems to me to imply that man as he really is differs a good deal from man as our categories of thought and our three-dimensional imaginations represent him; that the separateness… which we discern between individuals, is balanced, in absolute reality, by some kind of inter-inanimation of which we have no conception at all. It may be that the acts and sufferings of great archetypal individuals such as Adam and Christ are ours, not by legal fiction, metaphor, or causality, but in some much deeper fashion. There is no question, of course, of individuals melting down into a kind of spiritual continuum such as Pantheistic systems believe in; that is excluded by the whole tenor of our faith.'”

From Webster’s Unabridged, 1913 edition:

inanimate
, v. t.

[Pref. in- in (or intensively) + animate.]
 To animate. [Obs.] — Donne.

inanimation, n.

Infusion of life or vigor;
animation; inspiration.
[Obs.]
The inanimation of Christ
living and breathing within us.
Bp. Hall.

Related words…

Also from the 1913 Webster’s:

circumincession, n.

[Pref. circum- + L. incedere, incessum, to walk.]
(Theol.) The reciprocal existence in each other
of the three persons of the Trinity.

From an online essay:

perichoresis
, n.

“The term means mutual indwelling or, better, mutual interpenetration and refers to the understanding of both the Trinity and Christology. In the divine perichoresis, each person has ‘being in each other without coalescence’ (John of Damascus ca. 650). The roots of this doctrine are long and deep.”

—  Bert Waggoner

coinherence, n.

“In our human experience of personhood, at any rate in a fallen world, there is in each person an inevitable element of exclusiveness, of opaqueness and impenetrability.  But with the three divine persons it is not so.  Each is entirely ‘open’ to the others, totally transparent and receptive.  This transparency and receptivity is summed up in the Greek notion of perichoresis, which Gibbon once called ‘the deepest and darkest corner of the whole theological abyss.’  Rendered in Latin as circumincessio and in English usually as ‘coinherence,’ the Greek term means literally, cyclical movement, and so reciprocity, interchange, mutual indwelling.  The prefix peri bears the sense ‘around,’ while choresis is linked with chora, ‘room,’ space,’ ‘place’ or ‘container,’ and with chorein, to ‘go,’ ‘advance,’ ‘make room for’ or ‘contain.’  Some also see a connection with choros, ‘dance,’ and so they take perichoresis to mean ’round dance.’  Applied to Christ, the term signifies that his two natures, the divine and the human, interpenetrate one another without separation and without confusion.  Applied to the Trinity, it signifies that each person ‘contains’ the other two and ‘moves’ within them.  In the words of St Gregory of Nyssa, ‘All that is the Father’s is seen in the Son, and all that is the Son’s belongs also the Father. For the whole Son abides in the Father, and he has in his turn the whole Father abiding in himself.’ 

By virtue of this perichoresis, Father, Son and Holy Spirit ‘coinhere‘ in one another, each dwelling in the other two through an unceasing movement of mutual love – the ’round dance’ of the Trinity.”

— Timothy Ware, Bishop Kallistos of Diokleia,
    The Human Person as an Icon of the Trinity

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General — Tags: , — m759 @ 7:59 pm

The God Factor

Reba McEntire on
Make a Difference Day:

"Kids who may never get out of their town will be able to see the world through books. But I'm talking about my passion. What's yours?"

"There is the God factor…."

— NickyJett, Xanga comment

"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"

Many Dimensions,
by Charles Williams, 1931

For more on this theme
appropriate to Passion Week
Jews playing God — see

The image “http://www.log24.com/log/pix05/050322-Trio.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences Research Institute
at the Commonwealth Club, San Francisco,
Tuesday, March 22.  Wine and cheese
reception at 5:15 PM (San Francisco time).
 
For the meaning of the diamond,
see the previous entry.

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 pm

Relativity Blues

Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry.  Here is some related material.

In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).

(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)

On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:

"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."

One possible response to such statements was recently provided in some detail by a Princeton philosophy professor.  See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.

A more thoughtful response would take into account the following:

1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson

2. The physics underlying Einstein's remarks on free will, God, and dice
 
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels  Secret Passages and Broken Symmetries

4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University

5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.

From a review of Gilead by Jane Vandenburgh:  

"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"

For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries.  If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003,  and those of November 5, 2002.

From Many Dimensions (Eerdmans paperback, 1963, page 53):

"Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?"

A recent answer:

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Thursday, February 17, 2005

Thursday February 17, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:00 pm
Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

On the Lapis Philosophorum,
the Philosophers' Stone –

"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"
Many Dimensions,
by Charles Williams, 1931
(Eerdmans paperback,
April 1979, pp. 43-44)

"The lapis was thought of as a unity
and therefore often stands for
the prima materia in general."
Aion, by C. G. Jung, 1951
(Princeton paperback,
1979, p. 236)

"Its discoverer was of the opinion that
he had produced the equivalent of
the primordial protomatter
which exploded into the Universe."
The Stars My Destination,
by Alfred Bester, 1956
(Vintage hardcover,
July 1996, p. 216)

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

See also
The Diamond Archetype.

For more on modal theology, see

Kurt Gödel's Ontological Argument
and

 The Ontological Argument
 from Anselm to Gödel.

Wednesday, August 18, 2004

Wednesday August 18, 2004

Filed under: General — m759 @ 3:00 am

Drunk Bird


T. Charles Erickson
Shizuo Kakutani
in the 1980’s

Kakutani died yesterday.

“A drunk man will find his way home, but a drunk bird may get lost forever.”

— Shizuo Kakutani, quoted by J. Chang in Stochastic Processes (ps), p. 1-19.  Chang says the quote is from an R. Durrett book on probability.

Meaning:

A random walk in d dimensions is recurrent if d = 1 or d = 2, but transient if d is greater than or equal to 3.


From a web page on Kylie Minogue:

Turns out she’s a party girl
who loves Tequila:
“Time disappears with Tequila.  
  It goes elastic, then vanishes.”



Kylie sings
“Locomotion”

From a web page on Malcolm Lowry’s classic novel Under the Volcano

The day begins with Yvonne’s arrival at the Bella Vista bar in Quauhnahuac. From outside she hears Geoffrey’s familiar voice shouting a drunken lecture this time on the topic of the rule of the Mexican railway that requires that  “A corpse will be transported by express!” (Lowry, Volcano, p. 43).

For further literary details in memory of Shizuo Kakutani, Yale mathematician and father of book reviewer Michiko Kakutani, see

Santa Versus the Volcano.

Of course, Kakutani himself would probably prefer the anti-Santa, Michael Shermer.  For a refutation of Santa by this high priest of Scientism, see

Miracle on Probability Street

(Scientific American, July 26, 2004). 

Wednesday, March 3, 2004

Wednesday March 3, 2004

Filed under: General — Tags: , , — m759 @ 8:00 pm

Deep Play

In the previous entry, there was a reference to Carl Kaysen, former director of the Institute for Advanced Study at Princeton and father of Susanna Kaysen, author of Girl, Interrupted.

A search for further information on Carl Kaysen led to

Mark Turner, Cognitive Dimensions of Social Science: The Way We Think About Politics, Economics, Law, and Society, Oxford University Press, 2001.  For a draft of this work, click here.

Turner's book describes thought and culture in terms of what he calls "blends."  It includes a meditation on

Clifford Geertz, "Deep Play: Notes on the Balinese Cockfight," in Dædalus, Journal of the American Academy of Arts and Sciences, issue entitled, "Myth, Symbol, and Culture," Winter 1972, volume 101, number 1

That Turner bases weighty ruminations of what he is pleased to call "social science" on the properties of cockfights suggests that the academic world is, in some respects, even more bizarre than the mental hospital described by Kaysen's daughter.

Still, Turner's concept of "blends" is not without interest.

Here is a blend based on a diagram of the fields in which Turner and Kaysen père labor:

"politics, economics,
law, and society" (Turner)

and "economics, sociology,
politics and law" (Kaysen).

In the previous entry we abstracted from the nature of these academic pursuits, representing them simply as sets in a Venn diagram.  This led to the following religious icon, an example of a Turner "blend" —


The Jewel
in Venn's Lotus.

Here is another "blend," related both to the religious material in the previous entry and to Geertz's influential essay.

From my entry for
St. Patrick's Day, 2003
:

Summa Theologica

How can you tell there's an Irishman
present at a cockfight?
He enters a duck.
How can you tell a Pole is present?
He bets on the duck.
How can you tell an Italian is present?
The duck wins.

(Source: Blanche Knott,
Truly Tasteless Jokes)

Illustration for the entries
of Oct. 27, 2003:

El Pato-lógico and a

"dream of heaven."

Friday, September 19, 2003

Friday September 19, 2003

Filed under: General — m759 @ 3:57 am

The Mysteries of 26

My entry of May 26, 2003 —

Many Dimensions — Why 26? 

dealt with the question of whether this number, said to be of significance (as a number of dimensions) in theoretical physics, has any purely mathematical properties of interest.

That entry contained the above figure, a so-called Levi graph illustrating point/line incidence in the finite projective plane with 13 points and 13 lines, PG(2,3).

It turns out that in a paper of April 7, 2000, John H. Conway and Christopher S. Simons discussed a close connection between this plane and the Monster group.  See

26 Implies the Bimonster 

(Journal of Algebra. Vol. 235, no. 2.
MR 2001k:20028).

Conway had written about such a connection as early as 1985.

I apologize for not knowing about this sooner, and so misleading any mathematical readers about the number 26, which it seems does have considerable purely mathematical significance.

Thursday, September 4, 2003

Thursday September 4, 2003

Filed under: General — m759 @ 2:42 am

Monolith

“Music can name the unnameable
and communicate the unknowable.”

— Quotation attributed to Leonard Bernstein

“Finally we get to Kubrick’s ultimate trick….  His secret is in plain sight…. The film is the monolith. In a secret that seems to never have been seen by anyone: the monolith in the film has the same exact dimensions as the movie screen on which 2001 was projected.”

—  Alchemical Kubrick 2001, by Jay Weidner 

My entry of Saturday, August 30,
included the following illustration:

My entry of Monday, September 1,
concluded with the black monolith.

“There is little doubt that the black monolith
in 2001 is the Philosopher’s Stone.”

—  Alchemical Kubrick 2001, by Jay Weidner 

 The philosopher Donald Davidson
died on Saturday, August 30.

The New York Times says that as an undergraduate, Davidson “persuaded Harvard to let him put on ‘The Birds’ by Aristophanes and played the lead, Peisthetairos, which meant memorizing 700 lines of Greek. His friend and classmate Leonard Bernstein, with whom he played four-handed piano, wrote an original score for the production.”

Perhaps they are still making music together.

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 pm

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Tuesday, May 27, 2003

Tuesday May 27, 2003

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

Mental Health Month, Day 27:

Conspiracy Theory and
Solomon's Seal

In our journey through Mental Health Month, we have now arrived at day 27. This number, the number of lines on a non-singular cubic surface in complex projective 3-space, suggests it may be time to recall the following note (a sort of syllabus for an imaginary course) from August 1997, the month that the Mel Gibson film "Conspiracy Theory" was released.

Conspiracy Theory 101
August 13, 1997

Fiction:

(A) Masks of the Illuminati, by Robert Anton Wilson, Pocket Books, New York, 1981.  Freemasonry meets The Force (starring James Joyce and Albert Einstein).
(B) The Number of the Beast, by Robert A. Heinlein, Ballantine Books, New York, 1980.  "Pantheistic multiple solipsism" and transformation groups in n-dimensional space combine to yield "the ultimate total philosophy." (p. 438). 
(C) The Essential Blake, edited by Stanley Kunitz, MJF Books, New York, 1987.  "Fearful symmetry" in context.

Fact:

(1) The Cosmic Trigger, by Robert Anton Wilson, Falcon Press, Phoenix, 1986 (first published 1977).  Page 245 reveals that "the most comprehensive conspiracy theory," that of the physicist Sir Arthur Eddington, is remarkably similar to Heinlein's theory in (B) above.
(2) The Development of Mathematics, by E. T. Bell, 2nd. ed., McGraw-Hill, New York, 1945.  See the discussion of "Solomon's seal," a geometric configuration in complex projective 3-space.  This is as good a candidate as any for Wilson's "Holy Guardian Angel" in (A) above.
(3) Finite Projective Spaces of Three Dimensions, by J. W. P. Hirschfeld, Clarendon Press, Oxford, 1985.  Chapter 20 shows how to represent Solomon's seal in the 63-point 5-dimensional projective space over the 2-element field.  (The corresponding 6-dimensional affine space, with 64 points, is reminiscent of Heinlein's 6-dimensional space.)
 

See also China's 3,000-year-old "Book of Transformations," the I Ching, for more philosophy and lore of the affine 6-dimensional space over the binary field.

© 1997 S. H. Cullinane 

For a more up-to-date and detailed look at the mathematics mentioned above, see

Abstract Configurations
in Algebraic Geometry
,

by Igor Dolgachev.

"Art isn't easy." — Stephen Sondheim

Monday, May 26, 2003

Monday May 26, 2003

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

Mental Health Month, Day 26:

Many Dimensions,

Part III — Why 26?

At first blush, it seems unlikely that the number 26=2×13, as a product of only two small primes (and those distinct) has any purely mathematical properties of interest. (On the other hand, consider the number 6.)  Parts I and II of “Many Dimensions,” notes written earlier today, deal with the struggles of string theorists to justify their contention that a space of 26 dimensions may have some significance in physics.  Let them struggle.  My question is whether there are any interesting purely mathematical properties of 26, and it turns out, surprisingly, that there are some such properties. All this is a longwinded way of introducing a link to the web page titled “Info on M13,” which gives details of a 1997 paper by J. H. Conway*.

Info on M13

“Conway describes the beautiful construction of a discrete mathematical structure which he calls ‘M13.’  This structure is a set of 1,235,520 permutations of 13 letters. It is not a group. However, this structure represents the answer to the following group theoretic question:

Why do the simple groups M12 and L3(3) share some subgroup structure?

In fact, both the Mathieu group M12 and the automorphism group L3(3) of the projective plane PG(2,3) over GF(3) can be found as subsets of M13.  In addition, M13 is 6-fold transitive, in the sense that it contains enough permutations to map any two 6-tuples made from the thirteen letters into each other.  In this sense, M13 could pass as a parent for both M12 and L3(3).  As it is known from the classification of primitive groups that there is no finite group which qualifies as a parent in this sense.  Yet, M13 comes close to being a group.

To understand the definition of M13 let us have a look at the projective geometry PG(2,3)….

The points and the lines and the “is-contained-in” relation form an incidence structure over PG(2,3)….

…the 26 objects of the incidence structure [are] 13 points and 13 lines.”

Conway’s construction involves the arrangement, in a circular Levi graph, of 26 marks representing these points and lines, and chords representing the “contains/is contained in” relation.  The resulting diagram has a pleasingly symmetric appearance.

For further information on the geometry of the number 26, one can look up all primitive permutation groups of degree 26.  Conway’s work suggests we look at sets (not just groups) of permutations on n elements.  He has shown that this is a fruitful approach for n=13.  Whether it may also be fruitful for n=26, I do not know.

There is no obvious connection to physics, although the physics writer John Baez quoted in my previous two entries shares Conway’s interest in the Mathieu groups. 

 * J. H. Conway, “M13,” in Surveys in Combinatorics, 1997, edited by R. A. Bailey, London Mathematical Society Lecture Note Series, 241, Cambridge University Press, Cambridge, 1997. 338 pp. ISBN 0 521 59840 0.

Monday May 26, 2003

Filed under: General,Geometry — m759 @ 4:25 am

Mental Health Month, Day 26:

Many Dimensions,
Part II
— The Blue Matrix 

But seriously…

John Baez in July 1999:

"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."

Well, maybe.  I certainly hope so.  If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light.  Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory.  See the following from my journal:

Evariste Galois and the Rock that Changed Things, and

A Time to Gather Stones Together: Readings for Yom Kippur.

See, too, this web page on Marion Zimmer Bradley's fictional

Matrices, or Blue Star-Stones, and

the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.
 

Monday May 26, 2003

Filed under: General — m759 @ 3:14 am

Mental Health Month, Day 26:

Many Dimensions

John Baez on why bosonic string theory is said to require 26 dimensions —

“By now, if you’re a rigorous sort of pure mathematician, you must be suffering from grave doubts about the sanity of this whole procedure.”

Doubts?  Let us just say I prefer

G-string theory:

“The G-string is unique in that it combines the properties of all known string theories. It has 26-dimensional modes propagating to the left, 10-dimensional modes propagating to the right, and 2-dimensional modes just sitting around wondering what the hell is going on.”

Tuesday, May 13, 2003

Tuesday May 13, 2003

Filed under: General — m759 @ 12:06 pm

Cubist Catechism

For the birthday of Georges Braque

From A Wrinkle in Time, by Madeleine L’Engle:

“Now we will tesser, we will wrinkle again. Do you understand?”

“No,” Meg said flatly….

“Oh, dear,” Meg sighed. “I guess I am a moron. I just don’t get it.”

“That is because you think of space only in three dimensions,” Mrs. Whatsit told her….

Meg sighed. “Just explain it to me.”

“Okay,” Charles said. “What is the first dimension?”

“Well — a line.”

“Okay. And the second dimension?”

“Well, you’d square the line. A flat square would be in the second dimension.”

“And the third?”

“Well, you’d square the second dimension. Then the square wouldn’t be flat any more. It would have a bottom, and sides, and a top.”

“And the fourth?”

“Well, I guess if you want to put it into mathematical terms you’d square the square. But you can’t take a pencil and draw it the way you can the first three.”


Braque

Friday, January 3, 2003

Friday January 3, 2003

Filed under: General — m759 @ 3:33 pm
Tolkien is Eleventy-One Today!

In observance of this milestone, some links:

Monday, September 16, 2002

Monday September 16, 2002

Filed under: General — m759 @ 3:26 pm

A Time to Gather Stones Together
(Ecclesiastes 3:5)

Readings for Yom Kippur:

Friday, August 30, 2002

Friday August 30, 2002

Filed under: General,Geometry — Tags: , — m759 @ 2:30 am

For Mary Shelley, on her birthday: A Chain of Links The creator of Frankenstein might appreciate the following chain of thought. Lucifer.com Lucifer Media Corporation Lucifer Media Sites The Extropy Institute: International Transhumanist Solutions Why Super-Human Intelligence Would Be Equivalent To Precognition, by Marc Geddes:

"Consider the geometry of multiple dimensions as an analogy for mental abilities… …if there is a 4th dimension of intelligence, to us ordinary humans stuck with 3 dimensional reasoning, this 4th dimension would be indistinguishable from precognition. Post-humans would appear to us ordinary humans as beings which could predict the future in ways which would be inexplicable to us. We should label post-humans as 'Pre-Cogs.'

In the Steven Speilberg [sic]  film Minority Report, we encounter genetically engineered humans with precisely the abilities described above."

Internet Movie Database page on "Minority Report"

IMDb page on "Minority Report" author Philip K. Dick

IMDb biography of Philip K. Dick, where our chain of links ends.  Here Dick says that

"The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words."

On the other hand, Dick also says here that

"Reality is that which, when you stop believing in it, doesn't go away."

These two quotations summarize, on the one hand, the cynical, relativistic nominalism of the postmodernists and, on the other hand, the hard-nosed realism of the Platonists.

What does all this have to do with "the geometry of multiple dimensions"?

Consider the famous story for adolescents, A Wrinkle in Time, by Madeleine L'Engle.   The author, a well-meaning Christian, tries, like all storytellers,  to control her readers by controlling the meaning of words.   The key word in this book is "tesseract," a term from multi-dimensional geometry.   She insists that a tesseract has mystic properties and cannot be visualized.  She is wrong (at least about the visualizing).

See The Tesseract: A look into 4-dimensional space, by Harry J. Smith.

See also the many revealing comments in Harry J. Smith's Guestbook.

One of Smith's guests remarks, apropos of Smith's comments on St. Joseph, that he has his own connection with St. Augustine.

For a adult-level discussion of Augustine, time, eternity, and Platonism, see the website Time as a Psalm in St. Augustine, by A. M. Johnston.

See also the remark headlining Maureen Dowd's New York Times column of August 28, 2002, Saint Augustine's Day:

"I'm with Dick."

Whether the realist Dick or the nominalist Dick, she does not say.

As for precognition, see my series of journal notes below, which leads up to two intriguing errors in an Amazon.com site on the "Forbidden Planet" soundtrack.   The first two audio samples from this soundtrack are (wrongly) entitled "Birdland" and "Flamingo."  See also the West Wing episode rebroadcast on Wednesday, August 28, 2002,

The Black Vera Wang

C. J. Cregg (Allison Janney), who models a black Vera Wang dress in that episode, has the Secret Service codename Flamingo.

"…that woman in black She's a mystery She's everything a woman should be Woman in black got a hold on me"

(Foreigner 4 in my August 28 note below)

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