From the 1984 New Orleans film Tightrope—
Related material: Walking the Tightrope and Transgressing.
Sunday, February 3, 2013
Wednesday, January 2, 2013
PlanetMath link
Update of May 27, 2013:
The post below is now outdated. See
http://planetmath.org/cullinanediamondtheorem .
__________________________________________________________________
The brief note on the diamond theorem at PlanetMath
disappeared some time ago. Here is a link to its
current URL: http://planetmath.org/?op=getobj;from=lec;id=49.
Update of 3 PM ET Jan. 2, 2013—
Another item recovered from Internet storage:
Click on the Monthly page for some background.
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Tuesday, November 6, 2012
Bend Sinister
This morning's New York Times obituaries—
These suggest a look at Solving Nabokov's Lolita Riddle ,
by Joanne Morgan (Sydney: Cosynch Press, 2005).
That book discusses Lolita as a character like Lewis Carroll's Alice.
(The Red Queen and Alice of course correspond to figures in
the first two thumbnails above.)
From the obituary associated with the third thumbnail above:
"Front-page headlines combined concision and dark humor."
The title of this post, Bend Sinister , is not unlike such a headline.
It is the title of a novel by Nabokov (often compared with Orwell's 1984 )
that is discussed in the Lolita Riddle book.
Related material— The bend sinister found in Log24 searches
for Hexagram 14 and for the phrase Hands-On—
 |
 |
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Monday, November 5, 2012
Design Cubes
Continued from April 2, 2012.
Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties—
Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz cubes (see links below).
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Thursday, August 23, 2012
Chapman’s Homer
Louis Sahagun in today's Los Angeles Times—
The late Professor Marvin W. Meyer
"was our Indiana Jones," said James L. Doti,
president of Chapman University in Orange,
where Meyer held the Griset Chair in Bible
and Christian Studies and was director of
the Albert Schweitzer Institute.
Meyer reportedly died on August 16.
Thursday, August 16, 2012
|
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Thursday, August 16, 2012
Semiotics
"Two clichés make us laugh, but
a hundred clichés move us
because we sense dimly that the clichés
are talking among themselves and
celebrating a reunion."
"'Casablanca': Cult Movies and Intertextual Collage,"
by Umberto Eco in SubStance , Vol. 14, No. 2, Issue 47:
In Search of Eco's Roses (1985), pp. 3-12.
(This paper was presented at a symposium,
"Semiotics of the Cinema: The State of the Art,"
in Toronto on June 18, 1984.)
Journal article published by U. of Wisconsin Press.
Stable URL: http://www.jstor.org/stable/3685047.
Click image for some related material.
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Thursday, August 2, 2012
Logos
(Continued from December 26th, 2011)
Some material at math.stackexchange.com related to
yesterday evening's post on Elementary Finite Geometry—
Questions on this topic have recently been
discussed at Affine plane of order 4? and at
Turning affine planes into projective planes.
(For a better discussion of the affine plane of order 4,
see Affine Planes and Mutually Orthogonal Latin Squares
at the website of William Cherowitzo, professor at UC Denver.)
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Saturday, July 14, 2012
Lemma
For example—
A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—
|
… a "square-triangle" lemma:
(∀ t ∈ T , t is an n -replica )
[I.e., "Every triangle is an n -replica" |
For definitions, see the 1985 letter in Triangles Are Square.
(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)
A search today for related material yielded the following—
|
"Suppose that one side of a triangle has length n . Then it can be cut into n 2 congruent triangles which are similar to the original one and whose corresponding sides to the side of length n have lengths 1." |
This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite obvious.
At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.
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Sunday, July 8, 2012
Not Quite Obvious
"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."
— Steven H. Cullinane, letter in the
American Mathematical Monthly 1985 June-July issue
If the ancient Greeks had not been distracted by
investigations of triangular (as opposed to square )
numbers, they might have done something with this fact.
A search for occurrences of the phrase
"n2 [i.e., n 2 ] congruent triangles"
indicates only fairly recent (i.e., later than 1984) results.*
Some related material, updated this morning—
|
This suggests a problem—
What mappings of a square array of n 2 points to
In the figure above, whether |
* Update of July 15, 2012 (11:07 PM ET)—
Theorem on " rep-n 2 " (Golomb's terminology)
triangles from a 1982 book—
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Friday, June 22, 2012
Bowling in Diagon Alley
Josefine Lyche bowling (Facebook, June 12, 2012)
A professor of philosophy in 1984 on Socrates's geometric proof in Plato's Meno dialogue—
"These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling…."
— Ian Hacking in The New York Review of Books , Feb. 16, 1984
The same professor introducing a new edition of Kuhn's Structure of Scientific Revolutions—
"Paradigms Regained" (Los Angeles Review of Books , April 18, 2012)—
"That is the structure of scientific revolutions: normal science with a paradigm and a dedication to solving puzzles; followed by serious anomalies, which lead to a crisis; and finally resolution of the crisis by a new paradigm. Another famous word does not occur in the section titles: incommensurability. This is the idea that, in the course of a revolution and paradigm shift, the new ideas and assertions cannot be strictly compared to the old ones."
The Meno proof involves inscribing diagonals in squares. It is therefore related, albeit indirectly, to the classic Greek discovery that the diagonals of a square are incommensurable with its sides. Hence the following discussion of incommensurability seems relevant.
See also von Fritz and incommensurability in The New York Times (March 8, 2011).
For mathematical remarks related to the 10-dot triangular array of von Fritz, diagonals, and bowling, see this journal on Nov. 8, 2011— "Stoned."
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Thursday, March 22, 2012
Square-Triangle Theorem continued
Last night's post described a book by Alexander Soifer
on questions closely related to— and possibly
suggested by— a Miscellanea item and a letter to
the editor in the American Mathematical Monthly ,
June-July issues of 1984 and 1985.
Further search yields a series of three papers by
Michael Beeson on the same questions. These papers are
more mathematically presentable than Soifer's book.
Triangle Tiling I —
http://www.michaelbeeson.com/research/papers/TriangleTiling1.pdf
March 2, 2012
Triangle Tiling II —
http://www.michaelbeeson.com/research/papers/TriangleTiling2.pdf
February 18, 2012
Triangle Tiling III —
http://www.michaelbeeson.com/research/papers/TriangleTiling3.pdf
March 11, 2012
These three recent preprints replace some 2010 drafts not now available.
Here are the abstracts of those drafts—
"Tiling triangle ABC with congruent triangles similar to ABC"
(March 13, 2010),
"Tiling a triangle with congruent triangles"
(July 1, 2010).
Beeson, like Soifer, omits any reference to the "Triangles are square" item
of 1984 and the followup letter of 1985 in the Monthly .
Comments Off on Square-Triangle Theorem continued
Thursday, January 12, 2012
Triangles Are Square
Coming across John H. Conway's 1991*
pinwheel triangle decomposition this morning—
|
|
— suggested a review of a triangle decomposition result from 1984:
Figure A
(Click the below image to enlarge.)
The above 1985 note immediately suggests a problem—
What mappings of a square with c 2 congruent parts
to a triangle with c 2 congruent parts are "natural"?**
(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)
* Communicated to Charles Radin in January 1991. The Conway
decomposition may, of course, have been discovered much earlier.
** Update of Jan. 18, 2012— For a trial solution to the inverse
problem, see the "Triangles are Square" page at finitegeometry.org.
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Monday, December 5, 2011
The Shining (Norwegian Version)
A check tonight of Norwegian artist Josefine Lyche's recent activities
shows she has added a video to her web page that has for some time
contained a wall piece based on the 2×2 case of the diamond theorem —
The video (top left in screenshot above) is a tasteless New-Age discourse
that sounds frighteningly like the teachings of the late Heaven's Gate cult.
Investigating the source of the video on vimeo.com, I found the account of one "Jo Lyxe,"
who joined vimeo in September 2011. This is apparently a variant of Josefine Lyche's name.
The account has three videos—
- "High on RAM (OverLoad)"– Fluid running through a computer's innards
- "Death 2 Everyone"– A mystic vision of the afterlife
- "Realization of the Ultimate Reality (Beyond Form)"– The Blue Star video above
Lyche has elsewhere discussed her New-Age interests, so the contents of the videos
were not too surprising… except for one thing. Vimeo.com states that all three videos
were uploaded "2 months ago"— apparently when "Lyxe" first set up an account.*
I do not know, or particularly care, where she got the Blue Star video, but the other
videos interested me considerably when I found them tonight… since they are
drawn from films I discussed in this journal much more recently than "2 months ago."
"High on RAM (OverLoad)" is taken from the 1984 film "Electric Dreams" that I came across
and discussed here yesterday afternoon, well before re-encountering it again tonight.
And "Death 2 Everyone" (whose title** is perhaps a philosophical statement about inevitable mortality
rather than a mad terrorist curse) is taken from the 1983 Natalie Wood film "Brainstorm."
"Brainstorm" was also discussed here recently… on November 18th, in a post suggested by the
reopening of the investigation into Wood's death.
I had no inkling that these "Jo Lyxe" videos existed until tonight.
The overlapping content of Lyche's mental ramblings and my own seems rather surprising.
Perhaps it is a Norwegian mind-meld, perhaps just a coincidence of interests.
* Update: Google searches by the titles on Dec. 5 show that all three "Lyxe" videos
were uploaded on September 20 and 21, 2011.
** Update: A search shows a track with this title on a Glasgow band's 1994 album.
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Sunday, September 18, 2011
Anatomy of a Cube
R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott’s remarks.
For the group actions described by Cullinane, see “The Eightfold Cube” and
“A Simple Reflection Group of Order 168.”
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see “Cube Space, 1984-2003.”
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Saturday, June 25, 2011
The Fano Entity
The New York Times at 9 PM ET June 23, 2011—
ROBERT FANO: I’m trying to think briefly how to put it.
GINO FANO: "On the Fundamental Postulates"—
"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."
"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."
Also from the Times article quoted above…
"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”
— Tom Van Vleck
Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—
The art is also related to the mathematics of Gino Fano.
For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."
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Wednesday, June 15, 2011
Mischief
The New York Times today on a new show by tightrope artist Philippe Petit—
“He comes out of that really wonderful European tradition of street performance— it blends a boundary of what’s art and what’s life,” said Jay Wegman, the director of the Abrons Arts Center, who offered Mr. Petit the three-night run. “He’s also kind of mischievous, not in a threatening or evil way, but in a child’s way of teasing and having fun.”
For a much darker approach to street performance that also involves mischief and blended boundaries, see "Tightrope" (1984)—
Background: Men in Feminism , edited by Alice Jardine and
published by Taylor & Francis in 1987, "Walking the Tightrope
of Feminism and Male Desire," by Judith Mayne, page 64
See also yesterday's Another Opening and Football in this journal.
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Thursday, May 26, 2011
Prime Cubes
The title refers not to numbers of the form p 3, p prime, but to geometric cubes with p 3 subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p 3.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinate-array
have the same sorts of symmetries,
where p is any odd prime."
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Sunday, April 10, 2011
Finishing Up at Noon
From last October—
| Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage." – "Is Fiction the Art of Lying?"* by Mario Vargas Llosa, * The Web version's title has a misprint— |
A stitch in time…
Related material—
See also "Putting Mental Health on the Map at Harvard"—
Harvard Crimson , Friday, April 8, 2011, 2:09 AM—
They're outside the Science Center with their signs, their cheer, and their smiles. They've been introducing themselves over House lists, and they want you to ask questions. They're here for you. They're the Student Mental Heath Liaisons.
Harvard's SMHL crew—they pronounce it smile—have recently launched a new website and recruited more members in their effort to foster an informed and understanding environment on campus….
Mental Health Services, SMHL said, are not meant for "students who are really 'crazy.'" Everyone is entitled to a little help smiling.
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Friday, April 8, 2011
Windows
Roberta Smith in today's New York Times —
"… the argument that painting may ultimately be about
little more than the communication of some quality of
light and space, however abstract or indirect."
— Review of "Rooms With a View" at the Met
Lowry —
Malcolm Lowry, author of Under the Volcano
Hollywood —
Related material —
| Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage." – "Is Fiction the Art of Lying?"* by Mario Vargas Llosa, * The Web version's title has a misprint— |
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Tuesday, April 5, 2011
For Ned*
(A sequel to last night's "For Taylor")
On Joan Tewkesbury, who wrote the script for the 1975 film "Nashville"—
She urges writers to continue to generate new ideas
and new material. "Keep writing. The hardest thing
is to sell one script and not have another to follow it with."
One script— Yesterday's link titled "An Ordinary Evening in Tennessee"
Another— "A Point of Central Arrival"
Related material from last October—
| Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage." – "Is Fiction the Art of Lying?"* by Mario Vargas Llosa, * The Web version's title has a misprint— |
"You've got to pick up every stitch…"
* A former governor of Tennessee who died at 80 yesterday in Nashville
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Saturday, January 8, 2011
True Grid (continued)
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the Dolgachev-Artebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
|
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
|
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
|
Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, |
And from last October —
|
Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— |
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Friday, January 7, 2011
Coxeter and the Aleph
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 1984-2009.
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Friday, October 8, 2010
Starting Out in the Evening
… and Finishing Up at Noon
This post was suggested by last evening’s post on mathematics and narrative
and by Michiko Kakutani on Vargas Llosa in this morning’s New York Times.
Above: Frank Langella in Right: Johnny Depp in |
 |
“One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage.”
— “Is Fiction the Art of Lying?”* by Mario Vargas Llosa, New York Times essay of October 7, 1984
My own adventures in that realm— as reader, not author— may illustrate Llosa’s remark.
A nearby stack of paperbacks I haven’t touched for some months (in order from bottom to top)—
- Pale Rider by Alan Dean Foster
- Franny and Zooey by J. D. Salinger
- The Hobbit by J. R. R. Tolkien
- Le Petit Prince by Antoine de Saint Exupéry
- Literary Reflections by James A. Michener
- The Ninth Configuration by William Peter Blatty
- A Streetcar Named Desire by Tennessee Williams
- Nine Stories by J. D. Salinger
- A Midsummer Night’s Dream by William Shakespeare
- The Tempest by William Shakespeare
- Being There by Jerzy Kosinski
- What Dreams May Come by Richard Matheson
- Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig
- A Gathering of Spies by John Altman
- Selected Poems by Robinson Jeffers
- Hook— Tinkerbell’s Challenge by Tristar Pictures
- Rising Sun by Michael Crichton
- Changewar by Fritz Leiber
- The Painted Word by Tom Wolfe
- The Hustler by Walter Tevis
- The Natural by Bernard Malamud
- Truly Tasteless Jokes by Blanche Knott
- The Man Who Was Thursday by G. K. Chesterton
- Under the Volcano by Malcolm Lowry
What moral Vargas Llosa might draw from the above stack I do not know.
Generally, I prefer the sorts of books in a different nearby stack. See Sisteen, from May 25. That post the fanciful reader may view as related to number 16 in the above list. The reader may also relate numbers 24 and 22 above (an odd couple) to By Chance, from Thursday, July 22.
* The Web version’s title has a misprint— “living” instead of “lying.”
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Saturday, August 14, 2010
Iconic Notation
Continued from Friday the 13th—
Related material—
Cover art by Barclay Shaw reprinted from an earlier (1984) edition
A question from Ivan Illich
(founder of CIDOC, the Center for Intercultural Documentation,
in Cuernavaca, Mexico)—
"Who can be served by bridges to nowhere?"
For more about nowhere, see Utopia. See also http://outis.blogspot.com.
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Monday, June 21, 2010
Cube Spaces
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
group-actions theorem has not yet been ripped off.
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Sunday, June 20, 2010
Lovasz Wins Kyoto Prize
From a June 18 press release—
KYOTO, Japan, Jun 18, 2010 (BUSINESS WIRE) — The non-profit Inamori Foundation (President: Dr. Kazuo Inamori) today announced that Dr. Laszlo Lovasz will receive its 26th annual Kyoto Prize in Basic Sciences, which for 2010 focuses on the field of Mathematical Sciences. Dr. Lovasz, 62, a citizen of both Hungary and the United States, will receive the award for his outstanding contributions to the advancement of both the academic and technological possibilities of the mathematical sciences.
Dr. Lovasz currently serves as both director of the Mathematical Institute at Eotvos Lorand University in Budapest and as president of the International Mathematics Union. Among many positions held throughout his distinguished career, Dr. Lovasz also served as a senior research member at Microsoft Research Center and as a professor of computer science at Yale University.
Related material: Cube Space, 1984-2003.
See also “Kyoto Prize” in this journal—
The Kyoto Prize is “administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”
— – Montreal Gazette, June 20, 2008
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Wednesday, February 24, 2010
Transvections
A topic related to A Simple Reflection Group of Order 168—
Transvection groups over GF(2). See, for instance…
- Binary Coordinate Systems, by Steven H. Cullinane, 1984
- Classification of the Finite N-Generator Transvection Groups Over Z2, by Jizhu Nan and Jing Zhao, 2009, Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010), 185–202
- Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"
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Wednesday, November 4, 2009
Sinner or Saint?
As noted here yesterday, Claude Levi-Strauss may have died on Devil's Night, on Halloween, or on All Saints' Day. He was apparently a myth-transformer to the end.
The Independent says today he died on Sunday, All Saints' Day. Its eulogy, by Adam Kuper, is well-written, noting that linguist Roman Jakobson was a source of Levi-Strauss's theory of oppositions in myth, and observing that
"… binary oppositions tend to accumulate to form structures…."
Yes, they do. Examples:
I. The structures in the Diamond Puzzle…
Click on image for Jungian background.
II: The structure on a recent cover of Semiotica…
The Semiotica article by mathematical linguist Solomon Marcus is a defense of the Levi-Strauss canonic formula mentioned here yesterday.
It is available online for $40.
A less expensive, and possibly more informative, look at oppositions in linguistics is available for free online in a 1984 master's thesis (pdf, 8+ mb)–
"Language, Linguistics, and Philosophy: A Comparison of the Work of Roman Jakobson and the Later Wittgenstein, with Some Attention to the Philosophy of Charles Saunders Peirce," by Miles Spencer Kimball.
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Wednesday, August 19, 2009
Wednesday August 19, 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
— 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
The same example
at Wolfram.com:
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.
Comments Off on Wednesday August 19, 2009
Saturday, May 23, 2009
Saturday May 23, 2009
Release Date:
23 May 1984 (USA)
23 May 1984 (USA)
Plot:
“After arriving in India,
Indiana Jones is asked
by a desperate village
to find a mystical stone….”
Comments Off on Saturday May 23, 2009
Tuesday, January 6, 2009
Tuesday January 6, 2009
Archetypes, Synchronicity,
and Dyson on Jung
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 1984-2003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
In Chapter 3, “Sutures of Structures,” Taylor asks —
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
“Examples are the stained-glass
windows of knowledge.”
— Vladimir Nabokov
windows of knowledge.”
— Vladimir Nabokov
Comments Off on Tuesday January 6, 2009
Friday, December 19, 2008
Friday December 19, 2008
Inside the
White Cube
White Cube
Part I: The White Cube
Part II: Inside
Part III: Outside
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
|
From Chapter 3,
"Sutures* of Structures," p. 58: "What, then, is a frame, and what is frame work? This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'" * P. 61:
|
Comments Off on Friday December 19, 2008
Thursday, October 30, 2008
Thursday October 30, 2008
Readings for
Devil’s Night
| 1. Today’s New York Times review of Peter Brook’s production of “The Grand Inquisitor” 2. Mathematics and Theology 3. Christmas, 2005 4. Cube Space, 1984-2003 |
Comments Off on Thursday October 30, 2008
Friday, October 24, 2008
Friday October 24, 2008
“The Cube Space” is a name given to the eightfold cube in a vulgarized mathematics text, Discrete Mathematics: Elementary and Beyond, by Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3-space over the 2-element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the less-obvious generation of the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.
Comments Off on Friday October 24, 2008
Saturday, October 18, 2008
Saturday October 18, 2008
Allure
(continued from
July 3, 2006)
(continued from
July 3, 2006)
This morning’s New York Times
has an obituary for the father
of the paper’s executive editor,
Bill Keller:
For more on George Keller and on
the more colorful Levi Stubbs,
who also died on Friday,
see the Times‘s AP obituaries.
Keller’s son Bill has emphasized
what he calls the “allure” of the
Times‘s lifestyles coverage.
An example of such coverage–
a 2006 story on visual art in Mexico
that included a reference to…
For descriptions of such life,
I prefer the literary art of
Robert Stone– in particular,
Stone’s novel
A Flag for Sunrise.
Credit must be given to
the Times for an excellent
1981 review of that novel.
(This was well before
the younger Keller
joined the Times in 1984.)
My own views on life are
less like those of either Keller
than like those of Stone and
perhaps of Levi Stubbs, the
other father figure who
died on Friday.
Related material:
“Yes, you’ll be goin’ loco
down in Acapulco,
the magic down there
is so strong.”
— Levi Stubbs
Comments Off on Saturday October 18, 2008
Friday, August 22, 2008
Friday August 22, 2008
Tentative movie title:
Blockheads
Blockheads
The Kohs Block Design
Intelligence Test
Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.
Related material from
Notes on Finite Geometry:
Notes on Finite Geometry:
Other related material:
1. Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…
Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.
2. Wechsler Cubes of a different sort (Log24, May 25, 2008)
3. Manohla Dargis in today's New York Times:
"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….
In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."
4. The bohemian who named his son Azazel:
"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."
— Interview with Ken Jacobs, UC Berkeley, October 1999
5. For Dargis's "Aryan Youth types"–
From a Manohla Dargis
New York Times film review
of April 4, 2007
(Spy Wednesday) —
See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —
For Sarah Silverman —
and the 9/9/03 entry
Doonesbury,
August 21-22, 2008:
Comments Off on Friday August 22, 2008
Thursday, July 10, 2008
Thursday July 10, 2008
Something
From the current
issue of TIME:
“Religions are hardy. ‘Many a time
we have gotten all ready for the
funeral’ of one faith or another,
‘and found it postponed again,
on account of
the weather or something.'”
— Mark Twain
Twain was raised
as a Presbyterian
(the Calvinist tradition).
This year’s Twain award
for humor went to
George Carlin,
raised in
the Catholic tradition.
On learning he had won
the Twain award,
Carlin said,
“Thank you, Mr. Twain.
Have your people
call my people.”
Today’s Birthdays:
Born July 10, 1509 —
Comments Off on Thursday July 10, 2008
Saturday, June 21, 2008
Saturday June 21, 2008
for lifetime achievement
in arts and philosophy
this year goes to
Charles Taylor,
Montreal philosophy professor.
“The Kyoto Prize has been given in three domains since 1984:
advanced technology, basic sciences, and the arts and philosophy.
It is administered by the Inamori Foundation, whose president,
Kazuo Inamori, is founder and chairman emeritus of Kyocera and
KDDI Corporation, two Japanese telecommunications giants.”
 |
|
“The Kyocera brand symbol is composed of a corporate mark |
Related material —
Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
See also Talking of Michelangelo.
Comments Off on Saturday June 21, 2008
Saturday, January 19, 2008
Saturday January 19, 2008
In Memory of
Bobby Fischer
Bobby Fischer
Edward Rothstein has a piece on Bobby Fischer in today’s New York Times. The Rothstein opening:
“There may be only three human activities in which miraculous accomplishment is possible before adulthood: mathematics, music and chess.”
This echoes the opening of a classic George Steiner essay (The New Yorker, Sept. 7, 1968):
“There are three intellectual pursuits, and, so far as I am aware, only three, in which human beings have performed major feats before the age of puberty. They are music, mathematics, and chess.”
— “A Death of Kings,” reprinted in George Steiner: A Reader, Oxford University Press, 1984, pp. 171-178.
Despite its promising (if unoriginal) opening, the New York Times piece is mainly an attack on Fischer’s anti-Jewish stance. Rothstein actually has little of interest to say about what he calls the “glass-bead games” of music, mathematics, and chess. For a better-written piece on chess and madness, see Charles Krauthammer’s 2005 essay in TIME. The feuilletons of Rothstein and Krauthammer do not, of course, come close to the genuinely bead-game-like writing of Steiner.
Related material on
chess and religion:
Magical Thinking
(December 7th, 2005)
Comments Off on Saturday January 19, 2008
Sunday, October 14, 2007
Sunday October 14, 2007
The Dipolar God
|
"Logos and logic, crystal hypothesis,
— Wallace Stevens, |
Yesterday's meditation ("Simon's Shema") on the interpenetration of opposites continues:
Part I: The Jewel in the Lotus
"The fundamental conception of Tantric Buddhist metaphysics, namely, yuganaddha, signifies the coincidence of opposites. It is symbolized by the conjugal embrace (maithuna or kama-kala) of a god and goddess or a Buddha and his consort (signifying karuna and sunyata or upaya and prajna, respectively), also commonly depicted in Tantric Buddhist iconography as the union of vajra (diamond sceptre) and padme (lotus flower). Thus, yuganaddha essentially means the interpenetration of opposites or dipolar fusion, and is a fundamental restatement of Hua-yen theoretic structures."
— p. 148 in "Part II: A Whiteheadian Process Critique of Hua-yen Buddhism," in Process Metaphysics and Hua-Yen Buddhism: A Critical Study of Cumulative Penetration vs. Interpenetration (SUNY Series in Systematic Philosophy), by Steve Odin, State University of New York Press, 1982
Part II: The Dipolar God
And on p. 163 of Odin, op. cit., in "Part III: Theology of the Deep Unconscious: A Reconstruction of Process Theology," in the section titled "Whitehead's Dipolar God as the Collective Unconscious"–
"An effort is made to transpose Whitehead's theory of the dipolar God into the terms of the collective unconscious, so that now the dipolar God is to be comprehended not as a transcendent deity, but the deepest dimension and highest potentiality of one's own psyche."
Part III: Piled High and Deep
Odin obtained his Ph.D. degree from the Department of Philosophy at the State University of New York (SUNY) at Stony Brook in 1980. (See curriculum vitae (pdf).)
For an academic review of Odin's book, see David Applebaum, Philosophy East and West, Vol. 34 (1984), pp. 107-108.
It is perhaps worth noting, in light of the final footnote of Mark D. Brimblecombe's Ph.D. thesis "Dipolarity and God" quoted yesterday, that "tantra" is said to mean "loom." For some less-academic background on the Tantric iconography Odin describes, see the webpage "Love and Passion in Tantric Buddhist Art." For a fiction combining love and passion with the word "loom" in a religious context, see Clive Barker's Weaveworld. This fiction– which is, if not "supreme" in the Wallace Stevens sense, at least entertaining– may correspond to some aspects of the deep Jungian psychological reality discussed by Odin.
Actors portraying
Arendt and Heidegger
Arendt and Heidegger
Click on image for details.
Comments Off on Sunday October 14, 2007
Thursday, October 11, 2007
Thursday October 11, 2007
The Nobel Prize
in Literature
Background from 1947:
in Literature
this year goes to the author
of The Golden Notebook
and The Cleft.
Related material:
The Golden Obituary
and Cleavage —
Log24, Oct. 9, 2007 —
Background from 1947:
Further details:
Wheel
Quoted by physics writer
Heinz Pagels at the end of
The Cosmic Code:
“For the essence and the end
Of his labor is beauty… one beauty,
the rhythm of that Wheel….”
— Robinson Jeffers
|
“The Ferris wheel came into view again, just the top, silently burning high on the hill, almost directly in front of him, then the trees rose up over it. The road, which was terrible and full of potholes, went steeply downhill here; he was approaching the little bridge over the barranca, the deep ravine. Halfway across the bridge he stopped; he lit a new cigarette from the one he’d been smoking, and leaned over the parapet, looking down. It was too dark to see the bottom, but: here was finality indeed, and cleavage! Quauhnahuac was like the times in this respect, wherever you turned the abyss was waiting for you round the corner. Dormitory for vultures and city of Moloch! When Christ was being crucified, so ran the sea-borne, hieratic legend, the earth had opened all through this country…” — Malcolm Lowry, Under the Volcano, 1947. (Harper & Row reissue, 1984, p. 15) Comment by Stephen Spender: “There is a suggestion of Christ descending into the abyss for the harrowing of Hell. But it is the Consul whom we think of here, rather than of Christ. The Consul is hurled into this abyss at the end of the novel.” — Introduction to Under the Volcano Edward Gibbon, Decline and Fall of the Roman Empire, Chapter XXI — Gibbon, discussing the theology of the Trinity, defines perichoresis as “… the internal connection and spiritual penetration which indissolubly unites the divine persons59 ….
William Golding: “Simon’s head was tilted slightly up. His eyes could not break away and the Lord of the Flies hung in space before him. ‘What are you doing out here all alone? Aren’t you afraid of me?’ Simon shook. ‘There isn’t anyone to help you. Only me. And I’m the Beast.’ Simon’s mouth labored, brought forth audible words. ‘Pig’s head on a stick.’ ‘Fancy thinking the Beast was something you could hunt and kill!’ said the head. For a moment or two the forest and all the other dimly appreciated places echoed with the parody of laughter. ‘You knew, didn’t you? I’m part of you? Close, close, close!’ “ “Thought of the day: — Alice Woodrome, Good Friday, 2004 Anne Francis,
“Here was finality indeed, — Under the Volcano |
For further details of
the wheel metaphor, see
Rock of Ages
Comments Off on Thursday October 11, 2007
Tuesday, August 14, 2007
Tuesday August 14, 2007
Philip K. Dick,
1928 – 1982
on the cover of
a 1987 edition of
his 1959 novel
Time Out of Joint:
1928 – 1982
on the cover of
a 1987 edition of
his 1959 novel
Time Out of Joint:
Cover art by Barclay Shaw reprinted
from an earlier (1984) edition
Philip K. Dick as a
window wraith (see below)
The above illustration was suggested by yesterday's quoted New Yorker characterization by Adam Gopnik of Philip K. Dick–
"… the kind of guy who can't drink one cup of coffee without drinking six, and then stays up all night to tell you what Schopenhauer really said and how it affects your understanding of Hitchcock and what that had to do with Christopher Marlowe."
— as well as by the illustrations of Gopnik's characterization in Kernel of Eternity, and by the following passage from Gopnik's 2005 novel The King in the Window:
"What's a window wraith?"
"It's someone who once lived in the ordinary world who lives now in a window, and makes reflections of the people who pass by and look in."
"You mean you are a ghost?!" Oliver asked, suddenly feeling a little terrified.
"Just the opposite, actually. You see, ghosts come from another world and haunt you, but window wraiths are the world. We're the memory of the world. We're here for good. You're the ones who come and go like ghosts. You haunt us."
Related material: As noted, Kernel of Eternity, and also John Tierney's piece on simulated reality in last night's online New York Times. Whether our everyday reality is merely a simulation has long been a theme (as in Dick's novel above) of speculative fiction. Interest in this theme is widespread, perhaps partly because we do exist as simulations– in the minds of other people. These simulations may be accurate or may be– as is perhaps Gopnik's characterization of Philip K. Dick– inaccurate. The accuracy of the simulations is seldom of interest to the simulator, but often of considerable interest to the simulatee.
The cover of the Aug. 20 New Yorker in which the Adam Gopnik essay appears may also be of interest, in view of the material on diagonals in the Log24 entries of Aug. 1 linked to in yesterday's entry:
"Summer Reading,"
by Joost Swarte
Comments Off on Tuesday August 14, 2007
Saturday, December 9, 2006
Saturday December 9, 2006
Death on the Feast
of Saint Nicholas
Quotation from Log24 on
September 14, 2003–
of Saint Nicholas
Quotation from Log24 on
September 14, 2003–
Readings on Aesthetics for the
Feast of the Triumph of the Cross:
“We’re not here to stick a mirror on you. Anybody can do that, We’re here to give you a more cubist or skewed mirror, where you get to see yourself with fresh eyes. That’s what an artist does. When you paint the Crucifixion, you’re not painting an exact reproduction.”
— Julie Taymor on “Frida” (AP, 10/22/02)
|
“Saint Francis Borgia at the Deathbed of an Impenitent [above], painted by Francisco Goya (1746-1828) in 1788, is one of the most astonishing works in an oeuvre replete with remarkable images. In the decade and a half since its inclusion in Robert Rosenblum‘s survey* of nineteenth-century art, this canvas has become widely known among scholars and their students. Rosenblum, following a line of interpretation that dates back to the middle of the nineteenth century, uses this painting to support a symptomatic reading of Goya’s art, which he describes as ‘the most sharply accurate mirror of the collapse of the great religious and monarchic traditions of the West.'”
— Andrew Schulz in The Art Bulletin, Dec. 1, 1998 * 19th-Century Art, by H. W. Janson and Robert Rosenblum, 1984 |
on Wednesday,
the Feast of St. Nicholas.
the Feast of St. Nicholas.
For more on
St. Francis Borgia, see
In Lieu of Rosebud.
Comments Off on Saturday December 9, 2006
Friday, December 30, 2005
Friday December 30, 2005
Expression
Expression is an impossible word.
If you want to use it
I think you have to
explain it further…
— Ad Reinhardt,
Art as Art
“… an equation is a very
abstract expression
of knowledge
about something.”
— The Hidden Side
of Visualization
Well, yes.
For example:
Abstract Expression,
pixels on screen, 12/30/05
|
The Scope and Limits
of Quotation (pdf), in The Philosophy of Donald Davidson, ed. L. E. Hahn, Open Court Publishers, 1999, pp. 691-714, by Dr. Ernest Lepore, Rutgers University: “… an assumption I wish to reject, namely, that in quotation an abstract expression (shape) is denoted…. Since Davidson, however, only says ‘we may take [an expression] to be an abstract shape’ [1979, p.85, my emphasis], his theory is compatible with expressions being something else. We need only find something that can be instantiated by radically differently shaped objects. Whatever it is must be such that written tokens, spoken tokens, signed tokens, Braille tokens, Semaphore tokens, finger language tokens, and any other way in which words can be produced, can be instantiated by it…. Such entities might
Reference: Davidson, D., 1979, “Quotation,” in Inquiries into Truth and Interpretation , Oxford, Oxford University Press, 1984, pp.79-92. |
exist; if they do, they might ultimately play some role in the metaphysics of language.”For more on zero and other entities,
see Is Nothing Sacred?
For more on Davidson,
see Shema .
Comments Off on Friday December 30, 2005
Saturday, August 6, 2005
Saturday August 6, 2005
For André Weil on
the seventh anniversary
of his death:
the seventh anniversary
of his death:
A Miniature
Rosetta Stone
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
Comments Off on Saturday August 6, 2005
Thursday, August 4, 2005
Thursday August 4, 2005
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.
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
Comments Off on Thursday August 4, 2005
Tuesday, June 7, 2005
Tuesday June 7, 2005
The Sequel to Rhetoric 101:
101 101
“A SINGLE VERSE by Rimbaud,”
writes Dominique de Villepin,
the new French Prime Minister,
“shines like a powder trail
on a day’s horizon.
It sets it ablaze all at once,
explodes all limits,
draws the eyes
to other heavens.”
“Room 101 was the place where
your worst fears were realised
in George Orwell’s classic
Nineteen Eighty-Four.
writes Dominique de Villepin,
the new French Prime Minister,
“shines like a powder trail
on a day’s horizon.
It sets it ablaze all at once,
explodes all limits,
draws the eyes
to other heavens.”
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
“Room 101 was the place where
your worst fears were realised
in George Orwell’s classic
Nineteen Eighty-Four.
[101 was also]
Professor Nash’s office number
in the movie ‘A Beautiful Mind.'”
Professor Nash’s office number
in the movie ‘A Beautiful Mind.'”
Classics Illustrated —
Click on picture for details.
(For some mathematics that is actually
from 1984, see Block Designs
and the 2005 followup
The Eightfold Cube.)
Comments Off on Tuesday June 7, 2005
Monday, August 9, 2004
Monday August 9, 2004
Shape Note
A variation on the theme of the previous entry, Quartet.
|
The first |
Derek Taunt |
As in the previous entry, the illustration on the left is from a Log24 entry on the date of death of the person on the right.
Relevant quotations:
“It was rather like solving a crossword puzzle.”
— Derek Taunt, on breaking the Enigma Code
“… history is a pattern
Of timeless moments.”
“He [Dr. Taunt] and Angela [his wife] founded the Friends of Kettle’s Yard when the Arts Council cut its grant in 1984 and together organised countless fundraising activities for the museum and gallery.”
“How do we relate to the past? How are our memories affected by the cultural context that shapes our present? How many, and what kind of narratives compete in the representation of a historical moment? Rear View Mirror sets out to explore these questions and examine the devices we use to reconstruct events and people through different lenses….”
— On a future Kettle’s Yard exhibition
Time past and time future
What might have been
and what has been
Point to one end,
which is always present.
“The diamonds will be shining,
no longer in the rough.”
See the Log24 remarks on Jesus College— Taunt’s college– in a web page for June Carter Cash, The Circle is Unbroken.
Wednesday, July 7, 2004
Wednesday July 7, 2004
Beyond Geometry
(Title of current L. A. art exhibit)
|
What is the difference between topology and geometry? Geometry you learn in high school; topology in college. So, topology costs more. A bit more seriously…. |
“The greatest obstacle to discovery
is not ignorance —
it is the illusion of knowledge.”
— Daniel J. Boorstin,
American historian, educator, writer.
Source: The Washington Post,
“The Six O’Clock Scholar,”
by Carol Krucoff (29 Jan. 1984)
American historian, educator, writer.
Source: The Washington Post,
“The Six O’Clock Scholar,”
by Carol Krucoff (29 Jan. 1984)
For the illusion of knowledge,
see (for instance)
The Importance of Being Nothingness,
by Craig J. Hogan
(American Scientist, Sept.-Oct. 2001).
A bit more seriously…
“These cases are
neither harmless nor amusing.”
— Craig J. Hogan, op. cit.
For example:
“Thanks to Dr. Matrix
for honouring this website
with the Award for Science Excellence
on May 14, 2002 and selecting it
for prominent display in the categories
of Mathematics and Creative Minds.”
See also my notes
On Dharwadker’s Attempted Proof,
November 28, 2000, and
The God-Shaped Hole,
February 21, 2001.
Comments Off on Wednesday July 7, 2004
Saturday, June 26, 2004
Saturday June 26, 2004
Advanced Study
Herman Goldstine (left), shown in 1952
at the Institute for Advanced Study
with J. Robert Oppenheimer (center)
and John von Neumann (right).
Click on the picture above
for an obituary in today's New York Times
of Goldstine, who died on June 16, 2004.
Click on the picture below
for an event appropriate to
the date of Goldstine's death.
The event, a talk on black holes, took place at
the American Philosophical Society in Philadelphia.
Goldstine was executive director of the Society
from 1984 to 1997.
Comments Off on Saturday June 26, 2004
Saturday, May 22, 2004
Saturday May 22, 2004
Star Wars
In memory of Melvin J. Lasky, editor, 1958-1990, of the CIA-funded journal Encounter:
“Once called as lively, and as bitchy, as a literary cocktail party, Encounter published articles of unrivalled authority on politics, history and literature.”
Lasky died on Wednesday, May 19, 2004. From a journal entry of my own on that date:
This newly-digitized diagram is from a
paper journal note of October 21, 1999.
Note that the diagram’s overall form is that of an eight-point star. Here is an excerpt from a Fritz Leiber story dealing with such a star, the symbol of a fictional organization:
|
Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, “Look, Buster, do you want to live?” Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question. The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar. An X superimposed on a plus sign. It looked permanent. … “Here is how it stacks up: You’ve bought your way with something other than money into an organization of which I am an agent….” “It’s a very big organization,” she went on, as if warning me. “Call it an empire or a power if you like. So far as you are concerned, it has always existed and always will exist. It has agents everywhere, literally. Space and time are no barriers to it. Its purpose, so far as you will ever be able to know it, is to change, for its own aggrandizement, not only the present and the future, but also the past. It is a ruthlessly competitive organization and is merciless to its employees.” “I. G. Farben?” I asked grabbing nervously and clumsily at humor. She didn’t rebuke my flippancy, but said, “And it isn’t the Communist Party or the Ku Klux Klan, or the Avenging Angels or the Black Hand, either, though its enemies give it a nastier name.” “Which is?” I asked. “The Spiders,” she said. That word gave me the shudders, coming so suddenly. I expected the sigil to step off her forehead and scuttle down her face and leap at me—something like that. She watched me. “You might call it the Double Cross,” she suggested, “if that seems better.” — Fritz Leiber, |
From last year’s entry,
Indiana Jones and the Hidden Coffer,
of 6/14:
|
From Borges’s “The Aleph“:
From The Hunchback of Notre Dame:
Lena Olin and Harrison Ford |
Finally, from an excellent site
on the Knights Templar,
a quotation from Umberto Eco:
|
When all the archetypes burst out shamelessly, we plumb the depths of Homeric profundity. Two cliches make us laugh but a hundred cliches move us because we sense dimly that the cliches are talking among themselves, celebrating a reunion . . . Just as the extreme of pain meets sensual pleasure, and the extreme of perversion borders on mystical energy, so too the extreme of banality allows us to catch a glimpse of the Sublime. — “Casablanca: Cult Movies and Intertextual Collage” (1984) from Travels in Hyperreality |
Comments Off on Saturday May 22, 2004
Saturday, May 1, 2004
Saturday May 1, 2004
Fallen from Heaven
On today's stories:
Recall, gentle readers, the reference to Lucifer in last midnight's story, "The Devil and Wallace Stevens," and the reference in yesterday's story, "Notes," to the film "2010" (1984). Here is a quote from a review of the story behind that film:
"If the coming of Lucifer in this story doesn't set your pulse racing and your mind whirring, then I don't know what will."
For some of us — students of Stephen King and Malcolm Lowry — the coming of Lucifer is not such a surprising event. See
Comments Off on Saturday May 1, 2004
Saturday, April 10, 2004
Saturday April 10, 2004
Harrowing
“The Ferris wheel came into view again, just the top, silently burning high on the hill, almost directly in front of him, then the trees rose up over it. The road, which was terrible and full of potholes, went steeply downhill here; he was approaching the little bridge over the barranca, the deep ravine. Halfway across the bridge he stopped; he lit a new cigarette from the one he’d been smoking, and leaned over the parapet, looking down. It was too dark to see the bottom, but: here was finality indeed, and cleavage! Quauhnahuac was like the times in this respect, wherever you turned the abyss was waiting for you round the corner. Dormitory for vultures and city of Moloch! When Christ was being crucified, so ran the sea-borne, hieratic legend, the earth had opened all through this country …”
— Malcolm Lowry, Under the Volcano, 1947. (Harper & Row reissue, 1984, p. 15)
Comment by Stephen Spender:
“There is a suggestion of Christ descending into the abyss for the harrowing of Hell. But it is the Consul whom we think of here, rather than of Christ. The Consul is hurled into this abyss at the end of the novel.”
— Introduction to Under the Volcano
Edward Gibbon, Decline and Fall of the Roman Empire, Chapter XXI —
Gibbon, discussing the theology of the Trinity, defines perichoresis as
“… the internal connection and spiritual penetration which indissolubly unites the divine persons59 ….
59 … The
or ‘circumincessio,’ is perhaps the deepest and darkest corner of the whole theological abyss.”
“Whoever fights monsters should see to it that in the process he does not become a monster. And when you look long into an abyss, the abyss also looks into you.”
William Golding:
“Simon’s head was tilted slightly up. His eyes could not break away and the Lord of the Flies hung in space before him.
‘What are you doing out here all alone? Aren’t you afraid of me?’
Simon shook.
‘There isn’t anyone to help you. Only me. And I’m the Beast.’
Simon’s mouth labored, brought forth audible words.
‘Pig’s head on a stick.’
‘Fancy thinking the Beast was something you could hunt and kill!’ said the head. For a moment or two the forest and all the other dimly appreciated places echoed with the parody of laughter. ‘You knew, didn’t you? I’m part of you? Close, close, close!’ “
“Thought of the day:
You can catch more flies with honey than vinegar… if you’re into catchin’ flies.”
— Alice Woodrome, Good Friday, 2004
Anne Francis,
also known as
Honey West:
“Here was finality indeed,
and cleavage!”
— Under the Volcano
From the official
Anne Francis Web Site:
Come into my parlor….
For some background,
see the use of the word
“spider” in Under the Volcano:
WRIDER/ESPIDER:
THE CONSUL AS ARTIST IN
UNDER THE VOLCANO,
by Patrick A. McCarthy.
See, too, Why Me?
Tuesday, January 6, 2004
Tuesday January 6, 2004
720 in the Book
Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.
An article titled On Mathematical Imagination concludes by looking forward to
“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”
Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.
Hmm.
The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:
Abel’s Proof: An Essay
on the Sources and Meaning
of Mathematical Unsolvability
by Peter Pesic,
MIT Press, 2003
From a review:
“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….
Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations. The reader is left with little clarity on this sequel to the story….”
— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244
Here, it seems, is my epiphany:
“Elliptic modular functions suffice to solve all polynomial equations.”
Incidental Remarks
on Synchronicity,
Part I
Those who seek a star
on this Feast of the Epiphany
may click here.
Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.
Just how such equations can be solved is a less familiar story. I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions. Modular functions are also distantly related, via the topic of “moonshine” and via the “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.
Incidental Remarks
on Synchronicity,
Part II
There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.
Here is what I was able to find on the Web about Pesic’s claim:
From Wolfram Research:
From Solving the Quintic —
“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”
From Siegel Theta Function —
“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”
From Polynomial —
“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”
Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.
Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999.
From Angel Zhivkov,
Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:
“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker: in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function…. Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist. This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions. In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”
— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants“
Incidental Remarks
on Synchronicity,
Part III
From Music for Dunne’s Wake:
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
|
|
“720 in |
“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.
Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.
For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see
For the relevance of the time
of this entry, 10:10, see
|
Related recreational reading:
|
Labyrinth |
|
Comments Off on Tuesday January 6, 2004
Thursday, November 20, 2003
Thursday November 20, 2003
Chinese Theatre
Epic Records released a new Michael Jackson album, “Number Ones,” on Tuesday, Nov. 18, 2003.
From Those Were the Days:
On this date in…
“1984 – The largest crowd to see the unveiling of a Hollywood Walk-of-Fame star turned out as Michael Jackson got his piece of the sidewalk right in front of Mann’s Chinese Theatre in Los Angeles.
1971 – Isaac Hayes of Memphis, TN, got his first #1 hit as the ‘Theme from Shaft‘ began a two-week stay at the top of the charts.”
The above two Chinese characters
mean “Shaft One,” according to
The Source:
Hayes won an Oscar for best song.
Monday, October 27, 2003
Monday October 27, 2003
Dream of Heaven, continued
“…I am going up the hill on the grass behind juniper trees birches the road dusty she is coming up the other side yes there she is look it is who is it not Berty no Molly no a girl with red hair comes through the oak trees beautiful loves me puts out her hand kisses me we are kissing become one face floating in air with wings one fused face with wings Turner sunset and this and this and this and this and this WINGbeat and WINGbeat where whirled and well where whirled and well where whirled and well —”
— Great Circle, by Conrad Aiken, 1933.
Pp. 297-298 in paperback published by
Arbor House, New York, 1984.
For related material, see the poems of Conrad Aiken, the 1947 novel Under the Volcano by Malcolm Lowry, and the 1936 poem “Altarwise by Owl-Light,” by Dylan Thomas, whose birthday is today.
Surrealist postscripts:
The above dream contains a Turner sunset; a critic once called the work of Turner “pictures of nothing.” For details, see my entry of 8/23.
The time of this entry, 8:28, is a reference to the date, 8/28, of the Feast of St. Augustine, who was puzzled, as many still are, by the nature of time. For details, see my entry of 8/28.
Comments Off on Monday October 27, 2003
Tuesday, June 10, 2003
Tuesday June 10, 2003
The Triangular God
From the New York Times of June 10, 2003:
As Spinoza noted, “If a triangle could speak, it would say… that God is eminently triangular.”
— “Giving God a Break,” by Nicholas D. Kristof
Related material:
The figure above is by
Robert Anton Wilson.
From “The Cocktail Party,” Act One, Scene One, by T. S. Eliot:
UNIDENTIFIED GUEST [Sings]:
Tooryooly toory-iley
What’s the matter with One Eyed Riley?[Exit.]
JULIA: Edward, who is that dreadful man?
From T. S. Eliot, The Complete Poems and Plays, 1909-1950 (Harcourt, Brace and Company, 1952), page 144:
“The end is where we start from.”
From the end of that same book:
“And me be-in’ the One-Eyed Ri-ley”
For more on this song, see
Reilly’s Daughter (with midi tune),
See also my previous journal entry of
and the perceptive analysis of the Shakti-Shiva symbol that I quoted on May 25, 2003.
Here is a note from Sept. 15, 1984, for those who would like to
block that metaphor.
See also Block Designs from the Cabinet of Dr. Montessori and Sacerdotal Jargon.
Comments Off on Tuesday June 10, 2003
Wednesday, April 2, 2003
Wednesday April 2, 2003
Symmetries…. May 15, 1998
The following journal note, from the day after Sinatra died, was written before I heard of his death. Note particularly the quote from Rilke. Other material was suggested, in part, by Alasdair Gray's Glasgow novel 1982 Janine. The "Sein Feld" heading is a reference to the Seinfeld final episode, which aired May 14, 1998. The first column contains a reference to angels — apparently Hell's Angels — and the second column provides a somewhat more serious look at this theological topic.
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Sein Feld
1984 Janine
"But Angels love their own
"Logos means above all relation."
"Gesang ist Dasein…. |
Geometry and Theology
PA lottery May 14, 1998: |
|
|
Tape purchased 12/23/97:
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"In the middle of 1982 Janine there are pages in which Jock McLeish is fighting with drugs and alcohol, attempting to either die or come through and get free of his fantasies. In his delirium, he hears the voice of God, which enters in small print, pushing against the larger type of his ravings. Something God says is repeated on the first and last pages of Unlikely Stories, Mostly, complete with illustration and the words 'Scotland 1984' beside it. God's statement is 'Work as if you were in the early days of a better nation.' It is the inherent optimism in that statement that perhaps best captures the strength of Aladair Gray's fiction, its straightforwardness and exuberance." |
The group of all projectivities and correlations of 
For another look at angels, see "Winging It," by Christopher R. Miller, The New York Times Book Review Bookend page for Sunday, May 24, 1998. May 24 is the feast day of Sara (also known by the Hindu name Kali), patron saint of Gypsies.
For another, later (July 16, 1998) reply to Dyson, from a source better known than myself, see Why Religion Matters, by Huston Smith, Harper Collins, 2001, page 66.
Comments Off on Wednesday April 2, 2003
Friday, July 19, 2002
Old Xanga post numbers
This WordPress page from 9:22 PM ET on Aug. 17, 2016,
gives id numbers of old Xanga posts for Log24 and user m759.
It is backdated to July 19, 2002, the day before the first post in
this WordPress weblog, so it will not appear before other posts
in searches of the weblog.
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Saturday, December 07, 2002
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Friday, December 06, 2002
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Thursday, December 05, 2002
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Thursday, December 05, 2002
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Wednesday, December 04, 2002
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Monday, December 02, 2002
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Sunday, December 01, 2002
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Thursday, January 30, 2003
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Sunday, January 26, 2003
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Friday, January 24, 2003
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Wednesday, January 22, 2003
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Monday, January 20, 2003
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Saturday, January 18, 2003
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Thursday, January 16, 2003
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Tuesday, January 14, 2003
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Tuesday, January 14, 2003
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Sunday, January 12, 2003
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Saturday, January 11, 2003
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Friday, January 10, 2003
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Friday, January 10, 2003
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Thursday, January 09, 2003
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Tuesday, January 07, 2003
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Monday, January 06, 2003
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Sunday, January 05, 2003
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Sunday, January 05, 2003
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Saturday, January 04, 2003
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Friday, January 03, 2003
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Tuesday, February 25, 2003
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Tuesday, February 25, 2003
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Sunday, February 23, 2003
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Saturday, February 22, 2003
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Friday, February 21, 2003
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Thursday, February 20, 2003
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Tuesday, February 18, 2003
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Monday, February 17, 2003
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Sunday, February 16, 2003
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Saturday, February 15, 2003
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Thursday, February 13, 2003
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Wednesday, February 12, 2003
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Tuesday, February 11, 2003
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Monday, February 10, 2003
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Sunday, February 09, 2003
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Saturday, February 08, 2003
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Friday, February 07, 2003
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Thursday, February 06, 2003
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Wednesday, February 05, 2003
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Tuesday, February 04, 2003
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Monday, February 03, 2003
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Sunday, February 02, 2003
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Saturday, February 01, 2003
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Monday, March 31, 2003
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Saturday, March 29, 2003
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Friday, March 28, 2003
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Thursday, March 27, 2003
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Monday, March 24, 2003
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Saturday, March 22, 2003
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Friday, March 21, 2003
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Wednesday, March 19, 2003
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Tuesday, March 18, 2003
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Friday, March 14, 2003
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Thursday, March 13, 2003
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Wednesday, March 12, 2003
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Tuesday, March 11, 2003
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Monday, March 10, 2003
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Sunday, March 09, 2003
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Friday, March 07, 2003
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Thursday, March 06, 2003
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Wednesday, March 05, 2003
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Tuesday, March 04, 2003
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Sunday, March 02, 2003
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Wednesday, April 30, 2003
Tuesday, April 29, 2003
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Monday, April 28, 2003
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Sunday, April 27, 2003
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Friday, April 25, 2003
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Thursday, April 24, 2003
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Wednesday, April 23, 2003
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Tuesday, April 22, 2003
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Monday, April 21, 2003
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Sunday, April 20, 2003
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Saturday, April 19, 2003
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Friday, April 18, 2003
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Friday, April 18, 2003
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Thursday, April 17, 2003
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Wednesday, April 16, 2003
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Tuesday, April 15, 2003
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Tuesday, April 15, 2003
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Sunday, April 13, 2003
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Saturday, April 12, 2003
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Saturday, April 12, 2003
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Friday, April 11, 2003
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Wednesday, April 09, 2003
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Tuesday, April 08, 2003
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Monday, April 07, 2003
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Saturday, April 05, 2003
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Friday, April 04, 2003
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Thursday, April 03, 2003
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Wednesday, April 02, 2003
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Wednesday, May 28, 2003
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Tuesday, May 27, 2003
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Monday, May 26, 2003
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Sunday, May 25, 2003
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Saturday, May 24, 2003
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Friday, May 23, 2003
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Thursday, May 22, 2003
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Wednesday, May 21, 2003
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Monday, May 19, 2003
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Sunday, May 18, 2003
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Friday, May 16, 2003
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Thursday, May 15, 2003
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Thursday, May 15, 2003
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Wednesday, May 14, 2003
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Tuesday, May 13, 2003
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Monday, May 12, 2003
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Sunday, May 11, 2003
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Friday, May 09, 2003
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Friday, May 09, 2003
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Friday, May 02, 2003
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Thursday, May 01, 2003
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Sunday, June 29, 2003
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Friday, June 27, 2003
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Thursday, June 26, 2003
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Wednesday, June 25, 2003
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Monday, June 23, 2003
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Friday, June 20, 2003
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Wednesday, June 18, 2003
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Sunday, June 15, 2003
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Wednesday, June 11, 2003
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Tuesday, June 03, 2003
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Monday, June 02, 2003
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Sunday, June 01, 2003
<strong><font size="3">Thursday, May 29,
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Thursday, July 31, 2003
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Wednesday, July 30, 2003
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Tuesday, July 29, 2003
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Sunday, July 27, 2003
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Friday, July 25, 2003
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Thursday, July 24, 2003
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Friday, July 04, 2003
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Wednesday, August 27, 2003
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Friday, August 22, 2003
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Tuesday, August 19, 2003
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Monday, August 18, 2003
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Sunday, August 17, 2003
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Friday, August 08, 2003
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Thursday, August 07, 2003
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Friday, August 01, 2003
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Friday, August 01, 2003
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Tuesday, September 30, 2003
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Monday, September 29, 2003
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Sunday, September 28, 2003
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Friday, September 26, 2003
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Thursday, September 25, 2003
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Wednesday, September 24, 2003
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Tuesday, September 23, 2003
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Monday, September 22, 2003
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Thursday, September 18, 2003
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Monday, September 15, 2003
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Sunday, September 14, 2003
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Saturday, September 13, 2003
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Friday, September 12, 2003
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Thursday, September 11, 2003
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Wednesday, September 10, 2003
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Monday, September 08, 2003
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Monday, September 08, 2003
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Friday, September 05, 2003
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Thursday, September 04, 2003
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Wednesday, September 03, 2003
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Friday, October 31, 2003
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Wednesday, October 29, 2003
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Tuesday, October 28, 2003
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Monday, October 27, 2003
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Sunday, October 26, 2003
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Saturday, October 25, 2003
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Saturday, October 25, 2003
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Thursday, October 23, 2003
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Tuesday, October 21, 2003
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Thursday, October 16, 2003
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Wednesday, October 15, 2003
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Tuesday, October 14, 2003
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Sunday, October 12, 2003
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Saturday, October 11, 2003
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Friday, October 10, 2003
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Friday, October 10, 2003
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Wednesday, October 08, 2003
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Tuesday, October 07, 2003
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Sunday, October 05, 2003
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Saturday, October 04, 2003
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Friday, October 03, 2003
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Thursday, October 02, 2003
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Sunday, November 30, 2003
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Saturday, November 29, 2003
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Friday, November 28, 2003
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Tuesday, November 25, 2003
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Tuesday, November 25, 2003
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Monday, November 24, 2003
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Sunday, November 23, 2003
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Saturday, November 22, 2003
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Friday, November 21, 2003
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Thursday, November 20, 2003
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Wednesday, November 19, 2003
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Monday, November 17, 2003
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Sunday, November 16, 2003
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Saturday, November 15, 2003
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Friday, November 14, 2003
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Thursday, November 13, 2003
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Thursday, November 13, 2003
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Wednesday, November 12, 2003
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Tuesday, November 11, 2003
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Sunday, November 09, 2003
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Friday, November 07, 2003
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Thursday, November 06, 2003
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Wednesday, November 05, 2003
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Wednesday, November 05, 2003
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Tuesday, November 04, 2003
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Monday, November 03, 2003
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Sunday, November 02, 2003
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Wednesday, December 31, 2003
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Saturday, December 27, 2003
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Friday, December 26, 2003
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Monday, December 22, 2003
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Monday, December 22, 2003
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Sunday, December 21, 2003
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Saturday, December 20, 2003
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Saturday, December 20, 2003
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Friday, December 19, 2003
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Thursday, December 18, 2003
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Wednesday, December 17, 2003
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Tuesday, December 16, 2003
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Sunday, December 14, 2003
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Saturday, December 13, 2003
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Friday, December 12, 2003
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Thursday, December 11, 2003
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Wednesday, December 10, 2003
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Tuesday, December 09, 2003
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Monday, December 08, 2003
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Sunday, December 07, 2003
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Sunday, December 07, 2003
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Friday, December 05, 2003
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Saturday, January 31, 2004
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Thursday, January 29, 2004
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Wednesday, January 28, 2004
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Monday, January 26, 2004
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Monday, January 26, 2004
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Saturday, January 24, 2004
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Thursday, January 22, 2004
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Wednesday, January 21, 2004
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Tuesday, January 20, 2004
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Sunday, January 18, 2004
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Saturday, January 17, 2004
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Friday, January 16, 2004
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Wednesday, January 14, 2004
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Wednesday, January 14, 2004
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Tuesday, January 13, 2004
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Sunday, January 11, 2004
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Sunday, January 11, 2004
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Friday, January 09, 2004
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Thursday, January 08, 2004
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Tuesday, January 06, 2004
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Monday, January 05, 2004
Sunday, January 04, 2004
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Sunday, January 04, 2004
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Friday, January 02, 2004
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Thursday, January 01, 2004
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Sunday, February 29, 2004
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Saturday, February 28, 2004
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Thursday, February 26, 2004
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Thursday, February 26, 2004
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Wednesday, February 25, 2004
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Tuesday, February 24, 2004
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Monday, February 23, 2004
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Friday, February 20, 2004
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Thursday, February 19, 2004
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Wednesday, February 18, 2004
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Wednesday, February 18, 2004
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Tuesday, February 17, 2004
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Monday, February 16, 2004
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Saturday, February 14, 2004
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Thursday, February 12, 2004
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Wednesday, February 11, 2004
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Tuesday, February 10, 2004
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Monday, February 09, 2004
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Sunday, February 08, 2004
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Saturday, February 07, 2004
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Friday, February 06, 2004
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Thursday, February 05, 2004
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Tuesday, February 03, 2004
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Tuesday, February 03, 2004
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Monday, February 02, 2004
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Sunday, February 01, 2004
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Wednesday, March 31, 2004
76286330 3:33 PM
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Tuesday, March 30, 2004
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Tuesday, March 30, 2004
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Sunday, March 28, 2004
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Saturday, March 27, 2004
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Tuesday, March 23, 2004
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Monday, March 22, 2004
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Sunday, March 21, 2004
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Saturday, March 20, 2004
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Friday, March 19, 2004
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Thursday, March 18, 2004
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Wednesday, March 17, 2004
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Tuesday, March 16, 2004
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Monday, March 15, 2004
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Monday, March 15, 2004
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Sunday, March 14, 2004
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Saturday, March 13, 2004
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Saturday, March 13, 2004
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Friday, March 12, 2004
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Thursday, March 11, 2004
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Thursday, March 11, 2004
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Wednesday, March 10, 2004
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Sunday, March 07, 2004
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Saturday, March 06, 2004
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Friday, March 05, 2004
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Thursday, March 04, 2004
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Wednesday, March 03, 2004
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Wednesday, March 03, 2004
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Tuesday, March 02, 2004
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Friday, April 30, 2004
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Thursday, April 29, 2004
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Wednesday, April 28, 2004
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Tuesday, April 27, 2004
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Tuesday, April 27, 2004
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Monday, April 26, 2004
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Sunday, April 25, 2004
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Friday, April 23, 2004
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Thursday, April 22, 2004
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Wednesday, April 21, 2004
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Tuesday, April 20, 2004
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Monday, April 19, 2004
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Sunday, April 18, 2004
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Friday, April 16, 2004
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Thursday, April 15, 2004
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Wednesday, April 14, 2004
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Tuesday, April 13, 2004
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Sunday, April 11, 2004
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Saturday, April 10, 2004
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Friday, April 09, 2004
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Friday, April 09, 2004
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Thursday, April 08, 2004
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Wednesday, April 07, 2004
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Tuesday, April 06, 2004
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Monday, April 05, 2004
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Sunday, April 04, 2004
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Friday, April 02, 2004
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Thursday, April 01, 2004
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Monday, May 31, 2004
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Thursday, May 27, 2004
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Saturday, May 22, 2004
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Friday, May 21, 2004
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Friday, May 21, 2004
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Thursday, May 20, 2004
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Wednesday, May 19, 2004
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Saturday, May 15, 2004
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Saturday, May 15, 2004
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Friday, May 14, 2004
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Thursday, May 13, 2004
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Wednesday, May 12, 2004
Tuesday, May 11, 2004
Monday, May 10, 2004
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Saturday, May 08, 2004
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Friday, May 07, 2004
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Wednesday, May 05, 2004
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Tuesday, May 04, 2004
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Monday, May 03, 2004
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Sunday, May 02, 2004
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Saturday, May 01, 2004
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Saturday, May 01, 2004
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Tuesday, June 29, 2004
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Sunday, June 27, 2004
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Saturday, June 26, 2004
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Friday, June 25, 2004
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Tuesday, June 22, 2004
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Thursday, June 17, 2004
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Wednesday, June 16, 2004
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Tuesday, June 15, 2004
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Sunday, June 13, 2004
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Saturday, June 12, 2004
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Friday, June 11, 2004
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Tuesday, June 08, 2004
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Sunday, June 06, 2004
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Sunday, June 06, 2004
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Saturday, June 05, 2004
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Friday, June 04, 2004
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Thursday, June 03, 2004
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Tuesday, June 01, 2004
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Tuesday, June 01, 2004
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Saturday, July 31, 2004
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Thursday, July 29, 2004
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Wednesday, July 28, 2004
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Tuesday, July 27, 2004
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Monday, July 26, 2004
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Monday, July 26, 2004
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Sunday, July 25, 2004
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Saturday, July 24, 2004
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Friday, July 23, 2004
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Sunday, July 18, 2004
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Saturday, July 17, 2004
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Thursday, July 15, 2004
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Thursday, July 15, 2004
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Wednesday, July 14, 2004
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Monday, July 12, 2004
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Sunday, July 11, 2004
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Saturday, July 10, 2004
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Friday, July 09, 2004
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Thursday, July 08, 2004
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Thursday, July 08, 2004
Wednesday, July 07, 2004
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Friday, July 02, 2004
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Friday, July 02, 2004
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Thursday, July 01, 2004
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Tuesday, August 31, 2004
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Monday, August 30, 2004
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Sunday, August 29, 2004
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Saturday, August 28, 2004
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Sunday, August 22, 2004
Thursday, August 19, 2004
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Wednesday, August 18, 2004
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Tuesday, August 17, 2004
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Tuesday, August 17, 2004
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Monday, August 16, 2004
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Thursday, August 12, 2004
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Wednesday, August 11, 2004
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Tuesday, August 10, 2004
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Monday, August 09, 2004
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Monday, August 09, 2004
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Sunday, August 08, 2004
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Saturday, August 07, 2004
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Saturday, August 07, 2004
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Friday, August 06, 2004
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Wednesday, August 04, 2004
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Tuesday, August 03, 2004
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Tuesday, August 03, 2004
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Monday, August 02, 2004
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Sunday, August 01, 2004
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Thursday, September 30, 2004
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Wednesday, September 29, 2004
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Wednesday, September 29, 2004
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Tuesday, September 28, 2004
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Monday, September 27, 2004
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Sunday, September 26, 2004
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Saturday, September 25, 2004
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Thursday, September 23, 2004
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Wednesday, September 22, 2004
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Wednesday, September 22, 2004
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Tuesday, September 21, 2004
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Monday, September 20, 2004
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Saturday, September 18, 2004
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Friday, September 17, 2004
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Wednesday, September 15, 2004
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Tuesday, September 14, 2004
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Monday, September 13, 2004
Sunday, September 12, 2004
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Saturday, September 11, 2004
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Friday, September 10, 2004
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Monday, September 06, 2004
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Sunday, September 05, 2004
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<a name="1" target="_new"></a><big><b><font
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<a name="3" target="_new"></a><big><b><font
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Wednesday, September 01, 2004
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Friday, October 29, 2004
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Thursday, October 28, 2004
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Sunday, October 24, 2004
Saturday, October 23, 2004
Thursday, October 21, 2004
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Monday, October 18, 2004
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Wednesday, October 13, 2004
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Tuesday, October 12, 2004
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Tuesday, October 12, 2004
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Sunday, October 10, 2004
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Saturday, October 09, 2004
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Friday, October 08, 2004
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Thursday, October 07, 2004
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Wednesday, October 06, 2004
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Tuesday, October 05, 2004
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Monday, October 04, 2004
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Friday, October 01, 2004
Friday, October 01, 2004
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Friday, November 26, 2004
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Wednesday, November 24, 2004
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Sunday, November 21, 2004
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Sunday, November 21, 2004
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Saturday, November 20, 2004
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Thursday, November 11, 2004
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Wednesday, November 10, 2004
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Tuesday, November 09, 2004
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Monday, November 08, 2004
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Tuesday, November 02, 2004
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Wednesday, December 29, 2004
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Wednesday, December 22, 2004
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Sunday, December 19, 2004
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Thursday, December 16, 2004
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Wednesday, December 15, 2004
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Monday, December 13, 2004
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Friday, December 10, 2004
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Thursday, December 09, 2004
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Wednesday, December 08, 2004
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Tuesday, December 07, 2004
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Tuesday, December 07, 2004
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Monday, December 06, 2004
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Sunday, December 05, 2004
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Saturday, December 04, 2004
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Friday, December 03, 2004
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Thursday, December 02, 2004
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Saturday, January 29, 2005
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Wednesday, January 26, 2005
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Tuesday, January 25, 2005
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Sunday, January 23, 2005
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Thursday, January 20, 2005
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Wednesday, January 19, 2005
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Tuesday, January 18, 2005
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Sunday, January 16, 2005
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Friday, January 14, 2005
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Thursday, January 13, 2005
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Monday, January 10, 2005
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Sunday, January 09, 2005
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Saturday, January 08, 2005
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Saturday, January 08, 2005
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Thursday, January 06, 2005
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Tuesday, January 04, 2005
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Sunday, February 27, 2005
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Sunday, February 20, 2005
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Thursday, February 17, 2005
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Friday, February 11, 2005
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Wednesday, March 30, 2005
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Monday, March 28, 2005
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Sunday, March 27, 2005
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Saturday, March 26, 2005
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Wednesday, March 23, 2005
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Tuesday, March 22, 2005
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Sunday, March 20, 2005
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Thursday, March 17, 2005
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Wednesday, March 16, 2005
Tuesday, March 15, 2005
Tuesday, March 15, 2005
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Wednesday, March 02, 2005
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Tuesday, March 01, 2005
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Wednesday, May 04, 2005
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Tuesday, May 03, 2005
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Thursday, April 28, 2005
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Tuesday, April 26, 2005
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Thursday, April 14, 2005
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Thursday, April 07, 2005
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Wednesday, April 06, 2005
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Tuesday, April 05, 2005
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Monday, April 04, 2005
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Saturday, April 02, 2005
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Friday, April 01, 2005
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Tuesday, May 31, 2005
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Sunday, May 29, 2005
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Saturday, May 28, 2005
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Friday, May 27, 2005
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Thursday, May 26, 2005
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Wednesday, May 25, 2005
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Tuesday, May 24, 2005
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Monday, May 23, 2005
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Sunday, May 22, 2005
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Saturday, May 21, 2005
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Saturday, May 21, 2005
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Friday, May 20, 2005
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Thursday, May 19, 2005
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Wednesday, May 18, 2005
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Wednesday, May 18, 2005
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Saturday, May 14, 2005
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Friday, May 13, 2005
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Wednesday, May 11, 2005
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Monday, May 09, 2005
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Sunday, May 08, 2005
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Sunday, May 08, 2005
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Friday, May 06, 2005
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Wednesday, May 04, 2005
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Tuesday, May 03, 2005
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Monday, May 02, 2005
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Sunday, May 01, 2005
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Thursday, June 30, 2005
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Wednesday, June 29, 2005
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Monday, June 27, 2005
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Monday, June 27, 2005
Sunday, June 26, 2005
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Saturday, June 25, 2005
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Friday, June 24, 2005
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<a
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Thursday, June 01, 2006
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Monday, July 31, 2006
514305917 3:17 PM
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Sunday, July 30, 2006
513905059 2:02 PM
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Saturday, July 29, 2006
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Saturday, July 29, 2006
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Thursday, July 27, 2006
513003954 9:29 PM
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Wednesday, July 26, 2006
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Monday, July 24, 2006
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Sunday, July 23, 2006
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Saturday, July 22, 2006
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Friday, July 21, 2006
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Thursday, July 20, 2006
510208105 2:00 AM
Tuesday, July 18, 2006
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Monday, July 17, 2006
509357910 6:06 PM
Sunday, July 16, 2006
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Saturday, July 15, 2006
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Saturday, July 15, 2006
508370268 12:00 AM
Friday, July 14, 2006
Thursday, July 13, 2006
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Thursday, July 13, 2006
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Wednesday, July 12, 2006
507402580 9:00 AM
Tuesday, July 11, 2006
507237606 9:11 PM
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Tuesday, July 11, 2006
507143872 2:11 PM
Monday, July 10, 2006
506543837 2:48 AM
Sunday, July 09, 2006
506282052 11:00 AM
Saturday, July 08, 2006
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Friday, July 07, 2006
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Friday, July 07, 2006
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Thursday, July 06, 2006
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Wednesday, July 05, 2006
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Wednesday, July 05, 2006
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Monday, July 03, 2006
504270660 11:07 PM
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Monday, July 03, 2006
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Sunday, July 02, 2006
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Saturday, July 01, 2006
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Saturday, July 01, 2006
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Thursday, August 31, 2006
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Wednesday, August 30, 2006
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Tuesday, August 29, 2006
524130702 7:21 PM
Tuesday, August 29, 2006
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Monday, August 28, 2006
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Sunday, August 27, 2006
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Saturday, August 26, 2006
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Friday, August 25, 2006
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Thursday, August 24, 2006
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Wednesday, August 23, 2006
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Tuesday, August 22, 2006
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Monday, August 21, 2006
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Saturday, August 19, 2006
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Thursday, August 17, 2006
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Monday, August 14, 2006
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Monday, August 14, 2006
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Sunday, August 13, 2006
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Friday, August 11, 2006
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Friday, August 11, 2006
518205357 2:16 PM
Wednesday, August 09, 2006
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Tuesday, August 08, 2006
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Tuesday, August 08, 2006
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Monday, August 07, 2006
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Sunday, August 06, 2006
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Saturday, August 05, 2006
516078074 2:00 PM
Saturday, August 05, 2006
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Friday, August 04, 2006
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Friday, August 04, 2006
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Thursday, August 03, 2006
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Wednesday, August 02, 2006
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Tuesday, August 01, 2006
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Sunday, October 01, 2006
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Saturday, September 30, 2006
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Friday, September 29, 2006
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Thursday, September 28, 2006
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Tuesday, September 26, 2006
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Tuesday, September 26, 2006
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Monday, September 25, 2006
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Saturday, September 23, 2006
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Friday, September 22, 2006
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Thursday, September 21, 2006
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Wednesday, September 20, 2006
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Tuesday, September 19, 2006
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Monday, September 18, 2006
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Sunday, September 17, 2006
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Saturday, September 16, 2006
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Friday, September 15, 2006
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Thursday, September 14, 2006
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Wednesday, September 13, 2006
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Monday, September 11, 2006
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Sunday, September 10, 2006
527898122 2:56 PM
Sunday, September 10, 2006
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Saturday, September 09, 2006
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Friday, September 08, 2006
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Thursday, September 07, 2006
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Thursday, September 07, 2006
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Wednesday, September 06, 2006
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Tuesday, September 05, 2006
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Monday, September 04, 2006
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Monday, September 04, 2006
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Sunday, September 03, 2006
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Friday, September 01, 2006
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Tuesday, October 31, 2006
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Monday, October 30, 2006
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Sunday, October 29, 2006
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Saturday, October 28, 2006
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Friday, October 27, 2006
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Thursday, October 26, 2006
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Wednesday, October 25, 2006
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Tuesday, October 24, 2006
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Tuesday, October 24, 2006
540889575 2:56 PM
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Monday, October 23, 2006
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Sunday, October 22, 2006
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Sunday, October 22, 2006
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Saturday, October 21, 2006
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Friday, October 20, 2006
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Thursday, October 19, 2006
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Wednesday, October 18, 2006
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Tuesday, October 17, 2006
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Monday, October 16, 2006
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Sunday, October 15, 2006
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Saturday, October 14, 2006
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Saturday, October 14, 2006
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Friday, October 13, 2006
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Thursday, October 12, 2006
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Tuesday, October 10, 2006
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Tuesday, October 10, 2006
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Monday, October 09, 2006
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Saturday, October 07, 2006
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Friday, October 06, 2006
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Friday, October 06, 2006
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Thursday, October 05, 2006
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Wednesday, October 04, 2006
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Tuesday, October 03, 2006
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Monday, October 02, 2006
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Sunday, October 01, 2006
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Tuesday, November 28, 2006
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Monday, November 27, 2006
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Saturday, November 25, 2006
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Friday, November 24, 2006
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Wednesday, November 22, 2006
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Tuesday, November 21, 2006
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Monday, November 20, 2006
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Sunday, November 19, 2006
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Saturday, November 18, 2006
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Thursday, November 16, 2006
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Thursday, November 16, 2006
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Wednesday, November 15, 2006
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Monday, November 13, 2006
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Sunday, November 12, 2006
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Sunday, November 12, 2006
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Friday, November 10, 2006
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Friday, November 10, 2006
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Thursday, November 09, 2006
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Wednesday, November 08, 2006
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Tuesday, November 07, 2006
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Tuesday, November 07, 2006
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Monday, November 06, 2006
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Sunday, November 05, 2006
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Saturday, November 04, 2006
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Friday, November 03, 2006
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Thursday, November 02, 2006
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Wednesday, November 01, 2006
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Sunday, December 31, 2006
559841270 8:00 PM
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Friday, December 29, 2006
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Wednesday, December 27, 2006
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Monday, December 25, 2006
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Monday, December 25, 2006
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Saturday, December 23, 2006
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Thursday, December 21, 2006
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Wednesday, December 20, 2006
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Monday, December 18, 2006
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Thursday, December 14, 2006
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Wednesday, December 13, 2006
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Sunday, December 10, 2006
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Saturday, December 09, 2006
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Thursday, December 07, 2006
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Wednesday, December 06, 2006
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Tuesday, December 05, 2006
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Monday, December 04, 2006
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Sunday, December 03, 2006
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Saturday, December 02, 2006
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Saturday, December 02, 2006
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Friday, December 01, 2006
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Wednesday, January 31, 2007
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Monday, January 29, 2007
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Saturday, January 27, 2007
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Friday, January 26, 2007
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Wednesday, January 24, 2007
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Monday, January 22, 2007
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Sunday, January 21, 2007
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Saturday, January 20, 2007
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Friday, January 19, 2007
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Tuesday, January 09, 2007
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Sunday, January 07, 2007
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Saturday, January 06, 2007
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Friday, January 05, 2007
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Thursday, January 04, 2007
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Wednesday, January 03, 2007
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Monday, January 01, 2007
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Wednesday, February 28, 2007
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Tuesday, February 27, 2007
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Monday, February 26, 2007
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Sunday, February 25, 2007
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Sunday, February 25, 2007
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Tuesday, February 20, 2007
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Sunday, February 18, 2007
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Saturday, February 17, 2007
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Friday, February 16, 2007
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Thursday, February 15, 2007
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Tuesday, February 13, 2007
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Monday, February 12, 2007
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Sunday, February 11, 2007
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Saturday, February 10, 2007
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Friday, February 09, 2007
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Wednesday, February 07, 2007
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Saturday, February 03, 2007
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Saturday, February 03, 2007
Friday, February 02, 2007
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Thursday, February 01, 2007
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Saturday, March 31, 2007
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Friday, March 30, 2007
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Wednesday, March 28, 2007
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Saturday, March 24, 2007
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Thursday, March 22, 2007
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Wednesday, March 21, 2007
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Wednesday, March 21, 2007
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Monday, March 19, 2007
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Sunday, March 18, 2007
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Saturday, March 17, 2007
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Friday, March 16, 2007
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Thursday, March 15, 2007
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Wednesday, March 14, 2007
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Saturday, March 10, 2007
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Friday, March 09, 2007
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Thursday, March 08, 2007
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Wednesday, March 07, 2007
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Sunday, March 04, 2007
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Saturday, March 03, 2007
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Friday, March 02, 2007
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Thursday, March 01, 2007
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Monday, April 30, 2007
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Sunday, April 29, 2007
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Saturday, April 28, 2007
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Friday, April 27, 2007
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Wednesday, April 25, 2007
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Tuesday, April 24, 2007
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Monday, April 23, 2007
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Monday, April 23, 2007
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Sunday, April 22, 2007
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Saturday, April 21, 2007
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Friday, April 20, 2007
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Friday, April 20, 2007
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Thursday, April 19, 2007
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Wednesday, April 18, 2007
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Monday, April 16, 2007
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Sunday, April 15, 2007
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Sunday, April 15, 2007
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Saturday, April 14, 2007
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Saturday, April 14, 2007
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Friday, April 13, 2007
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Thursday, April 12, 2007
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Wednesday, April 11, 2007
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Monday, April 09, 2007
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Monday, April 09, 2007
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Sunday, April 08, 2007
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Saturday, April 07, 2007
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Friday, April 06, 2007
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Thursday, April 05, 2007
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Wednesday, April 04, 2007
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Tuesday, April 03, 2007
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Monday, April 02, 2007
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Sunday, April 01, 2007
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Thursday, May 31, 2007
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Wednesday, May 30, 2007
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Monday, May 28, 2007
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Sunday, May 27, 2007
Saturday, May 26, 2007
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Friday, May 25, 2007
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Thursday, May 24, 2007
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Wednesday, May 23, 2007
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Tuesday, May 22, 2007
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Monday, May 21, 2007
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Monday, May 21, 2007
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Sunday, May 20, 2007
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Saturday, May 19, 2007
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Friday, May 18, 2007
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Thursday, May 17, 2007
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Wednesday, May 16, 2007
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Tuesday, May 15, 2007
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Monday, May 14, 2007
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Sunday, May 13, 2007
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Sunday, May 13, 2007
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Saturday, May 12, 2007
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Friday, May 11, 2007
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Friday, May 11, 2007
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Thursday, May 10, 2007
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Wednesday, May 09, 2007
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Tuesday, May 08, 2007
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Monday, May 07, 2007
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Sunday, May 06, 2007
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Sunday, May 06, 2007
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Friday, May 04, 2007
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Thursday, May 03, 2007
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Tuesday, May 01, 2007
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Saturday, June 30, 2007
601100365 10:04 PM
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Thursday, June 28, 2007
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Wednesday, June 27, 2007
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Monday, June 25, 2007
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Sunday, June 24, 2007
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Saturday, June 23, 2007
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Friday, June 22, 2007
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Thursday, June 21, 2007
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Wednesday, June 20, 2007
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Tuesday, June 19, 2007
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Monday, June 18, 2007
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Monday, June 18, 2007
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Sunday, June 17, 2007
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Saturday, June 16, 2007
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Saturday, June 16, 2007
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Friday, June 15, 2007
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Thursday, June 14, 2007
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Thursday, June 14, 2007
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Wednesday, June 13, 2007
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Wednesday, June 13, 2007
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Tuesday, June 12, 2007
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Monday, June 11, 2007
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Sunday, June 10, 2007
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Sunday, June 10, 2007
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Saturday, June 09, 2007
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Friday, June 08, 2007
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Thursday, June 07, 2007
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Thursday, June 07, 2007
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Wednesday, June 06, 2007
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Tuesday, June 05, 2007
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Sunday, June 03, 2007
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Saturday, June 02, 2007
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Tuesday, July 31, 2007
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Monday, July 30, 2007
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Monday, July 30, 2007
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Sunday, July 29, 2007
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Sunday, July 29, 2007
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Saturday, July 28, 2007
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Thursday, July 26, 2007
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Wednesday, July 25, 2007
606111937 9:00 AM
Tuesday, July 24, 2007
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Monday, July 23, 2007
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Saturday, July 21, 2007
605340686 9:45 AM
Thursday, July 19, 2007
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Thursday, July 19, 2007
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Wednesday, July 18, 2007
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Tuesday, July 17, 2007
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Monday, July 16, 2007
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Sunday, July 15, 2007
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Saturday, July 14, 2007
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Friday, July 13, 2007
603712325 7:00 AM
Thursday, July 12, 2007
603629730 7:00 PM
603588597 1:00 PM
Wednesday, July 11, 2007
603247518 2:45 AM
Tuesday, July 10, 2007
603081833 9:00 AM
603070402 8:00 AM
603069912 7:59 AM
603003210 12:01 AM
Monday, July 09, 2007
603003081 11:59 PM
602729011 2:35 AM
Saturday, July 07, 2007
602529877 2:22 PM
602398088 1:48 AM
602383121 12:00 AM
Friday, July 06, 2007
602375889 11:07 PM
602359542 8:00 PM
602357485 7:47 PM
602175079 12:26 AM
602173734 12:18 AM
Friday, July 06, 2007
602172268 12:06 AM
Thursday, July 05, 2007
602143009 7:11 PM
602097762 12:48 PM
Tuesday, July 03, 2007
601723043 9:29 PM
Monday, July 02, 2007
601494979 8:28 PM
Monday, July 02, 2007
601432920 12:03 PM
Sunday, July 01, 2007
601293940 10:31 PM
601243649 1:27 PM
601130231 2:06 AM
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<div class="blogheader">Friday, August 31, 2007
613358969 10:10 PM
Wednesday, August 22, 2007
611673223 11:00 PM
611572141 10:31 AM
Tuesday, August 21, 2007
611446544 3:29 PM
611350451 8:14 AM
611344571 8:00 AM
Monday, August 20, 2007
611155585 8:01 AM
Sunday, August 19, 2007
610972092 8:19 AM
Saturday, August 18, 2007
610897230 7:20 PM
Tuesday, August 14, 2007
610054662 12:00 PM
Monday, August 13, 2007
609832401 11:07 AM
Sunday, August 12, 2007
609619788 9:00 AM
Saturday, August 11, 2007
609532971 10:00 PM
Friday, August 10, 2007
609282854 10:31 AM
Thursday, August 09, 2007
609085797 12:00 PM
Tuesday, August 07, 2007
608725736 6:25 PM
Tuesday, August 07, 2007
608610911 8:00 AM
Monday, August 06, 2007
608410039 8:00 AM
Sunday, August 05, 2007
608321060 7:00 PM
608227620 9:00 AM
Saturday, August 04, 2007
608037417 8:00 AM
Friday, August 03, 2007
607979954 10:09 PM
607941337 2:02 PM
607861289 8:09 AM
Wednesday, August 01, 2007
607579945 9:29 PM
607568556 6:19 PM
Wednesday, August 01, 2007
607455961 8:00 AM
Sunday, September 30, 2007
618944496 2:00 PM
618870750 8:28 AM
618834293 3:14 AM
Saturday, September 29, 2007
618659874 3:09 AM
Friday, September 28, 2007
618490463 6:25 AM
Thursday, September 27, 2007
618423240 4:00 PM
618292623 6:29 AM
Tuesday, September 25, 2007
617947916 8:00 AM
Monday, September 24, 2007
617704736 1:06 AM
Sunday, September 23, 2007
617692818 11:07 PM
Sunday, September 23, 2007
617555085 5:51 AM
617548823 5:01 AM
Saturday, September 22, 2007
617377554 6:25 AM
617377300 6:23 AM
617377167 6:22 AM
Friday, September 21, 2007
617206110 8:28 AM
Wednesday, September 19, 2007
616810021 5:00 AM
Saturday, September 15, 2007
616192373 8:00 PM
Thursday, September 13, 2007
615683763 3:57 AM
Thursday, September 13, 2007
615675861 2:02 AM
Wednesday, September 12, 2007
615630908 5:01 PM
Tuesday, September 11, 2007
615287767 12:07 AM
Monday, September 10, 2007
615195800 11:07 AM
Saturday, September 08, 2007
614881336 7:11 PM
Saturday, September 08, 2007
614857110 2:02 PM
Friday, September 07, 2007
614712034 9:29 PM
614680955 2:02 PM
Thursday, September 06, 2007
614448043 11:00 AM
Sunday, September 02, 2007
613711378 5:11 PM
Wednesday, October 31, 2007
624616664 8:28 PM
Tuesday, October 30, 2007
Monday, October 29, 2007
624137721 7:20 AM
Sunday, October 28, 2007
623967292 7:59 AM
Thursday, October 25, 2007
623439304 9:19 AM
Wednesday, October 24, 2007
623365906 11:11 PM
623285567 11:01 AM
623258985 9:26 AM
Sunday, October 21, 2007
622727625 10:31 AM
Thursday, October 18, 2007
622251512 3:14 PM
622198809 11:07 AM
Wednesday, October 17, 2007
622007807 10:00 AM
Tuesday, October 16, 2007
621824383 10:00 AM
Monday, October 15, 2007
Sunday, October 14, 2007
621464914 11:00 AM
Saturday, October 13, 2007
621253189 9:22 AM
Friday, October 12, 2007
621072729 9:10 AM
621011874 12:00 AM
Thursday, October 11, 2007
620997954 9:26 PM
620981910 5:01 PM
620943547 12:00 PM
Tuesday, October 09, 2007
620482237 4:09 AM
Monday, October 08, 2007
Sunday, October 07, 2007
620194402 12:07 PM
Friday, October 05, 2007
619784656 9:00 AM
Thursday, October 04, 2007
Wednesday, October 03, 2007
619494781 3:09 PM
Monday, October 01, 2007
619055443 7:20 AM
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