Mathematics and Narrative continued Narrative: xxx Mathematics: "It must be remarked that these 8 heptads are the key to an elegant proof...." -- Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference, (July 2000), Springer, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97 Mathematics: "Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and O6 + (2)." -- J. Sutherland Frame in "Orthogonal Groups over GF(2) and Related Graphs," Springer Lecture Notes in Mathematics vol. 642, Theory and Applications of Graphs (Proceedings, Michigan, May 11–15, 1976), edited by Y. Alavi and D. R. Lick, pp. 174-185 "One has O+(6) ≅ S8, the symmetric group of order 8!...." -- "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint. This paper gives some context in superstring theory for the following work of Frame: [F1] J.S. Frame, The classes and representations of the group of 27 lines and 28 bitangents, Annali Posted 2/28/2009 at 8:00 AM |
Time and Chance continued Today's Pennsylvania lottery numbers suggest the following meditations... Midday: Lot 497, Bloomsbury Auctions May 15, 2008-- Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."* Evening: 5/07, 2008, in this journal-- "Forms of the Rock." Related material: A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009. The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space." A related quotation by Lefebvre (cf. 2/22 2009): "... an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered... the space... of classical perspective and geometry...."This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910. In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.† * Ending on Stephen King's birthday, 2008 † Mardi Gras Posted 2/27/2009 at 7:35 PM |
Lasting Significance Wittgenstein's Lasting Significance, edited by Max Kölbel and Bernhard Weiss, published by Routledge, 2004-- Page 168: "Wittgenstein told Norman Malcolm that 'a serious and good philosophical work could be written that would consist entirely of jokes (without being facetious)' (Malcolm 1999: 64)." Malcolm, N. (1999) "Wittgenstein: A Memoir," in F.A. Flowers (ed.) Portraits of Wittgenstein, vol. 3, Bristol: Thoemmes Press, pp. 60-112The lasting significance here is perhaps in the page numbers. Posted 2/27/2009 at 10:12 AM |
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Truth and Consequences: From Roger Cohen to Alain Badiou to Wallace Stevens
"The Event of Truth," European Graduate School video: Quoted by Badiou at European Graduate School, August 2002:
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Midnight "Dead time lasts for one hour-- from half an hour before midnight to half an hour after midnight. The half-hour before midnight is for doin' good. The half-hour after midnight is for doin' evil...." -- Glenna Whitley, "Voodoo Justice" From the Curriculum Vitae of Patrick McGee: "Theory and the Common from Marx to Badiou (Palgrave 2009, scheduled for March 31 publication)" Thanks for the warning.
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"Dead time lasts for one hour-- from half an hour before midnight to half an hour after midnight. The half-hour before midnight is for doin' good." -- Glenna Whitley, "Voodoo Justice," The New York Times, March 20, 1994 Posted 2/25/2009 at 11:30 PM |
Ideas of Reference for Ash Wednesday Happy trails to you... These numbers from yesterday (Mardi Gras, 2009) are random, yet have a particular meaning for me and perhaps one other person. -- Google Book Search Posted 2/25/2009 at 12:00 PM |
Hollywood Nihilism Meets Pantheistic Solipsism Tina Fey to Steve Martin at the Oscars: "Oh, Steve, no one wants to hear about our religion ... that we made up."
Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"-- noting, however, that "This article has multiple issues." As, indeed, does pantheistic solipsism-- a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein. Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up-- namely, the properties of space. Heinlein: "Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three-- you seem to be smart at such projections." I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it." Stevens:
Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:
This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism. More positively... in pure mathematics... For instance, the 6-dimensional affine space (or the corresponding 5-dimensional projective space) over the two-element Galois field can be viewed as an illustration of Stevens's metaphor in "The Rock." Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three. Here the 6-dimensional affine space contains the 63 points of PG(5, 2), plus the origin, and the 3-dimensional affine space contains as its 8 points Conwell's eight "heptads," as in Generating the Octad Generator. Posted 2/24/2009 at 1:00 PM |
McGee Thanks the Academy: "The ulterior motive behind this essay ["Reading Authority," above], the purpose for which I seize this occasion, concerns the question or problem of authority. I stress at the outset my understanding of authority as the constructed repository of value or foundation of a system of values, the final effect of fetishism-- in this case, literary fetishism. [Cf. Marx, Das Kapital] Reading-- as in the phrase 'reading authority'-- should be grasped as the institutionally determined act of constructing authority...."Wikipedia: "[In Peter Pan] Smee is Captain Hook's right-hand man... Barrie describes him as 'Irish' and 'a man who stabbed without offence'...."Background: In yesterday's morning entry, James Joyce as Jesuit, with "Dagger Definitions." A different Smee appears as an art critic in yesterday's afternoon entry "Design Theory."-- Smee Stabs Without Offence: A Jesuit in Portrait of the Artist as a Young Man: "When may we expect to have something from you on the esthetic question?" Beckett Bethicketted: Posted 2/23/2009 at 2:22 PM |
Themes and Variations The Boston Globe today on a current Museum of Fine Arts exhibit of works collected by one Horace Brock-- "Designed objects, Brock writes, can be broken down into 'themes' and 'transformations.' A theme is a motif, such as an S-curve; a transformation might see that curve appear elsewhere in the design, but stretched, rotated 90 degrees, mirrored, or otherwise reworked. Aesthetic satisfaction comes from an apprehension of how those themes and transformations relate to each other, or of what Brock calls their 'relative complexity.' Basically-- and this is the nub of it-- 'if the theme is simple, then we are most satisfied when its echoes are complex... and vice versa.'" Related material:Theme and Variations See also earlier tributes to Hollywood Game Theory and Hollywood Religion: For some variations on the above checkerboard theme, see Finite Relativity and A Wealth of Algebraic Structure. Posted 2/22/2009 at 4:07 PM |
Design at Harvard: Natural or Unnatural? From the Harvard Call for Entries: DEADLINE FEBRUARY 27, 2009 8PM EST "According to Henri Lefebvre, the social production of space has three components: spatial practice, the representation of space, and the space of representation. The latter two are integral to both design and the review process." Posted 2/22/2009 at 11:00 AM |
The Graduate Today's New York Times: The Times goes on to say... "A native of Buffalo, Ms. Jacobus graduated from Le Moyne College in Syracuse." She died yesterday. A quotation from yesterday's entries may be relevant:
Related material: From a previous appearance of the Eastwood meditation in this journal: Click on image for details. Posted 2/21/2009 at 10:10 AM |
The Cross of Constantine mentioned in this afternoon's entry "Emblematizing the Modern" was the object of a recent cinematic chase sequence (successful and inspiring) starring Mira Sorvino at the Metropolitan Museum of Art. In memory of Dr. Hunter S. Thompson, dead by his own hand on this date four years ago -- Click for details. There is another sort of object we may associate with a different museum and with a modern Constantine ... See "Art Wars for MoMA" (Dec. 14, 2008). This object, modern rather than medieval, is the ninefold square: It may suit those who, like Rosalind Krauss (see "Emblematizing"), admire the grids of modern art but view any sort of Christian cross with fear and loathing. For some background that Dr. Thompson might appreciate, see notes on Geometry and Death in this journal, June 1-15, 2007, and the five Log24 entries ending at 9 AM Dec. 10. 2006, which include this astute observation by J. G. Ballard: "Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom." Selah. Posted 2/20/2009 at 11:01 PM |
A Kind of Cross "For every kind of vampire, there is a kind of cross." -- Thomas Pynchon in Gravity's Rainbow Click for source. Related material: A memorial service held at 2 PM today at the U.S. Space & Rocket Center in Huntsville, Alabama, and today's previous entry. Posted 2/20/2009 at 6:00 PM |
Emblematizing the Modern
The Cross of Descartes
There is a reason, apart from her ethnic origins, that Rosalind Krauss (cf. 9/13/06) rejects, with a shudder, the cross as a key to "the Pandora's box of spiritual reference that is opened once one uses it." The rejection occurs in the context of her attempt to establish not the cross, but the grid, as a religious symbol:
Related material: Time Fold and Weyl on objectivity and frames of reference. See also Stambaugh on The Formless Self as well as A Study in Art Education and Jung and the Imago Dei. Posted 2/20/2009 at 2:01 PM |
A Sunrise for Sunrise "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." --Rosalind Krauss, "Grids" Yesterday's entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo) may be used to create a graphic design-- a Greek cross, which appears also in crossword puzzles: Illustration by Paul Rand (born Peretz Rosenbaum) A more sophisticated example of the ninefold square in graphic design:
From Paul-Rand.com Posted 2/19/2009 at 7:07 AM |
Raiders of the Lost Well
-- Olga Raggio, a graduate of the Vatican library school and the University of Rome who, at one point in her almost 60 years with the Metropolitan Museum of Art, organized "The Vatican Collections," a blockbuster show. Dr. Raggio died on January 24. The next day, "The Last Templar," starring Mira Sorvino, debuted on NBC. "The story, involving the Knights Templar, the Vatican, sunken treasure, the fate of Christianity and a decoding device that looks as if it came out of a really big box of medieval Cracker Jack, is the latest attempt to combine Indiana Jones derring-do with 'Da Vinci Code' mysticism." Sorvino in "The Last Templar" at the Church of the Lost Well: "One highlight of Secretary of State Hillary Clinton's first overseas trip will be a stop in China. Her main mission in Beijing will be to ensure that US-China relations under the new Obama administration get off to a positive start." -- Stephanie Ho, Voice of America Beijing bureau chief, today Symbol of The Positive, "Stephanie started at the Voice of America as an intern in 1991. She left briefly to attend film school in London in 2000. Although she didn't finish, she has always wanted to be a film school dropout, so now she's living one of her dreams. Stephanie was born in Ohio and grew up in California. She has a bachelor's degree in Asian studies with an emphasis on Chinese history and economics, from the University of California at Berkeley." "She is fluent in As is Mira Sorvino. Those who, like Clinton, Raggio, and In so doing, they of course (See Rosalind Krauss, Professor Krauss Posted 2/18/2009 at 11:30 AM |
Diamond-Faceted: Transformations of the Rock
A mathematical version of this poetic concept appears in a rather cryptic note from 1981 written with Stevens's poem in mind: For some explanation of the groups of 8 and 24 motions referred to in the note, see an earlier note from 1981. For the Perlis "diamond facets," see the Diamond 16 Puzzle. For a much larger group of motions, see Solomon's Cube. As for "the mind itself" and "possibilities for human thought," see Geometry of the I Ching. Posted 2/17/2009 at 1:06 PM |
Related material:
For transformations of a more specifically religious nature, see the remarks on Richard Strauss, "Death and Transfiguration," (Tod und Verklärung, Opus 24) in Mathematics and Metaphor on July 31, 2008, and the entries of August 3, 2008, related to the death of Alexander Solzhenitsyn. Posted 2/15/2009 at 11:00 AM |
The Devil in the Details Here are clearer pictures of the Einstein-Gutkind letter discussed here February 7. The pictures are from the Bloomsbury Auctions site. The Bloomsbury Auctions caption for these images is as follows: 303. Einstein (Albert, theoretical physicist, 1879-1955) Autograph Letter signed to Eric B. Gutkind, in German, 1½pp. & envelope, 4to, Princeton, 3rd January 1954, thanking him for a copy of his book and expressing his view of God and Judaism, [The word God is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honorable, but still primitive legends which are nevertheless pretty childish... . For me the Jewish religion like all other religions is an incarnation of the most childish superstitions. And the Jewish people to whom I gladly belong and with whose mentality I have a deep affinity have no different quality for me than all other people...], folds, slightly browned ; and a photograph of Gutkind, v.s., v.d.Here is a close reading of the part of the letter itself that Bloomsbury gives in English, transcribed from the above images. Line-by-line transcription of paragraph 2, starting at line 4 of that paragraph: Phrases by Stambaugh that do not appear in the German text are highlighted.... Das Wort Gott ist für mich nichts als Ausdruck Stambaugh, a philosophy professor, is the author of a work on Buddhism, The Formless Self. For some related material on young men who "go crying 'The world is myself, life is myself'" in May, see Wallace Stevens's "The Pediment of Appearance." Posted 2/14/2009 at 9:29 PM |
"Somewhere in the heart of Rome" -- Sinatra, 1954 USA Today: Geithner has some success with world stage debut Click for commentary. Posted 2/14/2009 at 12:00 PM |
Fire and Ice Prologue from Answers.com: bombardier The member of a combat aircraft crew who operates the bombsight and drops the bombs.
Today in History, by The Associated Press On this date... in 1945, during World War II, Allied planes began bombing the German city of Dresden. For the rest of the story, see Kurt Vonnegut and Robert Frost. Posted 2/13/2009 at 7:36 PM |
DENNIS OVERBYE "From the grave, Albert Einstein poured gasoline on the culture wars between science and religion this week. A letter the physicist wrote in 1954 to the philosopher Eric Gutkind, in which he described the Bible as 'pretty childish'...."This morning's New York Times: The plane crashed at about 10:20 PM. Yesterday evening in Springfield (as scheduled): 6:20 PM THE PRESIDENT arrives in Springfield, IL 7:00 PM THE PRESIDENT delivers remarks at the 102nd Abraham Lincoln Association Annual Banquet 8:30 PM THE PRESIDENT departs from Springfield, IL
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Happy birthday to King Friday XIII and friend: Yesterday, by the way, was Georgia Day in Savannah. "I Put a Spell on You" -- Nina Simone, title of autobiograpy "The voodoo priestess looked across the table at her wealthy client, a man on trial for murder: 'Now, you know how dead time works. Dead time lasts for one hour-- from half an hour before midnight to half an hour after midnight. The half-hour before midnight is for doin' good. The half-hour after midnight is for doin' evil....'" -- Glenna Whitley, "Voodoo Justice," The New York Times, March 20, 1994 Posted 2/13/2009 at 12:00 AM |
Headliners Today, many observe the 200th anniversary of the birth of two noted philosophers of death: Charles Darwin and Abraham Lincoln. A fitting headline: FAUST VIVIFIES DEATH (Harvard Crimson, February 7, 2008) Happy birthday, Cotton Mather. Robert Stone, A Flag for Sunrise: "Our secret culture is as frivolous as a willow on a tombstone. It's a wonderful thing-- or it was. It was strong and dreadful, it was majestic and ruthless. It was a stranger to pity. And it's not for sale, ladies and gentlemen." Posted 2/12/2009 at 11:11 AM |
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"Brams... uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios...." (The number-pairs here reflect relative values of the situations the author assigns to SB and to P.) Related Material on theology and drama -- the two Log24 entries on Stephen King's birthday, 2008. Posted 2/10/2009 at 11:07 PM |
Coming Soon! Trailer: "Now, here's my plan..." "'What plan?' asked Bert Ely, an Alexandria, Va., banking consultant. 'The devil is in the details, and the details are hiding in the bushes or deep underground.' The Dow, which was down only about 70 points before Geithner's speech, fell sharply as soon as he began talking." -- Walter Hamilton in The Los Angeles Times today Posted 2/10/2009 at 7:11 PM |
The Vision Thing The British Academy Awards last night showed two Paul Newman clips: "Sometimes nothin' can be a real cool hand." "Boy, I got vision and the rest of the world wears bifocals." Related material: This journal, September 2008. As for bifocals...
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The Sound of Silence See also yesterday's entry on philosophy professor Joan Stambaugh and the fabrication of a now-famous saying falsely attributed to Einstein-- that the Bible is "pretty childish." Stambaugh advocates a Zen form of nihilism. The 4x4 space illustrated above is a Western form of the the Sunyata, or emptiness, discussed by Stambaugh in The Formless Self. It appeared in this journal on the feast day this year of St. John Neumann. Posted 2/8/2009 at 11:00 AM |
DENNIS OVERBYE Einstein did not, at least in the place alleged, call the Bible "childish." Proof: The image of the letter is from the Sept./Oct. 2008 Search Magazine. By the way, today is the birthday of G. H. Hardy. Here is an excerpt from his thoughts on childish things: "What 'purely aesthetic' qualities can we distinguish in such theorems as Euclid's or Pythagoras's?.... In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions." "Space: what you -- James Joyce, Ulysses Posted 2/7/2009 at 2:02 PM |
Eternal City Today's New York Times: "Olga Raggio was born in Rome on Feb. 5, 1926, to a Russian mother and an Italian father. She earned a diploma from the Vatican library school in 1947 and a Ph.D. from the University of Rome in 1949." "... Raggio, an internationally known scholar and curator who in almost 60 years with the Metropolitan Museum of Art organized some of its best-known exhibitions, scoured the world for treasure and coaxed rarely seen artworks from places as far flung as the Vatican and as close at hand as a New Jersey abbey, died on Jan. 24 in the Bronx. She was 82 and lived in Manhattan." Quoted here on the date of Raggio's death: "Death is not earnest in the same way the eternal is. To the earnestness of death belongs precisely that capacity for awakening, that resonance of a profound mockery which, detached from the thought of the eternal, is an empty and often brash jest, but together with the thought of the eternal is just what it should be, utterly different from the insipid solemness which least of all captures and holds a thought with tension like that of death." -- Soren Kierkegaard, Works of Love, Harper Torchbooks, 1964, p. 324 Related material: February 2, 3, and 4 as well as Posted 2/6/2009 at 4:00 AM |
Through the Looking Glass: A Sort of Eternity From the new president's inaugural address: "... in the words of Scripture, the time has come to set aside childish things." The words of Scripture:
"through a glass"-- [di’ esoptrou]. By means of a mirror [esoptron]. Childish things: © 2005 The Institute for Figuring Photo by Norman Brosterman fom the Inventing Kindergarten exhibit at The Institute for Figuring (co-founded by Margaret Wertheim) Not-so-childish: Three planes through the center of a cube that split it into eight subcubes: Through a glass, darkly: A group of 8 transformations is generated by affine reflections in the above three planes. Shown below is a pattern on the faces of the 2x2x2 cube that is symmetric under one of these 8 transformations-- a 180-degree rotation: (Click on image for further details.) But then face to face: A larger group of 1344, rather than 8, transformations of the 2x2x2 cube is generated by a different sort of affine reflections-- not in the infinite Euclidean 3-space over the field of real numbers, but rather in the finite Galois 3-space over the 2-element field. Galois age fifteen, drawn by a classmate. These transformations in the Galois space with finitely many points produce a set of 168 patterns like the one above. For each such pattern, at least one nontrivial transformation in the group of 8 described above is a symmetry in the Euclidean space with infinitely many points. For some generalizations, see Galois Geometry. Related material:
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Overkill In memory of James Joyce and of Patrick McGoohan. who both died on a January 13th -- Related material: The phrase "Habitat Global Village" in the previous entry. Marshall McLuhan was apparently the originator of the phrase "global village." The phrase, coined by McLuhan, a Catholic, should be associated more with Rome than with Americus, Georgia. "The association is the idea." -- Ian Lee, The Third Word War Posted 2/4/2009 at 1:23 PM |
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Everything and Nothing "I know what 'nothing' means...." -- Joan Didion, Play It As It Lays, Farrar, Straus and Giroux, 1990 paperback, page 214 "In 1935, near the end of a long affectionate letter to his son George in America, James Joyce wrote: 'Here I conclude. My eyes are tired. For over half a century they have gazed into nullity, where they have found a lovely nothing.'" -- Lionel Trilling, "James Joyce in His Letters," Commentary, 45, no. 2 (Feb. 1968), abstract "The quotation is from The Letters of James Joyce, Volume III, ed. Richard Ellman (New York, 1966), p. 359. The original Italian reads 'Adesso termino. Ho gli occhi stanchi. Da più di mezzo secolo scrutano nel nulla dove hanno trovato un bellissimo niente.'" -- Lionel Trilling: Criticism and Politics, by William M. Chace, Stanford U. Press, 1980, page 198, Note 4 to Chapter 9 "Space: what you damn well have to see." -- James Joyce, Ulysses "What happens to the concepts of space and direction if all the matter in the universe is removed save a small finite number of particles?" -- "On the Origins of Twistor Theory," by Roger Penrose "... we can look to the prairie, the darkening sky, the birthing of a funnel-cloud to see in its vortex the fundamental structure of everything..." -- Against the Day, by Thomas Pynchon (See previous entry.) "A strange thing then happened." -- L. Frank Baum Posted 2/3/2009 at 7:59 AM |
Against the Day is a novel by Thomas Pynchon published on Nov. 21, 2006, in hardcover, and in paperback on Oct. 30, 2007 (Devil's Night). Perhaps the day the title refers to is one of the above dates... or perhaps it is--
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"For every kind of vampire, there is a kind of cross." -- Gravity's Rainbow Quaternion Happy St. Brigid's Day. Posted 2/1/2009 at 9:00 AM |