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Wednesday, May 28, 2003 |
Mental Health Month, Day 28: The Eightfold Way and For a continuation of the mathematical and religious themes in yesterday's entry, click on the figure below.
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Tuesday, May 27, 2003 |
Mental Health Month, Day 27: Conspiracy Theory and In our journey through Mental Health Month, we have now arrived at day 27. This number, the number of lines on a non-singular cubic surface in complex projective 3-space, suggests it may be time to recall the following note (a sort of syllabus for an imaginary course) from August 1997, the month that the Mel Gibson film "Conspiracy Theory" was released. Conspiracy Theory 101 Fiction: Fact: See also China's 3,000-year-old "Book of Transformations," the I Ching, for more philosophy and lore of the affine 6-dimensional space over the binary field. © 1997 S. H. Cullinane For a more up-to-date and detailed look at the mathematics mentioned above, see Abstract Configurations by Igor Dolgachev. "Art isn't easy." -- Stephen Sondheim
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Monday, May 26, 2003 |
Mental Health Month, Day 26: Many Dimensions, At first blush, it seems unlikely that the number 26=2x13, as a product of only two small primes (and those distinct) has any purely mathematical properties of interest. (On the other hand, consider the number 6.) Parts I and II of "Many Dimensions," notes written earlier today, deal with the struggles of string theorists to justify their contention that a space of 26 dimensions may have some significance in physics. Let them struggle. My question is whether there are any interesting purely mathematical properties of 26, and it turns out, surprisingly, that there are some such properties. All this is a longwinded way of introducing a link to the web page titled "Info on M13," which gives details of a 1997 paper by J. H. Conway*. "Conway describes the beautiful construction of a discrete mathematical structure which he calls ' In fact, both the Mathieu group To understand the definition of The points and the lines and the "is-contained-in" relation form an incidence structure over ...the 26 objects of the incidence structure [are] 13 points and 13 lines." Conway's construction involves the arrangement, in a circular Levi graph, of 26 marks representing these points and lines, and chords representing the "contains/is contained in" relation. The resulting diagram has a pleasingly symmetric appearance. For further information on the geometry of the number 26, one can look up all primitive permutation groups of degree 26. Conway's work suggests we look at sets (not just groups) of permutations on n elements. He has shown that this is a fruitful approach for n=13. Whether it may also be fruitful for n=26, I do not know. There is no obvious connection to physics, although the physics writer John Baez quoted in my previous two entries shares Conway's interest in the Mathieu groups. * J. H. Conway, "M13," in Surveys in Combinatorics, 1997, edited by R. A. Bailey, London Mathematical Society Lecture Note Series, 241, Cambridge University Press, Cambridge, 1997. 338 pp. ISBN 0 521 59840 0. |
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Monday, May 26, 2003 |
Mental Health Month, Day 26: Many Dimensions, But seriously... John Baez in July 1999: "...it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries." Well, maybe. I certainly hope so. If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light. Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory. See the following from my journal: Evariste Galois and the Rock that Changed Things, and A Time to Gather Stones Together: Readings for Yom Kippur. See, too, this web page on Marion Zimmer Bradley's fictional Matrices, or Blue Star-Stones, and the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives. |
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Monday, May 26, 2003 |
Mental Health Month, Day 26: John Baez on why bosonic string theory is said to require 26 dimensions -- "By now, if you're a rigorous sort of pure mathematician, you must be suffering from grave doubts about the sanity of this whole procedure." Doubts? Let us just say I prefer "The G-string is unique in that it combines the properties of all known string theories. It has 26-dimensional modes propagating to the left, 10-dimensional modes propagating to the right, and 2-dimensional modes just sitting around wondering what the hell is going on." |
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Sunday, May 25, 2003 |
From the web page Amande: Le Christ et la Vierge apparurent souvent entourés d'une auréole en forme d'amande: la mandorle. Étymologiquement, le mot amande est une altération de amandala, qui dérive lui-même du latin classique amygdala.... L'amande a... une connotation symbolique, celle du sexe féminin. Elle figure souvent la vulve. Elle est alors en analogie avec la yoni du vocabulaire de l'hindouisme, la vulve ou la matrice, représentée par une amande ou une noix coupée en deux. Screenshot of the online Ariel the Hutt and Princess Amygdala by Horia Cristescu and The intersection of two geometric forms (lines, triangles, circles, etc.) represents forces that are even more intense than those generated by the simple forms. Such an interpenetration indicates a high level in the dynamic interaction of the correspondent energies. The empty spaces generated by such combinations are described as very efficient operational fields of the forces emanating from the central point of the YANTRA. That is why we can very often encounter representations of MANTRAS in such spaces. YANTRA and MANTRA are complementary aspects of SHIVA and their use together is much more efficient than the use of one alone.
AMEN. |
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Sunday, May 25, 2003 |
— ART WARS — Mental Health Month, Day 25: Matrix of the Death God Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham. (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.) For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week. See, too, The Matrix of Abraham, illustrated below. This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933. The Matrix of Abraham A summary of the religious import of the above from Princeton University Press: "Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center.... The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned." -- Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72 Other appearances of this religious icon on the Web: A less religious approach to the icon may be found on page 393 of R. D. Carmichael's Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956). This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.
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Saturday, May 24, 2003 |
Mental Health Month, Day 24: The Sacred Day of On this day, Gypsies from all over Europe gather in Provence for the sacred day of St. Sarah, also known as Kali. Various representations of Kali exist; there is a novel about the ways men have pictured her:
From the prologue to She was old when the earth was young. She stood atop Cemetery Ridge when Pickett made his charge, and she was there when the six hundred rode into the Valley of Death. She was at Pompeii when Mount Vesuvius blew, and she was in the forests of Siberia when the comet hit. She hunted elephant with Selous and buffalo with Cody, and she was there the night the high wire broke beneath the Flying Wallendas. She was at the fall of Troy and the Little Bighorn, and she watched Manolete and Dominguez face the brave bulls in the bloodstained arenas of Madrid.... She has no name, no past, no present, no future. She wears only black, and though she has been seen by many men, she is known to only a handful of them. You'll see her -- if you see her at all -- just after you've taken your last breath. Then, before you exhale for the final time, she'll appear, silent and sad-eyed, and beckon to you. She is the Dark Lady, and this is her story. The above is one of the best descriptions of Kali I know of in literature; another is in a short story by Fritz Leiber, "Damnation Morning." It is not coincidental that one collection of Leiber's writings is called "Dark Ladies."
My journal note "Biblical Proportions" was in part inspired by Leiber.
Frank Sinatra may have pictured her as Ava Gardner. I think I saw her the night Sinatra died... hence my entries of March 31 and April 2, 2003.
It is perhaps not irrelevant that Kali is, among other things, a mother goddess, and that my entry "Raiders of the Lost Matrix" of May 20 deals with this concept and with the number 24. The above religious symbol (see "Damnation Morning") pictures both the axes of symmetry of the square¹ and a pattern with intriguing combinatorial properties². It also is the basis of a puzzle³ I purchased on August 29, 1997 -- Judgment Day in Terminator 2. Linda Hamilton as Sarah Connor in that film is an excellent representation of the Dark Lady, both as mother figure and as Death Goddess. Background music: "Bit by bit..." -- Stephen Sondheim... See Sondheim and the Judgment Day puzzle in my entry of May 20. The Lottery Covenant. ¹ A. W. Joshi, Elements of Group Theory for Physicists, Third Edition, Wiley, 1982, p. 5 ² V. K. Balakrishnan, Combinatorics, McGraw-Hill, 1995, p. 180 |
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Friday, May 23, 2003 |
Mental Health Month, Day 23: The Prime Cut Gospel On Christmas Day, 1949, Lee Marvin, Sissy Spacek in "Prime Cut" Exercises for Mental Health Month: Read this discussion of the phrase, suggested by Spacek's date of birth, "God's gift to men." Read this discussion of the phrase "the same yesterday, today, and forever," suggested by the previous reading. Read the more interesting of these discussions of the phrase "the eternal in the temporal." Read this discussion of eternal, or "necessary," truths versus other sorts of alleged "truths." Read this discussion of unimportant mathematical properties of the prime number 23. Read these discussions of important properties of 23: Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, "Tactical Configurations," and J. H. Conway, "Three Lectures on Exceptional Groups," pp. 215-247 in Finite Simple Groups (Oxford, 1969), edited by M. B. Powell and G. Higman, Academic Press, London, 1971..... Reprinted as Ch. 10 in Sphere Packings, Lattices, and Groups Read this discussion of what might be called "contingent," or "literary," properties of the number 23. Read also the more interesting of these discussions of the phrase "the 23 enigma." Having thus acquired some familiarity with both contingent and necessary properties of 23... Read this discussion of Aquinas's third proof of the existence of God. Note that the classic Spacek film "Prime Cut" was released in 1972, the year that Spacek turned 23: |
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Friday, May 23, 2003 |
Götterdämmerung As our celebration of Wagner's May 22 birthday draws to a close, let us recall that on this date in 1966 the Beatles released "Paperback Writer" in the US. Perhaps our most notable paperback writer is now Stephen King; in honor of a recurring theme in his Hearts in Atlantis, our site music today is "Twilight Time." |
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Thursday, May 22, 2003 |
Seek and Ye Shall Find: On the Mystical Properties On this date in history: May 22, 1942: Unabomber Theodore John Kaczynski is born in the Chicago suburb of Evergreen Park, Ill., to Wanda Kaczynski and her husband Theodore R. Kaczynski, a sausage maker. His mother brings him up reading Scientific American. From the June 2003 Scientific American: "Seek and ye shall find." - Michael Shermer From my note Mark of April 25, 2003: "Tell me of runes to grave -- A. E. Housman, quoted by G. H. Hardy in A Mathematician's Apology "Here, as examples, are one rune and one bastion.... (illustrations: the Dagaz rune and the Nike bastion of the Acropolis).... Neither the rune nor the bastion discussed has any apparent connection with the number 162... But seek and ye shall find." Here is a connection to runes: Mayer, R.M., "Runenstudien," Beiträge zur Geschichte der deutschen Sprache und Literatur 21 (1896): pp. 162 - 184. Here is a connection to Athenian bastions from a UN article on Communist educational theorist Dimitri Glinos: "Educational problems cannot be scientifically solved by theory and reason alone...." (D. Glinos (1882-1943), Dead but not Buried, Athens, Athina, 1925, p. 162) "Schools are.... not the first but the last bastion to be taken by... reform...." "...the University of Athens, a bastion of conservatism and counter-reform...." I offer the above with tongue in cheek as a demonstration that mystical numerology may have a certain heuristic value overlooked by fanatics of the religion of Scientism such as Shermer. For a more serious discussion of runes at the Acropolis, see the photo on page 16 of the May 15, 2003, New York Review of Books, illustrating the article "Athens in Wartime," by Brady Kiesling. |
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Thursday, May 22, 2003 |
Mental Health Month: "And now what you've all been waiting for... -- Colin Hay as Zac in the film "Cosi" "When I sought those who would sympathize with my plans, I had only you, the friends of my particular art, my most personal work and creation, to turn to." -- Wagner's address at the ceremony for the laying of the foundation stone of the Festival Theater in Bayreuth, May 22 (Wagner's birthday), 1872 "The new computer package DISCRETA which was created in Bayreuth is in the process of permanent development." -- "A Computer Approach to the Enumeration of Block Designs Which Are Invariant With Respect to a Prescribed Permutation Group" The above is a preprint from Dresden. See, too, the work of Bierbrauer, who received his doctorate at Mainz in 1977 and taught at Heidelberg from 1977 to 1994. Bierbrauer's lecture notes give a particularly good background for the concepts involved in my Diamond Theory, in the tradition of Witt and Artin. See Introduction to Group Theory by Jürgen Bierbrauer, 138 pp., PostScript |
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Wednesday, May 21, 2003 |
The 401 Club: Commentary on
"There are dark comedies. There are screwball comedies. But there aren't many dark screwball comedies. And if Nora Ephron's Lucky Numbers is any indication, there's a good reason for that." See also the dark screwball comedy starring Pat Robertson and Michael Eisner, 3:57 pm |
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Tuesday, May 20, 2003 |
Mental Health Month: Here are the evening lottery numbers for Pennsylvania, the Keystone state, drawn on Monday, May 19, 2003: 401 and 1993. This, by the sort of logic beloved of theologians, suggests we find out the significance of the divine date 4/01/1993. It turns out that April 1, 1993, was the date of the New York opening of the Stephen Sondheim retrospective "Putting It Together." For material related to puzzles, games, Sondheim, and Mental Health Month, see Notes on The figures below illustrate some recurrent themes in these notes. “Not games. Puzzles. Big difference. That’s a whole other matter. All art — symphonies, architecture, novels — it’s all puzzles. The fitting together of notes, the fitting together of words have by their very nature a puzzle aspect. It’s the creation of form out of chaos. And I believe in form.” |
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Tuesday, May 20, 2003 |
Raiders of the Lost Matrix "In general, a matrix... is something that provides support or structure, especially in the sense of surrounding and/or shaping. It comes from the Latin word for 'womb,' itself derived from the Latin word for 'mother,' which is mater [as in alma mater]." -- Wikipedia For a mystical interpretation of the above matrix as it relates to the Hebrew words at the center of the official Yale seal, see Talmud.
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Monday, May 19, 2003 |
DAY OF THE MOTHER SHIP I just saw the John Travolta film "Phenomenon" for the first time. (It was on the ABC Family Channel from 8 to 11.) Why is it that tellers of uplifting stories (like Zenna Henderson, in "Day of the Mother Ship, Part I," or the authors of "Phenomenon" or the Bible) always feel they have to throw in some cockamamie and obviously false miracles to hold people's attention? On May 11 (Mother's Day), Mother Nature got my attention with a mighty wind waving the branches of nearby trees, just before a tornado watch was issued for the area I was in. This made me recall a Biblical reference I had come across in researching references to "Our Lady of the Woods" for my Beltane (May 1) entry. ...And his heart was moved, and the heart of his people, as the trees of the wood are moved with the wind. This is what I thought of on May 11 watching branches swaying in the wind on Mother's Day -- which some might regard as a festival of Our Lady of the Woods. John Travolta in "Phenomenon" sees a very similar scene partway through the picture; then, at the end, explains to his girlfriend how the swaying branches made him feel -- without mentioning the branches -- by asking her to describe how she would cradle and rock a child in her arms. At the very end of the film, she herself is reminded of his question by the swaying branches of another tree. Events like these are miracle enough for me.
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Sunday, May 18, 2003 |
Phaedrus Lives! Fans of Zen and the Art of Motorcycle Maintenance may recall that it is a sort of elegy for an earlier self named Phaedrus who vanished with the recovery of mental health. Since this is Mental Health Month, the following observations seem relevant. Reading another weblog's comments today, I found the following remark: "...the mind is an amazing thing and it can create patterns and interconnections among things all day it you let it, regardless of whether they are real connections." This, of course, prompted me to look for patterns and interconnections. The first thing I thought of was the fictional mathematician in "A Beautiful Mind" establishing an amazing -- and, within the fiction, real -- connection between the pattern on a colleague's tie and the reflections from a glass. A web search led to a really real connection.... i.e., to a lengthy listserver letter from an author named Christopher Locke, whose work is new to me but also strangely familiar.... I recognize in his writing both some of my own less-than-mentally-healthy preoccupations and also what might be called the spirit of Phaedrus, from Zen and the Art. Here is a link to a cache I made of the Locke letter and a follow-up he wrote detailing his sources: One part of Locke's letter seems particularly relevant in light of yesterday's entries related to the death of June Carter Cash: "Will the circle be unbroken? Amen. Concluding Unscientific Postscript from Sir Arthur Quiller-Couch ("Q"), quoting Socrates's remarks to the original Phaedrus: ‘By Hera,’ says Socrates, ‘a fair resting-place, full of summer sounds and scents! This clearing, with the agnus castus in high bloom and fragrant, and the stream beneath the tree so gratefully cool to our feet! Judging from the ornaments and statues, I think this spot must be sacred to Acheloüs and the Nymphs. This quotation illustrates a connection between Jesus (College) -- from my entry of 3:33 PM Thursday -- and a Nymph -- from my entry of 11:44 PM Friday. See, too, Q's quoting of Socrates's prayer to Pan, as well as the cover of the May 19, 2003, New Yorker: For a discussion of the music
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Friday, May 16, 2003 |
Highballs "If you can bounce high, Magazine purchased at A Whiff of Camelot – New York Times, Song title from the "Gatsby's Restaurant" From The Great Gatsby, Chapter Four: "Highballs?" asked the head waiter. Mimi Beardsley, JFK playmate, On JFK's plane trips: Apparently there was some function.... "Don't forget the coffee!" |
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Friday, May 16, 2003 |
Enough Commentary on the May 15 death of In light of yesterday's Jesus College entry While walking out one evening ◊ an "old shape-note gospel song that A.P. Carter found and rearranged. The song itself had been written and copyrighted back in 1897, with composer credits to C.W. Bryan." |
STAR WARS 
The Triangle (TRIKONA)
