Mental Health Month, Day 28:
The Eightfold Way and
Solomon's Seal
For a continuation of the mathematical and religious themes in yesterday's entry, click on the figure below.
Month: May 2003
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Mental Health Month, Day 27:
Conspiracy Theory and
Solomon's Seal
In our journey through Mental Health Month, we have now arrived at day 27. This number, the number of lines on a non-singular cubic surface in complex projective 3-space, suggests it may be time to recall the following note (a sort of syllabus for an imaginary course) from August 1997, the month that the Mel Gibson film "Conspiracy Theory" was released.
Conspiracy Theory 101
August 13, 1997
Fiction:
(A)
Masks of the Illuminati, by Robert Anton Wilson, Pocket Books, New York, 1981. Freemasonry meets The Force (starring James Joyce and Albert Einstein).
(B)
The Number of the Beast, by Robert A. Heinlein, Ballantine Books, New York, 1980. "Pantheistic multiple solipsism" and transformation groups in n-dimensional space combine to yield "the ultimate total philosophy." (p. 438).
(C)
The Essential Blake, edited by Stanley Kunitz, MJF Books, New York, 1987. "Fearful symmetry" in context.
Fact:
(1)
The Cosmic Trigger, by Robert Anton Wilson, Falcon Press, Phoenix, 1986 (first published 1977). Page 245 reveals that "the most comprehensive conspiracy theory," that of the physicist Sir Arthur Eddington, is remarkably similar to Heinlein's theory in (B) above.
(2)
The Development of Mathematics, by E. T. Bell, 2nd. ed., McGraw-Hill, New York, 1945. See the discussion of "Solomon's seal," a geometric configuration in complex projective 3-space. This is as good a candidate as any for Wilson's "Holy Guardian Angel" in (A) above.
(3)
Finite Projective Spaces of Three Dimensions, by J. W. P. Hirschfeld, Clarendon Press, Oxford, 1985. Chapter 20 shows how to represent Solomon's seal in the 63-point 5-dimensional projective space over the 2-element field. (The corresponding 6-dimensional affine space, with 64 points, is reminiscent of Heinlein's 6-dimensional space.)
See also China's 3,000-year-old "Book of Transformations," the I Ching, for more philosophy and lore of the affine 6-dimensional space over the binary field.
© 1997 S. H. Cullinane
For a more up-to-date and detailed look at the mathematics mentioned above, see
Abstract Configurations
in Algebraic Geometry,
by Igor Dolgachev.
"Art isn't easy." -- Stephen Sondheim
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Mental Health Month, Day 26:
Many Dimensions,
Part III — Why 26?
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 '
M13 .' This structure is a set of 1,235,520 permutations of 13 letters. It is not a group. However, this structure represents the answer to the following group theoretic question:
Why do the simple groupsM12 andL3(3) share some subgroup structure?
In fact, both the Mathieu group
M12 and the automorphism groupL3(3) of the projective planePG(2,3) overGF(3) can be found as subsets ofM13. In addition,M13 is 6-fold transitive, in the sense that it contains enough permutations to map any two 6-tuples made from the thirteen letters into each other. In this sense,M13 could pass as a parent for bothM12 andL3(3). As it is known from the classification of primitive groups that there is no finite group which qualifies as a parent in this sense. Yet,M13 comes close to being a group.
To understand the definition of
M13 let us have a look at the projective geometryPG(2,3)....
The points and the lines and the "is-contained-in" relation form an incidence structure over
PG(2,3)....
...the 26 objects of the incidence structure [are] 13 points and 13 lines."
Conway's construction involves the arrangement, in a circular Levi graph, of 26 marks representing these points and lines, and chords representing the "contains/is contained in" relation. The resulting diagram has a pleasingly symmetric appearance.
For further information on the geometry of the number 26, one can look up all primitive permutation groups of degree 26. Conway's work suggests we look at sets (not just groups) of permutations on n elements. He has shown that this is a fruitful approach for n=13. Whether it may also be fruitful for n=26, I do not know.
There is no obvious connection to physics, although the physics writer John Baez quoted in my previous two entries shares Conway's interest in the Mathieu groups.
* J. H. Conway, "M13," in Surveys in Combinatorics, 1997, edited by R. A. Bailey, London Mathematical Society Lecture Note Series, 241, Cambridge University Press, Cambridge, 1997. 338 pp. ISBN 0 521 59840 0.
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Mental Health Month, Day 26:
Many Dimensions,
Part II — The Blue Matrix
But seriously...
John Baez in July 1999:
"...it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."
Well, maybe. I certainly hope so. If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light. Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory. See the following from my journal:
Evariste Galois and the Rock that Changed Things, and
A Time to Gather Stones Together: Readings for Yom Kippur.
See, too, this web page on Marion Zimmer Bradley's fictional
Matrices, or Blue Star-Stones, and
the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.
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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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STAR WARS
opened on this date in 1977.
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
New York Times, May 25, 2003:
Ariel the Hutt and Princess Amygdala
by Horia Cristescu and
Dan Bozaru
The Triangle (TRIKONA)
The triangle (TRIKONA) is the symbol of SHAKTI , the feminine energy or aspect of Creation. The triangle pointing down represents the YONI , the feminine sexual organ and the symbol of the supreme source of the Universe, and when the triangle is pointing upwards it signifies intense spiritual aspiration, the sublimation of one's nature into the most subtle planes and the element of fire (AGNI TATTVA). The fire is always oriented upwards, thus the correlation with the upward triangle - SHIVA KONA. On the other hand, the downward pointing triangle signifies the element of water which always tends to flown and occupy the lowest possible position. This triangle is known as SHAKTI KONA.
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.
The Six Points Star (SHATKONA)
A typical combination often found in the graphical structure of a YANTRA is the superposition of two triangles, one pointing upwards and the other downwards, forming a star with six points (SHATKONA), also known as David's Star. This form symbolically represents the union of PURUSHA and PRAKRITI or SHIVA-SHAKTI, without which there could be no Creation.
AMEN.
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— 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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Mental Health Month, Day 24:
The Sacred Day of
Kali, the Dark Lady
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
The Dark Lady,
by Mike Resnick.
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.
Sarah Connor
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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Mental Health Month, Day 23:
The Prime Cut Gospel
On Christmas Day, 1949,
Mary Elizabeth Spacek was born in Texas.
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:
- R. D. Carmichael's 1937 discussion of the linear fractional group modulo 23 in
Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, "Tactical Configurations," and
- Conway's 1969 discussion of the same group in
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:
1949
+ 23
1972
Essay question:
If Jesus was God's gift to man, and (as many men would agree) so was the young Sissy Spacek (also born on Christmas Day), was young Sissy's existence in her 23rd year contingent or necessary? If the latter, should she be recognized as a Person of the Trinity? Quaternity? N-ity?
Talk amongst yourselves. -
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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