Annals of Fiction:
Sunday, April 12, 2009 3:09 AM
"The aim of Conway’s game M13 is to get the hole at the top point and all counters in order 1,2,…,12 when moving clockwise along the circle." --Lieven Le Bruyn
The illustration is from the weblog entry by Lieven Le Bruyn quoted below. The colored circles represent 12 of the 13 projective points described below, the 13 radial strokes represent the 13 projective lines, and the straight lines in the picture, including those that form the circle, describe which projective points are incident with which projective lines. The dot at top represents the "hole."
From "The Mathieu Group M12 and Conway’s M13-Game" (pdf), senior honors thesis in mathematics by Jeremy L. Martin under the supervision of Professor Noam D. Elkies, Harvard University, April 1, 1996--
"Let P3 denote the projective plane of order 3. The standard construction of P3 is to remove the zero point from a three-dimensional vector space over the field F3 and then identify each point x with -x, obtaining a space with
Conway  proposed the following game.... Place twelve numbered counters on the points... of P3 and leave the thirteenth point... blank. (The empty point will be referred to throughout as the "hole.") Let the location of the hole be p; then a primitive move of the game consists of selecting one of the lines containing the hole, say
There is an obvious characterization of a move as a permutation in S13, operating on the points of P3. By limiting our consideration to only those moves which return the hole to its starting point.... we obtain the Conway game group. This group, which we shall denote by GC, is a subgroup of the symmetric group S12 of permutations of the twelve points..., and the group operation of GC is concatenation of paths. Conway  stated, but did not prove explicitly, that GC is isomorphic to the Mathieu group M12. We shall subsequently verify this isomorphism.
The set of all moves (including those not fixing the hole) is given the name M13 by Conway. It is important that M13 is not a group...."
 John H. Conway, "Graphs and Groups and M13," Notes from New York Graph Theory Day XIV (1987), pp. 18–29.
Another exposition (adapted to Martin's notation) by Lieven le Bruyn (see illustration above):
"Conway’s puzzle M13 involves the 13 points and 13 lines of P3. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the 'hole' (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture [above] the lines are represented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway's M13 puzzle having the hole at the top position contains the sporadic simple Mathieu group M12 as a subgroup."
Sunday, April 12, 2009 2:02 AM
On Holy Saturday:
Saturday, April 11, 2009
--Elias Canetti, The Human Province
Posted by David Lavery at 1:00 AM
Friday, April 10, 2009 8:00 AM
Good Friday Meditation:
Consider the following question in a paper cited by V. S. Varadarajan:
"There is a pleasantly discursive
treatment of Pontius Pilate's
'What is truth?'."
-- H. S. M. Coxeter, 1987,
introduction to Trudeau's
remarks on the "Story Theory"
of truth as opposed to the
"Diamond Theory" of truth in
The Non-Euclidean Revolution
Thursday, April 9, 2009 7:11 PM
Thursday, April 9, 2009 12:12 AM
Mathematics and Narrative:
Wednesday, April 8, 2009 8:00 PM
Annals of Religion:
"Plato's Ghost evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost...."
-- Princeton University Press on Plato's Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"She's a brick house..."
-- Plato's Ghost according to
Log24, April 2007
"First of all, I'd like
to thank the Academy."
-- Remark attributed to Plato
Wednesday, April 8, 2009 12:12 AM
Snow White Variations:Good's Singularity
"He was born as Isidore Jacob Gudak to a Jewish family in London. In his publications he was called I. J. Good. He studied mathematics at Jesus College, Cambridge, graduating in 1938. He did research work under G.H. Hardy and Besicovitch before moving to Bletchley Park in 1941 on completing his doctorate.
Wikipedia states that "I. J. Good's vanity car license plate,
hinting at his spylike wartime work, was
|"Some say the symbol
of Apple Computers,
the apple with a bite out of it,
is a nod to Alan Turing."
-- from "Alan Turing and
the Apple" at Flickr, uploaded
on Epiphany 2006 by guano
Tuesday, April 7, 2009 5:24 PM
ART WARS continued:
Bright Star and Dark Lady
"Mexico is a solar country -- but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child."
-- Octavio Paz,
The same story on
May 11, 2005
with a different
Sunday, April 5, 2009 7:35 AM
Sunday, April 5, 2009 5:01 AM
Annals of Fantasy--
THE SNAKE. I can talk of many things. I am very wise. It was I who whispered the word to you that you did not know. Dead. Death. Die.
EVE [shuddering] Why do you remind me of it? I forgot it when I saw your beautiful hood. You must not remind me of unhappy things.
THE SERPENT. Death is not an unhappy thing when you have learnt how to conquer it.
EVE. How can I conquer it?
THE SERPENT. By another thing, called birth.
EVE. What? [Trying to pronounce it] B-birth?
THE SERPENT. Yes, birth.
EVE. What is birth?
THE SERPENT. The serpent never dies. Some day you shall see me come out of this beautiful skin, a new snake with a new and lovelier skin. That is birth.
EVE. I have seen that. It is wonderful.
THE SERPENT. If I can do that, what can I not do? I tell you I am very subtle. When you and Adam talk, I hear you say 'Why?' Always 'Why?' You see things; and you say 'Why?' But I dream things that never were; and I say 'Why not?' I made the word dead to describe my old skin that I cast when I am renewed. I call that renewal being born.
EVE. Born is a beautiful word.
THE SERPENT. Why not be born again and again as I am, new and beautiful every time?
EVE. I! It does not happen: that is why.
THE SERPENT. That is how; but it is not why. Why not?
EVE. But I should not like it. It would be nice to be new again; but my old skin would lie on the ground looking just like me; and Adam would see it shrivel up and--
THE SERPENT. No. He need not. There is a second birth.
EVE. A second birth?THE SERPENT. Listen. I will tell you a great secret....
Saturday, April 4, 2009 9:48 PM
Live from New York, it's...
Saturday, April 4, 2009 7:01 PM
ART WARS continued:
Saturday, April 4, 2009 8:00 AM
Mathematics and Narrative, continued:
|Adapted (for HTML) from the opening
paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups
M22, M23, M24..."--
"[A]... uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge  (see also Conwell ) gives of certain isomorphisms between classical groups, namely
where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example , Kapitel II).
The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau...."
4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910).
5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954).
7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.
Friday, April 3, 2009 5:24 AM
Mr. Holland's Week continues:
Thursday, April 2, 2009 8:00 PM
Annals of Philosophy:
Thursday, April 2, 2009 12:25 PM
Time and Chance, continued:
Wednesday, April 1, 2009 7:59 PM
Annals of Cinema:
Wednesday, April 1, 2009 5:01 PM
Annals of Literature: