Thursday, January 18, 2024
Wednesday, February 2, 2022
Conway’s Game vs. Pure Geometry
A question attributed to John Horton Conway
about configurations in his Game of Life —
"Indeed, is there a Godlike still-life,
one that can only have existed
for all time . . . . ?"
A simple answer … but not from Conway's Game —
"Before time began, there was the Cube." — Optimus Prime
Related remarks: Ogdoad.
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Thursday, August 31, 2017
A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
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Thursday, October 10, 2024
Automata Studies: Sevenfold Studio Work
Motifs for Conway:
Later . . .
The above reposting was suggested in part by
the word "sevenfold" in Milton —
From the above nineteenth-century text, a verse by Spenser, adapted —
"Bodied, heard, souled, seen."
— might well be applied to a noted brother and sister, as in Petronius:
"… dum frater sororis suae automata per clostellum miratur …."
Detail from the Instagram of Emma Watson —
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Thursday, August 15, 2024
Geometry Exercise
As G. M. Conwell pointed out in a 1910 paper, the group of all
40,320 permutations of an 8-element set is the same, in an
abstract sense, as the group of all collineations and dualities
of PG(3,2), the projective 3-space over the 2-element field.
This suggests we study the geometry related to the above group's
actions on the 105 partitions of an 8-set into four separate 2-sets.
Note that 105 equals 15×7 and also 35×3.
In such a study, the 15 points of PG(3,2) might correspond (somehow)
to 15 pairwise-disjoint seven-element subsets of the set of 105 partitions,
and the 35 lines of PG(3,2) might correspond (somehow) to 35 pairwise-
disjoint three-element subsets of the set of 105 partitions.
Exercise: Is this a mere pipe dream?
A search for such a study yields some useful background . . .
.
Taylor's Index of Names includes neither Conwell nor the
more recent, highly relevant, names Curtis and Conway .
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Wednesday, May 29, 2024
The Strong Law of Small Shapes*
Two examples:
The above note led to a letter from John H. Conway, which in turn
led to the following . . .
* The title refers to a well-known 1988 article by Richard K. Guy.
A shape from the date of Guy's reported death —
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Monday, November 6, 2023
Letter from Birmingham Grid
"Just as Socrates felt that it was necessary to create a tension in the mind
so that individuals could rise from the bondage of myths and half-truths
to the unfettered realm of creative analysis and objective appraisal . . . ."
See also today's previous post, from "Terminator Zero: Rise of the Chatbots."
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Saturday, October 21, 2023
Monday, October 16, 2023
A Harlan Kane Rite Aid Special: Chapter 11
From a search in this journal for "Chapter 11" —
Inner structure —
The above three images share the same
vector-space structure —
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
|
a + b |
a + b + c |
a + b + d |
a + b + |
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
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Monday, March 13, 2023
Frontiers of Artificial Mathematics
Previous posts have shown ChatGPT answering the question
"What is the diamond theorem?" with references to Thurston
and, later, to Conway. Today it is Penrose's turn.
Related search results (click to enlarge) —
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Sunday, March 5, 2023
Annals of Artificial Mathematics
The response of ChatGPT to a question about my work
continues to evolve. It now credits Conway, not Thurston,*
for the diamond theorem.
The paragraph beginning "The theorem states" appears** to be based
on the following 24 patterns — which number only 8, if rotated or
reflected patterns are considered equivalent.
* For Thurston in an earlier ChatGPT response to the same question,
see a Log24 post of Feb. 25.
** The illustration above is based on the divison of a square into
four smaller subsquares. If the square is rotated by 45 degrees,
it becomes a diamond that can be, in the language of ChatGPT,
divided into "four smaller diamonds ."
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Saturday, November 5, 2022
Under the Bridge, Over the Dam … Whatever.
Bridge
Dam
April 11, 2020, was the dies natalis ,
in the Catholic sense,
of John Horton Conway.
Whatever.
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Thursday, February 3, 2022
Through the Asian Looking Glass
The Miracle Octad Generator (MOG) —
The Conway-Sloane version of 1988:
See also the 1976 R. T. Curtis version, of which the Conway-Sloane version
is a mirror reflection —
“There is a correspondence between the two systems
of 35 groups, which is illustrated in Fig. 4 (the MOG or
Miracle Octad Generator).”
—R.T. Curtis, “A New Combinatorial Approach to M24,”
Mathematical Proceedings of the Cambridge Philosophical
Society (1976), 79: 25-42
Curtis’s 1976 Fig. 4. (The MOG.)
The Guy Embedding (named for M.J.T., not Richard K., Guy) states that
the MOG is naturally embedded in the codewords of the extended binary
Golay code, if those codewords are generated in lexicographic order.
The above reading order for the MOG 4×6 array —
down the columns, from left to right — yields the Conway-Sloane MOG.
Since that is a mirror image of the original Curtis MOG, the reading order
yielding that MOG is down the columns, from right to left.
"Traditionally, Chinese, Japanese, Vietnamese and Korean are written vertically
in columns going from top to bottom and ordered from right to left, with each
new column starting to the left of the preceding one." — Wikipedia
The Asian reading order has certain artistic advantages:
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Saturday, January 29, 2022
On the Diamond-Theorem Group* of Order 322,560
Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.
This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.
* See a Log24 post of June 1, 2013.
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Saturday, June 26, 2021
Annals of Experimental Theology . . .
See as well . . .
April 11, 2020, was the dies natalis ,
in the Catholic sense,
of John Horton Conway.
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Thursday, May 13, 2021
Annals of Experimental Theology
The Axiomatic Method:
"We hold these truths to be self-evident…."
Other methods:
"In Gauss we trust." (See below.)
But perhaps not so much in Princeton . . .
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Wednesday, April 28, 2021
Blocks in a Box
In Scientific American today —
For a more sophisticated approach to the phrase
“blocks in a box,” search for “the 759 blocks” and
then see box759.wordpress.com.
The mathematics there is based on an apparently
less sophisticated example of “blocks in a box” —
See also Cube Space in this journal.
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Monday, December 28, 2020
Childermas
Related material for innocents — Siobhan Roberts
on Conway’s Game of Life in today’s New York Times .
Those desiring greater literary depth may consult
this journal’s Gameplayers.
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Thursday, August 27, 2020
The Complete Extended Binary Golay Code
All 4096 vectors in the code are at . . .
http://neilsloane.com/oadir/oa.4096.12.2.7.txt.
Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —
As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.
(Of course, any permutation of the 24 columns above produces
a version of the code qua code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)
See my post of July 13, 2020 —
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
For some related results, Google the twelfth generator:
* Sloane’s list is of the codewords as the rows of an orthogonal array —
See also http://neilsloane.com/oadir/.
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Saturday, August 8, 2020
A Natural Diagram
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Monday, July 13, 2020
The Lexicographic Octad Generator (LOG)*
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
By Steven H. Cullinane, July 13, 2020
Background —
The Miracle Octad Generator (MOG)
of R. T. Curtis (Conway-Sloane version) —
A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in
"Constructing the Extended Binary Golay Code"
Ben Adlam
Harvard University
August 9, 2011:
000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100
Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator vectors
(as in the Conway-Sloane SPLAG version above, in
which Curtis's earlier heavy bricks are reflected in
their vertical axes) —
01 02 03 04 05 . . . 20 21 22 23 24 --> 01 05 09 13 17 21 02 06 10 14 18 22 03 07 11 15 19 23 04 08 12 16 20 24 0000 0000 0000 0000 1111 1111 --> 0000 11 0000 11 0000 11 0000 11 as in the MOG. 0000 0000 0000 1111 0000 1111 --> 0001 01 0001 01 0001 01 0001 01 as in the MOG. 0000 0000 0011 0011 0011 0011 --> 0000 00 0000 00 0011 11 0011 11 as in the MOG. 0000 0000 0101 0101 0101 0101 --> 0000 00 0011 11 0000 00 0011 11 as in the MOG. 0000 0000 1001 0110 0110 1001 --> 0010 01 0001 10 0001 10 0010 01 as in the MOG. 0000 0011 0000 0011 0101 0110 --> 0000 00 0000 11 0101 01 0101 10 as in the MOG. 0000 0101 0000 0101 0110 0011 --> 0000 00 0101 10 0000 11 0101 01 as in the MOG. 0000 1001 0000 0110 0011 1010 --> 0100 01 0001 00 0001 11 0100 10 as in the MOG. 0001 0001 0001 0001 0111 1000 --> 0000 01 0000 10 0000 10 1111 10 as in the MOG. 0010 0001 0001 0010 0001 1101 --> 0000 01 0000 01 1001 00 0110 11 as in the MOG. 0100 0001 0001 0100 0100 1110 --> 0000 01 1001 11 0000 01 0110 00 as in the MOG. 1000 0001 0001 0111 0010 0100 --> 10 00 00 00 01 01 00 01 10 01 11 00 as in the MOG (heavy brick at center).
Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:
And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code, found (as I just did)
that weight-8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG. (Lexicographic codes have apparently been
known since 1960, the MOG since the early 1970s.)
* Addendum at 4 AM ET the next day —
See also Logline (Walpurgisnacht 2013).
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Saturday, May 2, 2020
Turyn’s Octad Theorem: The Next Level*
From the obituary of a game inventor who reportedly died
on Monday, February 25, 2013 —
” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”
* For the theorem, see Wolfram Neutsch, Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)
Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .
“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….
To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”
— From page 761
“The theorem of Turyn” is on page 764 —
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Thursday, April 23, 2020
Dead Poet Joke
From the subtitles of the recent Kristen Stewart film “Underwater” —
427
00:30:26,144 –> 00:30:27,476
He’d always say
he had a new joke,
428
00:30:27,478 –> 00:30:29,445
and then he’d tell
the same stupid joke.
429
00:30:29,447 –> 00:30:32,785
I was… laughing at that joke.
430
00:30:34,053 –> 00:30:35,685
Yeah, what was it?
431
00:30:35,687 –> 00:30:38,654
What did the fish say when
it bumped into the brick wall?
April 11 was the dies natalis , in the Catholic sense, of John Horton Conway.
Related material: Other posts containing the phrase “brick wall.”
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Sunday, April 12, 2020
Blackboard Jungle Continues.
From a post this morning by Peter J. Cameron
in memory of John Horton Conway —
| ” This happened at a conference somewhere in North America. I was chairing the session at which he was to speak. When I got up to introduce him, his title had not yet been announced, and the stage had a blackboard on an easel. I said something like ‘The next speaker is John Conway, and no doubt he is going to tell us what he will talk about.’ John came onto the stage, went over to the easel, picked up the blackboard, and turned it over. On the other side were revealed five titles of talks. He said, ‘I am going to give one of these talks. I will count down to zero; you are to shout as loudly as you can the number of the talk you want to hear, and the chairman will judge which number is most popular.’ “ |
From Log24 on August 21, 2014 —
Thursday, August 21, 2014
NoxFiled under: Uncategorized — m759 @ 1:00 AM ( A sequel to Lux ) “By groping toward the light we are made to realize — Arthur Koestler, The Call Girls: A Tragi-Comedy ,
Robin Williams and the Stages of Math i) shock & denial A related description of the process — “You know how sometimes someone tells you a theorem, — Tom Leinster yesterday at The n-Category Café |
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Sunday, March 15, 2020
The “Octad Group”
The phrase “octad group” discussed here in a post
of March 7 is now a domain name, “octad.group,”
that leads to that post. Remarks by Conway and
Sloane now quoted there indicate how the group
that I defined in 1979 is embedded in the large
Mathieu group M24.
Related literary notes — Watson + Embedding.
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Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From "Mathieu Moonshine and Symmetry Surfing" —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
"This presentation of the symmetry groups Gi is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group
G = (Z2)4 ⋊ A8 .
It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O9 = {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."
This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the "octad group" in 1993 —
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Saturday, June 29, 2019
That’s “Merry” … And Quite Contrary
"John Horton Conway is a cross between
Archimedes, Mick Jagger and Salvador Dalí."
— The Guardian paraphrasing Siobhan Roberts,
John Horton Conway and his Leech lattice doodle
in The Guardian . Photo: Hollandse Hoogte/Eyevine.
. . . .
"In junior school, one of Conway’s teachers had nicknamed him 'Mary'.
He was a delicate, effeminate creature. Being Mary made his life
absolute hell until he moved on to secondary school, at Liverpool’s
Holt High School for Boys. Soon after term began, the headmaster
called each boy into his office and asked what he planned to do with
his life. John said he wanted to read mathematics at Cambridge.
Instead of 'Mary' he became known as 'The Prof'. These nicknames
confirmed Conway as a terribly introverted adolescent, painfully aware
of his own suffering." — Siobhan Roberts, loc. cit.
From the previous post —
See as well this journal on the above Guardian date —
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Thursday, April 4, 2019
For Harlan Kane . . .
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Tuesday, December 25, 2018
Simply
|
"So to obtain the isomorphism from L2(7) onto L3(2) we simply
— Sphere Packings, Lattices and Groups , |
Compare and contrast —
.gif)
This post was suggested by a New York Times headline today —
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Friday, December 14, 2018
Small Space Odyssey
References in recent posts to physical space and
to mathematical space suggest a comparison.
Physical space is well known, at least in the world
of mass entertainment.
Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.
A figure from each space —
The source of the Conway-Sloane brick —
Quote from a mathematics writer —
“Looking carefully at Golay’s code is like staring into the sun.”
The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.
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Sunday, June 10, 2018
Pieces of April
This journal on April 16, 2018 —
Happy birthday to Pope Emeritus Benedict XVI.
Related material from another weblog in a post also dated April 16, 2018 —
"As I write this, it’s April 5, midway through the eight-day
festival of Passover. During this holiday, we Jews air our
grievances against the ancient Pharaoh who enslaved
and oppressed us, and celebrate the feats of strength
with which the Almighty delivered us from bondage —
wait a minute, I think I’m mixing up Passover with Festivus."
. . . .
"Next month: Time and Tesseracts."
From that next post, dated May 16, 2018 —
"The tesseract entered popular culture through
Madeleine L’Engle’s 'A Wrinkle in Time' . . . ."
The post's author, James Propp, notes that
" L’Engle caused some of her readers confusion
when one of the characters … the prodigy
Charles Wallace Murray [sic ] , declared 'Well, the fifth
dimension’s a tesseract.' "
Propp is not unfamiliar with prodigies:
"When I was a kid living in the Long Island suburbs,
I sometimes got called a math genius. I didn’t think
the label was apt, but I didn’t mind it; being put in
the genius box came with some pretty good perks."
— "The Genius Box," a post dated March 16, 2018
To me, Propp seems less like Charles Wallace
and more like the Prime Coordinator —
For further details, see the following synchronicity checks:
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Tuesday, April 17, 2018
A Necessary Possibility*
"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."
— Rodolphe Gasché, The Tain of the Mirror ,
Harvard University Press, 1986
An inscription from 2010 —
An inscription from 1984 —
|
American Mathematical Monthly, June-July 1984, p. 382 MISCELLANEA, 129 Triangles are square
"Every triangle consists of n congruent copies of itself" |
* See also other Log24 posts mentioning this phrase.
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Monday, April 16, 2018
The Brooklyn Game
"Can you bring me some players?"
— Molly Bloom in "Molly's Game"
Happy birthday to Pope Emeritus Benedict XVI.
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Sunday, April 15, 2018
Colorado Olympiad
Or: Personalities Before Principles
Personalities —
Principles —
This journal on April 28, 2004 at 7:00 AM.
Backstory —
Square Triangles in this journal.
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Friday, November 24, 2017
The Matrix Meets the Grid
The Matrix —
The Grid —
Picturing the Witt Construction —
"Read something that means something." — New Yorker ad
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
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Thursday, September 14, 2017
Found …
( A sequel to the previous post, Lost )
From a link, "A Little Boy and a Little Girl," found in a Log24
search for Andersen + Atlantic —
"A few flakes of snow were falling, and one of them, rather larger
than the rest, alighted on the edge of one of the flower boxes.
This snow-flake grew larger and larger, till at last it became
the figure of a woman, dressed in garments of white gauze,
which looked like millions of starry snow-flakes linked together.
She was fair and beautiful, but made of ice—
shining and glittering ice." — "The Snow Queen"
Related material —
Analogue of the little boy from "The Snow Queen" in "Equals" (2015) —
"Nice piece of ice." — Brendan Fraser in
"The Mummy: Tomb of the Dragon Emperor" (2008).
See also the concept that everything adds up to nothing in
"The Zero Theorem" (2013) …
… and the Conway-Norton-Ryba theorem (2017).
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Tuesday, September 5, 2017
Florence 2001
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
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Thursday, December 22, 2016
Wednesday, November 9, 2016
Great Again
Song suggested by Kellyanne Conway's remarks
in a CNN story today —
"We always felt that Hillary Clinton promising to
put coal miners out of work, or steel workers,
that wasn't going to go well in a place like
Pennsylvania. Michigan, Wisconsin, the same thing,"
she said. "So it just all started to come together."
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Wednesday, May 25, 2016
Framework
"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
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Monday, May 2, 2016
Subjective Quality
The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.
Siobhan Roberts recently wrote a book about the first of these
authors, Conway. I just discovered that last fall she also had an
article about the second author, Sloane, published:
"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.
Meanwhile, in this journal …
Log24 on that same date, Oct. 22, 2015 —
Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective quality, as opposed to the objective quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here last October.
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Sunday, May 1, 2016
Sunday Appetizer from 1984
Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —
“What would an organic Christian Sabbath look like today?”
The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —
|
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich |
That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."
The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the
1988 first edition of Sphere Packings, Lattices and Groups ,
by J. H. Conway and N. J. A. Sloane —
A passage from the 1984 document —
Comments Off on Sunday Appetizer from 1984
Wednesday, November 4, 2015
Group Theory at Princeton
This is from the program of
Finite Simple Groups: Thirty Years of the Atlas and Beyond —
Celebrating the Atlases and Honoring John Conway
November 2-5, 2015 at Princeton University
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Saturday, August 1, 2015
In Memoriam
Death of an academic on
the feast of St. Ignatius Loyola, 2015 —
"Cheuse was involved in a serious car accident
on July 14, 2015 on California State Route 17
while driving from Olympic Valley to Santa Cruz,
California. He was reported to be in a coma on
July 20, 2015 with injuries including fractured ribs,
cervical vertebrae, and an acute subdural hematoma.
He died on July 31, 2015 from his injuries at the age
of 75." — Wikipedia
Also on July 14 …
See as well Cheuse on Santa Cruz …
Home Away From Home in Santa Cruz
There are towns you are born into,
and there are towns you grow into.
Related artistic image —
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Tuesday, July 14, 2015
Perspective
Today's Huffington Post has a review of the
new book on John Horton Conway. The reviewer
is Colm Mulcahy. For some perspective, see
a search for Mulcahy in this journal.
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Wednesday, November 26, 2014
A Tetrahedral Fano-Plane Model
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by
Burkard Polster (Springer, 1998).
"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"
—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya
For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."
The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay"
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.
Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.
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Monday, October 6, 2014
Reviews
From the MacTutor biography of Otto Neugebauer:
“… two projects which would be among the most important
contributions anyone has made to mathematics. He persuaded
Springer-Verlag to publish a journal reviewing all mathematical
publications, which would complement their reviewing journals
in other topics. In 1931 the first issue of Zentralblatt für Matematik
appeared, edited by Neugebauer.” [Mathematical Reviews was
the other project.]
Neugebauer appeared in Sunday morning’s post In Nomine Patris .
A review from Zentralblatt appeared in the Story Creep link from
this morning’s post Mysterious Correspondences.
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Tuesday, June 17, 2014
Finite Relativity
Anyone tackling the Raumproblem described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:
The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper. Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—
This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:
An explanation of the apparent falsity in Curtis's 1989 paper:
By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads that resulted later from the Conway coordinates,
as in the images below.
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Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the Turyn-Curtis construction
from the University of Cambridge —
— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.
A third construction of Curtis's 35 4×6 1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22-March 23 —
Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.
* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
Comments Off on Three Constructions of the Miracle Octad Generator
Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
Comments Off on The 4×4 Relativity Problem
Sunday, December 15, 2013
Sermon
Odin's Jewel
Jim Holt, the author of remarks in yesterday's
Saturday evening post—
"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."
— Vanity Fair Daily , July 16, 2012
Backstory: Odin + Jewel in this journal.
See also Odin on the Kyoto school —
For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —
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Saturday, December 14, 2013
Beautiful Mathematics
The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.
Some material relevant to the title adjective:
| "For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books |
Some relevant links—
- Strangeness and inevitability
- Simply defined abstractions
- Hidden quirks and complexities
- Seemingly unrelated structures
- Mysterious correspondences
- Uncanny patterns
- The rigor of logic
- Beethoven quartet
The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links. See also a post of
Jan. 31, 2014.
Update of March 9, 2014 —
The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).
Comments Off on Beautiful Mathematics
Sunday, May 19, 2013
Sermon
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the Conway-Sloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
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Tuesday, April 2, 2013
Rota in a Nutshell
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
Comments Off on Rota in a Nutshell
Saturday, January 5, 2013
Vector Addition in a Finite Field
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
Comments Off on Vector Addition in a Finite Field
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
| a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Comments Off on The Galois Tesseract
Monday, July 16, 2012
Mapping Problem continued
Another approach to the square-to-triangle
mapping problem (see also previous post)—
For the square model referred to in the above picture, see (for instance)
- Picturing the Smallest Projective 3-Space,
- The Relativity Problem in Finite Geometry, and
- Symmetry of Walsh Functions.
Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.
This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.
Update of 9:35 AM ET July 16, 2012:
Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —
— share the following vector-space structure —
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
| a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
Comments Off on Mapping Problem continued
Sunday, May 6, 2012
Triality continued
This post continues the April 9 post
commemorating Élie Cartan's birthday.
That post mentioned triality .
Here is John Baez reviewing
On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry
by John H. Conway and Derek A. Smith
(A.K. Peters, Ltd., 2003)—
"In this context, triality manifests itself
as the symmetry that cyclically permutes
the Hurwitz integers i , j , and k ."
Related material— Quaternion Acts in this journal
as well as Finite Geometry and Physical Space.
Comments Off on Triality continued
Monday, January 23, 2012
How It Works
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.
Comments Off on How It Works
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.
Comments Off on Triangles Are Square
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
Wednesday, August 31, 2011
Odin’s Day
Today is Wednesday.
O.E. Wodnesdæg "Woden's day," a Gmc. loan-translation of L. dies Mercurii "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin . — Online Etymology Dictionary
Above: Anthony Hopkins as Odin in the 2011 film "Thor"
Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—
"The Tesseract* was the jewel of Odin's treasure room."
Weaving also played Agent Smith in The Matrix Trilogy.
The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.
"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."
— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com
For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—
Today's New York Times has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—
"He treated it like a puzzle."
Sometimes that's the best you can do.
* See also tesseract in this journal.
** For a different arrangement of 13 things, see the cube's 13 axes in this journal.
Comments Off on Odin’s Day
Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
Comments Off on The Schwartz Notes
Saturday, August 7, 2010
Camp Inception
With a nod to Larry Doyle's "Sleeper Camp"—
From the Mathcamp Reunion Schedule for Saturday, July 24, 2010—
2:30-3:30 PM — John Conway Colloquium
3:30-5:30 PM — Relays: Heaven, Hell, and Purgatory!
In this journal on Saturday, July 24—
Playing with Blocks (noon) and The Leonardo Code (1 PM).
A happy Mathcamper defines Heaven, Hell, and Purgatory—
…Heaven and Hell relays. your team starts in hell, when you get one right, one person can go to heaven and work on heaven questions, but first they have to pass through purgatory. aka this means entertain the people running purgatory. for me this meant dancing in the middle of the gym. i danced and sung the YMCA, which they deemed sufficient (thankfully).
— Imaginary Thoughts and Irrational Ideas weblog
Note in the Mathcamp schedule the Friday night Shabbat dinner and the religious activity on Sunday— a "mini-puzzle hunt."
Comments Off on Camp Inception
Friday, May 14, 2010
Competing MOG Definitions
A recently created Wikipedia article says that “The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space….” (Clearly any array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a well-known one that preserves a certain incidence property. See Eightfold Geometry.)
From the 1976 paper defining the MOG—
“There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator).” —R.T. Curtis, “A New Combinatorial Approach to M24,” Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42
Curtis’s 1976 Fig. 4. (The MOG.)
The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—
I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35-sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about “Curtis’s original way of finding octads in the MOG [Cur2]” indicate that the correspondence definition was the one Curtis used in 1973—
Here the picture of “the 35 standard sextets of the MOG”
is very like (modulo a reflection) Curtis’s 1976 picture
of the MOG as a correspondence between two 35-sets.
A later paper by Curtis does use the array definition. See “Further Elementary Techniques Using the Miracle Octad Generator,” Proceedings of the Edinburgh Mathematical Society (1989) 32, 345-353.
The array definition is better suited to Conway’s use of his hexacode to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases “vector space structure in the standard square” and “parallel 2-spaces” (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper. See my own page on the MOG at finitegeometry.org.
Comments Off on Competing MOG Definitions
Sunday, May 3, 2009
Sunday May 3, 2009
Annals of
Sacred Geometry
Sacred Geometry
(The phrase “sacred geometry”
is of course anathema to most
mathematicians, to whom
nothing is sacred.)
From “The Geometric
Art of John Michell“:
From this morning’s
New York Times:
| John Michell, Counterculture Author Who Cherished Idiosyncrasy, Dies at 76
By DOUGLAS MARTIN Mr. Michell, a self-styled Merlin of the 1960s English counterculture, inspired disciples like the Rolling Stones with a deluge of writings…. |
Michell, who wrote on Glastonbury
(a site associated with King Arthur)
and on sacred geometry, seems to
have had a better education than
most sacred-geometry enthusiasts.
He is said to have studied at
Eton and at Trinity College,
Cambridge.
(a site associated with King Arthur)
and on sacred geometry, seems to
have had a better education than
most sacred-geometry enthusiasts.
He is said to have studied at
Eton and at Trinity College,
Cambridge.
He is not to be
confused with an earlier
Trinity figure, mathematician
John Henry Michell,
who died at 76 on the third
day of February in 1940.
Related material:
See the Log24 entry
from the date of death
of the later Michell —
April 24 —
and, in light of the later
Michell’s interest in
geometry and King Arthur,
the Log24 remarks for
Easter Sunday this year
(April 12).
These remarks include the
following figure by
Sebastian Egner related,
if only through myth,
to Arthur’s round table —
— and the classic Delmore Schwartz
poem “Starlight Like Intuition
Pierced the Twelve.”
Which of the two John Michells
(each a Merlin figure of sorts)
would be more welcome in
Camelot is open to debate.
Comments Off on Sunday May 3, 2009
Sunday, April 12, 2009
Sunday April 12, 2009
Where Entertainment
Is God, continued
Is God, continued
Dialogue from the classic film Forbidden Planet—
"… Which makes it a gilt-edged priority that one of us gets into that Krell lab and takes that brain boost."
— Taken from a video (5:18-5:24 of 6:09) at David Lavery's weblog in the entry of Tuesday, April 7.
(Cf. this journal on that date.)
Thanks to Professor Lavery for his detailed notes on his viewing experiences.
My own viewing recently included, on the night of Good Friday, April 10, the spiritually significant film Indiana Jones and the Kingdom of the Crystal Skull.
The mystic circle of 13 aliens at the end of that film, together with Leslie Nielsen's Forbidden Planet remark quoted above, suggests the following:
|
"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 [3] 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 [3] 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…." [3] 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." |
For the religious significance of the circle of 13 (and the "hole"), consider Arthur and the 12 knights of the round table, et cetera.
But seriously…
Comments Off on Sunday April 12, 2009
Saturday, August 2, 2008
Saturday August 2, 2008
Prattle
There is an article in today’s Telegraph on mathematician Simon Phillips Norton– co-author, with John Horton Conway, of the rather famous paper “Monstrous Moonshine” (Bull. London Math. Soc. 11, 308–339, 1979).
“Simon studies one of the most complicated groups of all: the Monster. He is, still, the world expert on it ….
Simon tells me he has a quasi-religious faith in the Monster. One day, he says, … the Monster will expose the structure of the universe.
… although Simon says he is keen for me to write a book about him and his work on the Monster and his obsession with buses, he doesn’t like talking, has no sense of anecdotes or extended conversation, and can’t remember (or never paid any attention to) 90 per cent of the things I want him to tell me about in his past. It is not modesty. Simon is not modest or immodest: he just has no self-curiosity. To Simon, Simon is a collection of disparate facts and no interpretative glue. He is a man without adjectives. His speech is made up almost entirely of short bursts of grunts and nouns.
This is the main reason why we spent three weeks together …. I needed to find a way to make him prattle.”
Those in search of prattle and interpretive glue should consult Anthony Judge’s essay ““Potential Psychosocial Significance of Monstrous Moonshine: An Exceptional Form of Symmetry as a Rosetta Stone for Cognitive Frameworks.” This was cited here in Thursday’s entry “Symmetry in Review.” (That entry is just a list of items related in part by synchronicity, in part by mathematical content. The list, while meaningful to me and perhaps a few others, is also lacking in prattle and interpretive glue.)
Those in search of knowledge, rather than glue and prattle, should consult Symmetry and the Monster, by Mark Ronan. If they have a good undergraduate education in mathematics, Terry Gannon‘s survey paper “Monstrous Moonshine: The First Twenty-Five Years” (pdf) and book– Moonshine Beyond the Monster— may also be of interest.
Comments Off on Saturday August 2, 2008
Thursday, May 24, 2007
Monday, October 16, 2006
Monday October 16, 2006
Characters
Two items from a Wikipedia watchlist today:
1. User Loyola added a list of central characters to the article on The Glass Bead Game.
2. A dialogue between the Wikipedia characters Prof02 and Charles Matthews continues.
Item 2 seems almost to echo item 1.
The Bead Game, a classic novel by Hermann Hesse, is, in part, a commentary on German cultural history, and the Prof02-Matthews dialogue concerns the Wikipedia article on Erich Heller, a noted scholar of German cultural history.
Matthews is an expert on the game of Go. The Bead Game article says that
“The Game derives its name from the fact that it was originally played with tokens, perhaps analogous to those of an abacus or the game Go….
Although invented after Hesse’s death, Conway’s Game of Life can be seen as an example of a Go-like glass bead game with surprisingly deep properties; since it can encode Turing machines, it contains in some sense everything.”
For some related thoughts on cellular automata (i.e., Conway’s game) and Go, see The Field of Reason with its links Deep Game, And So To Bed.
For some related thoughts on Turing, see the November 2006 Notices of the American Mathematical Society (special issue on Turing).
For some related religious reflections, see Wolfram’s Theory of Everything and the Gameplayers of Zan, as well as the Log24 entries of last Halloween.
Comments Off on Monday October 16, 2006
Wednesday, November 30, 2005
Wednesday November 30, 2005
For St. Andrew’s Day
“The miraculous enters…. When we investigate these problems, some fantastic things happen….”
— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)
The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.
A picture of the Miracle Octad Generator, with my comments, is available online.
Related material:
Mathematics and Narrative.
Comments Off on Wednesday November 30, 2005
Thursday, August 25, 2005
Thursday August 25, 2005
Analogical
Train of Thought
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
Train of Thought
Part I: The 24-Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
| From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota |
Like footprints erased in the sand….
Part II: Discrete Space
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that space-time should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete Space-Time
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete space-time structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
Disclaimer:
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.
Part III: Quaternions
in a Discrete Space
in a Discrete Space
Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Comments Off on Thursday August 25, 2005
Sunday, February 20, 2005
Sunday February 20, 2005
Relativity Blues
Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry. Here is some related material.
In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).
(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)
On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:
"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."
One possible response to such statements was recently provided in some detail by a Princeton philosophy professor. See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.
A more thoughtful response would take into account the following:
1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson
2. The physics underlying Einstein's remarks on free will, God, and dice
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels Secret Passages and Broken Symmetries
4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University
5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.
From a review of Gilead by Jane Vandenburgh:
"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"
For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries. If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003, and those of November 5, 2002.
From Many Dimensions (Eerdmans paperback, 1963, page 53):
"Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?"
A recent answer:
Modal Theology
"We symbolize logical necessity
with the box 
)
and logical possibility
with the diamond 
).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by 
?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Comments Off on Sunday February 20, 2005
Sunday, November 30, 2003
Sunday November 30, 2003
The Proof and the Lie
A mathematical lie has been circulating on the Internet.
It concerns the background of Wiles’s recent work on mathematics related to Fermat’s last theorem, which involves the earlier work of a mathematician named Taniyama.
This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as “elliptic curves” and “modular forms.”
The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, “The Proof,” in October 1997 and repeated in a book based on the program and in a Scientific American article, “Fermat’s Last Stand,” by Simon Singh and Kenneth Ribet, in November 1997.
“… elliptic curves and modular forms… are from opposite ends of the mathematical spectrum, and had previously been studied in isolation.”
— Site on Simon Singh’s 1997 book Fermat’s Last Theorem
“JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that’s so hard to explain.
BARRY MAZUR: So, let me explain. Over here, you have the elliptic world, the elliptic curves, these doughnuts. And over here, you have the modular world, modular forms with their many, many symmetries. The Shimura-Taniyama conjecture makes a bridge between these two worlds. These worlds live on different planets. It’s a bridge. It’s more than a bridge; it’s really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.
KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous….”
— Transcript of NOVA program, “The Proof,” October 1997
The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:
“Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn’t believe that the two are related.”
— Ivars Peterson, “Curving Beyond Fermat,” November 1999
The lie has now contaminated university mathematics courses, as well as popular accounts:
“Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together.”
— Site on Fermat’s last theorem by undergraduate K. V. Binns
Authors like Singh who wrote about Wiles’s work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat’s last theorem.
A 1996 page in Daney’s site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms. Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.
Here is what Daney says:
“Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/r -> C. j is therefore an example of what is called a modular function. We’ll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions.”
“Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996″
Update of Dec. 2, 2003
For the relationship between modular functions and modular forms, see (for instance) Modular Form in Wikipedia.
Some other relevant quotations:
From J. S. Milne, Modular Functions and Modular Forms:
“The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenth-century] theory of elliptic functions.”
The next quote, also in a nineteenth-century context, relates elliptic functions to elliptic curves.
From Elliptic Functions, a course syllabus:
“Elliptic functions parametrize elliptic curves.”
Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama’s 1955 conjecture), between elliptic curves and modular forms.
Another quote from Milne, to summarize:
“From this [a discussion of nineteenth-century mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms.”
Serge Lang apparently agrees:
“Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.”
— Editorial description of Lang’s Elliptic Functions (second edition, 1987)
Update of Dec. 3, 2003
“The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.”
— Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp. (The review appeared in Bulletin of the American Mathematical Society, January 1994.)
Comments Off on Sunday November 30, 2003
Friday, September 19, 2003
Friday September 19, 2003
The Mysteries of 26
My entry of May 26, 2003 —
dealt with the question of whether this number, said to be of significance (as a number of dimensions) in theoretical physics, has any purely mathematical properties of interest.
That entry contained the above figure, a so-called Levi graph illustrating point/line incidence in the finite projective plane with 13 points and 13 lines, PG(2,3).
It turns out that in a paper of April 7, 2000, John H. Conway and Christopher S. Simons discussed a close connection between this plane and the Monster group. See
(Journal of Algebra. Vol. 235, no. 2.
MR 2001k:20028).
Conway had written about such a connection as early as 1985.
I apologize for not knowing about this sooner, and so misleading any mathematical readers about the number 26, which it seems does have considerable purely mathematical significance.
Comments Off on Friday September 19, 2003
Monday, May 26, 2003
Monday May 26, 2003
Mental Health Month, Day 26:
Part III — Why 26?
At first blush, it seems unlikely that the number 26=2×13, 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 ‘ Why do the simple groups 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.
Comments Off on Monday May 26, 2003
Friday, May 23, 2003
Friday May 23, 2003
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.
Comments Off on Friday May 23, 2003