From Encyclopedia of Mathematics —
The above images from the history of mathematics might be
useful at some future point for illustrating academic hurly-burly.
Related reading . . .
Monday, January 15, 2024
Foursquare Variations
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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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Friday, November 4, 2022
Annals of Theology: The Coordinatization Being
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Thursday, November 3, 2022
Dazzled or Baffled?
"If you can't dazzle 'em with brilliance,
baffle 'em with bullshit." — Folk saying
Brilliance —
Bullshit —
A New York Times book review on All Saints' Day 2002 —
"Like every villain, Sill has an origin story. This one involves
the murder of his father and the assassination of Martin Luther King Jr.
As vengeance, Sill resolves to destroy America using a weapon
called a 'complex projective plane orbiter.' What does that mean?
Don’t worry about it."
Related reading: The Aloha Grid .
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Monday, February 7, 2022
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, April 17, 2021
Monday, March 15, 2021
The Abstract Signature
Caption: "I notice the signatures are never abstract." —
Abstract Art
Abstract Signature
From Encyclopedia of Mathematics —
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Saturday, November 28, 2020
Star Quality: “Mirror Guy”
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Saturday, August 8, 2020
A Natural Diagram
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Wednesday, July 22, 2020
Card
“The pattern of the thing precedes the thing.
I fill in the gaps of the crossword at any spot
I happen to choose. These bits I write on
index cards until the novel is done.”
— Vladimir Nabokov, interview,
Paris Review No. 41 (Summer-Fall 1967).
Another story —
Related material: Mathematics as a Black Art.
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Friday, July 17, 2020
Theoretical as Well as Practical Value
See also The Lexicographic Octad Generator (LOG) (July 13, 2020)
and Octads and Geometry (April 23, 2020).
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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, 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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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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Monday, November 27, 2017
The Golay Code via Witt’s Construction
Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053.
http://scholarworks.sjsu.edu/etd_theses/4053.
See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG 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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Thursday, November 16, 2017
Tuesday, June 7, 2016
Art and Space…
Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —
The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.
From this journal on St. Bridget's Day this year —
"Possible title:
A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24"
The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.
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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 —
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Monday, February 1, 2016
Historical Note
Possible title:
A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24
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Thursday, January 8, 2015
Gitterkrieg
From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —
"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72 I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."
On an earlier talk by Nebe at Oberwolfach in 2011 —
"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."
Nebe's Oberwolfach slides include one on
"The history of Turyn's construction" —
Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's
"Miracle Octad Generator."
Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.
See Turyn in this journal.
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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 —
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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).
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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.
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