Log24

Saturday, November 28, 2020

Star Quality: “Mirror Guy”

Filed under: General — Tags: — m759 @ 12:30 PM

Related material from pure mathematicsM. J. T.  Guy

Saturday, August 8, 2020

A Natural Diagram

Filed under: General — Tags: , — m759 @ 7:54 PM

See also other posts now tagged
       Natural Diagram .

Related remarks by J. H. Conway —

Wednesday, July 22, 2020

Card

Filed under: General — Tags: , — m759 @ 2:48 AM

“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.

Friday, July 17, 2020

Theoretical as Well as Practical Value

Filed under: General — Tags: , , — m759 @ 12:25 AM

See also The Lexicographic Octad Generator (LOG) (July 13, 2020)
and Octads and Geometry (April 23, 2020).

Monday, July 13, 2020

The Lexicographic Octad Generator (LOG)*

Filed under: General — Tags: , , — m759 @ 5:43 PM

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 mid-1970s.)

* Addendum at 4 AM ET  the next day —
See also Logline  (Walpurgisnacht 2013).

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — Tags: — m759 @ 6:32 PM

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 O= {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 —

Thursday, April 4, 2019

For Harlan Kane . . .

Filed under: General — m759 @ 12:00 PM

The Aloha Grid

Some less demanding reading Mysteries of the Rectangle .

Tuesday, December 25, 2018

Simply

Filed under: General — m759 @ 11:42 AM
 

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: General,Geometry — Tags: — m759 @ 2:08 PM

Reference

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.

Friday, November 24, 2017

The Matrix Meets the Grid

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 2:00 PM

The Matrix —

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Thursday, November 16, 2017

A Line at Infinity

Filed under: G-Notes,General,Geometry — m759 @ 12:00 PM

Tuesday, June 7, 2016

Art and Space…

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

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.

Wednesday, May 25, 2016

Framework

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"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 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

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.

Monday, May 2, 2016

Subjective Quality

Filed under: General,Geometry — m759 @ 6:01 AM

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.

Sunday, May 1, 2016

Sunday Appetizer from 1984

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich
unter http://​www.​encyclopediaofma​th.​org/​index.​php/​
Cullinane_​diamond_​theorem
und http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

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 —

Monday, February 1, 2016

Historical Note

Filed under: General,Geometry — Tags: , — m759 @ 6:29 AM

Possible title

A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24

Thursday, January 8, 2015

Gitterkrieg

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

(Continued)

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,
was the basis for Curtis's octad-generator construction.

See Turyn in this journal.

Tuesday, June 17, 2014

Finite Relativity

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

Continued.

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.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, 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 RelativityGalois 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,  not  by hand (such as Turyn’s), 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 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

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—

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).

Wednesday, November 30, 2005

Wednesday November 30, 2005

Filed under: General,Geometry — m759 @ 1:00 AM

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.


 Cartoon by S.Harris

Related material:
Mathematics and Narrative.

Powered by WordPress