Log24

This morning at 11:44 I happened upon …

This was published as …

Toshiyuki Katsura, Shigeyuki Kondo, Ichiro Shimada,
"On the supersingular K3 surface in characteristic 5 with Artin invariant 1,"
Michigan Mathematical Journal , vol. 63, issue 4 (Dec. 2014), 803–844.

Related material from later today —

See also earlier Log24 remarks on the Hoffman-Singleton graph
and a remark on geometry for Princeton.

Click to enlarge:

For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .

For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.

For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.

Symplectic.

Related material:

From the website of the American Mathematical Society today,
a column by John Baez that was falsely backdated to Sept. 1, 2015 —

Compare and contrast this Baez column 
with the posts in the above
Log24 search for "Symplectic."

Updates after 9 PM ET Sept. 17, 2015 —

Related wrinkles in time: 

Baez's preceding Visual Insight  post, titled 
"Tutte-Coxeter Graph," was dated Aug. 15, 2015.
This seems to contradict the AMS home page headline
of Sept. 5, 2015, that linked to Baez's still earlier post
"Heawood Graph," dated Aug. 1. Also, note the 
reference in "Tutte-Coxeter Graph" to Baez's related 
essay — dated August 17, 2015 — 

• "A Wrinkle in the Mathematical Universe" 
   at The n-Category Café .

The phrase “the permutation group S_n” refers to a
particular  group of permutations that act on an
n -element set N— namely, all  of them. For a given n ,
there are, in general, many  permutation groups that
act on N.  All but one are smaller than S_n .

In other words, the phrase “the permutation group S_n”
does not  imply that “S_n ” is a symbol for a structure
associated with n  called “the  permutation group.”
It is instead a symbol for “the symmetric  group,” the largest
of (in general) many permutation groups that act on N.

This point seems to have escaped John Baez.

For two misuses by Baez of the phrase “permutation group” at the
n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

Continued from June 17, 2013
(John Baez as a savior for atheists):

As an atheists-savior, I prefer Galois…

The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois  geometry —

See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:

For the genesis of the figure,
see The Geometry of Logic.

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.

Grid from a post linked to in yesterday's 24 Hour DeLillo—

A Study in Art Education

For an example of this grid as slow art , consider the following—

"One can show that the binary tetrahedral group
is isomorphic to the special linear group SL(2,3)—
the group of all 2×2 matrices over the finite field F₃
with unit determinant." —Wikipedia

As John Baez has noted, these two groups have the same structure as the geometric 24-cell.

For the connection of the grid to the groups and the 24-cell, see Visualizing GL(2,p).

Related material—

The 3×3 grid has been called a symbol of Apollo (Greek god of reason and of the sun).

"This is where we sat through his hushed hour,
a torchlit sky, the closeness of hills barely visible
at high white noon." — Don DeLillo, Point Omega

"… the best way to understand a group is to
see it as the group of symmetries of something."

— John Baez, p. 239, Bulletin (New Series) of the
American Mathematical Society , Vol. 42, No. 2,
April 2005, book review on pp. 229–243
electronically published on January 26, 2005

"Imagine yourself as a gem cutter,
turning around this diamond…."

— Ibid ., p. 240

See also related material from Log24.

A search for some background on Dmitri Tymoczko, the subject of yesterday's evening entry on music theory, shows that his name and mine once both appeared in the same web page— "This Week's Finds in Mathematical Physics (Week 234)," by John Baez, June 12, 2006 (linked to by the Wikipedia article on transformational music theory).

In that page, Baez speculates on the possibility of a connection between music theory and Mathieu groups and says—

"For a pretty explanation of M₂₄, also try this:

Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html."

I know of no connection* between the groups I discussed there and music theory. For some background on Tymoczko's work, see the helpful survey "Exploring Musical Space," by Julian Hook (Science  magazine, 7 July 2006).

* Apart, that is, from the tesseract (see Geometry of the 4 × 4 Square) shown by Tymoczko in a 2010 lecture—

This is perhaps "Chopin's tesseract" from section 8.5 of Tymoczko's new book
A Geometry of Music  (Oxford University Press, 2011).

A new NY Times column:

Today's New York Times
re-edited for philosophers:

See also

Eightfold Symmetry,

John Baez's paper
Duality in Logic and Physics
(for a May 29 meeting at Oxford),

The Shining of May 29, and

Lubtchansky's Key, with its links
to Duelle (French, f. adj., dual)
and Art Wars for Trotsky's Birthday.

"[The applet] Syntheme illustrates ways of partitioning the 12 vertices of an icosahedron into 3 sets of 4, so that each set forms the corners of a rectangle in the Golden Ratio. Each such rectangle is known as a duad. The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges, so there are 15 duads.

Each partition of the vertices into duads is known as a syntheme. There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while the other 10 consist of duads that share a common line of intersection."

— Greg Egan, Syntheme

The above note shows
duads and synthemes related
to the diamond theorem.

See also John Baez's essay
"Some Thoughts on the Number 6."
That essay was written 15 years
ago today– which happens
to be the birthday of
Sir Laurence Olivier, who,
were he alive today, would
be 100 years old.

"Is it safe?"

Baez Link

John Baez’s latest This Week’s Finds
(Week 234, June 12, 2006) has a link
to my “Geometry of the 4×4 Square” at
http://finitegeometry.org/sc/16/geometry.html.

Like footprints erased in the sand….

Log24, May 27, 2004 —

  “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

Beyond Geometry

(Title of current L. A. art exhibit)

“The greatest obstacle to discovery
is not ignorance —
  it is the illusion of knowledge.”

Mental Health Month, Day 26:

Many Dimensions,

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*.

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.

Mental Health Month, Day 26:

Many Dimensions,
Part II — The Blue Matrix 

But seriously…

John Baez in July 1999:

"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."

Well, maybe.  I certainly hope so.  If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light.  Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory.  See the following from my journal:

Evariste Galois and the Rock that Changed Things, and

A Time to Gather Stones Together: Readings for Yom Kippur.

See, too, this web page on Marion Zimmer Bradley's fictional

Matrices, or Blue Star-Stones, and

the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.