Log24

A Unified Field —

Click the above image for further details.

See also a search in this journal for Jorie Graham.

Related dramatic dialogue for Emma Stone and 
Joaquin Phoenix, actors in "Irrational Man" —

"Are you  aware of what's going on at that  table?"

Philosophical backstory by Hans Christian Andersen —

"He was quite frightened, and he tried to repeat the Lord's Prayer;
but all he could do, he was only able to remember the multiplication table."

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure 
clearly is of fundamental importance.  

It is, however, perhaps too  small  to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array…

… hence as the traditional Chinese  Holy Field.

Marketing the Holy Field

The above illustration of China's  Holy Field occurred in the context of
Log24 posts on Child Buyers.   For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

   For aficionados of mathematics and narrative —

Illustration from
"The Galois Quaternion— A Story"

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(3²), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(3²) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 9² + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

 

Translation Plane
for Rosh Hashanah

Figure A

From the website of Priv.-Doz. Dr. H. Klein, Arbeitsgruppe Geometrie, Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel — The Translation Plane of Order Nine There are exactly four projective planes of order nine, and one of these planes is a non-Desarguesian translation plane. Theorem. Up to isomorphism, there exists exactly one non-Desarguesian translation plane of order 9. This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.   As it turns out, the coordinatizing quasifield is a nearfield. Moreover the non-Desarguesian translation plane of order 9 has Lenz-Barlotti type IVa.3.

Two versions of the defining spreadset for this plane are shown in Figure A.  In the left part of Fig. A, the matrices of Dr. Klein are altered by the use of “2” instead of “-1” (since these are the same, modulo 3).  In the right part of Fig. A, the corresponding figures from my 1985 note Visualizing GL(2, p) are shown.

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.