Field GF(9) « Search Results

Tuesday, February 13, 2018

A Titan of the Field

Filed under: General,Geometry — Tags: Lynden-Bell — m759 @ 9:45 am

On the late Cambridge astronomer Donald Lynden-Bell —

"As an academic at a time when students listened and lecturers lectured, he had the disconcerting habit of instead picking on a random undergraduate and testing them on the topic. One former student, now a professor, remembered how he would 'ask on-the-spot questions while announcing that his daughter would solve these problems at the breakfast table'.

He got away with it because he was genuinely interested in the work of his colleagues and students, and came to be viewed with great affection by them. He also got away with it because he was well established as a titan of the field."

— The London Times on Feb. 8, 2018, at 5 PM (British time)

Related material —

Two Log24 posts from yesteday, Art Wars and The Void.

See as well the field GF(9) …

… and the 3×3 grid as a symbol of Apollo
(an Olympian rather than a Titan) —

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Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: Conceptualist Minimalism, Fields, The Coxeter Aleph — m759 @ 6:29 pm

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

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Monday, February 20, 2012

Coxeter and the Relativity Problem

Filed under: General,Geometry — Tags: The Coxeter Aleph — m759 @ 12:00 pm

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"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

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of the primitive root λ or 10 are

λ = 10, λ² = 21, λ³ = 22, λ⁴ = 02,
λ⁵ = 20, λ⁶ = 12, λ⁷ = 11, λ⁸ = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ⁴, λ⁵, λ⁷.)"

Coxeter's Figure 10 yields...

The Aleph

The details:

(Click to enlarge)

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

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Thursday, September 8, 2011

Starring the Diamond

Filed under: General,Geometry — m759 @ 2:02 pm

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 8₃
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6—

The Diamond Star

The above is a version of a figure from Configurations and Squares.

Yesterday's post related the the Pappus configuration to this figure.

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.

Some excerpts from Coxeter—

The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—

(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—

"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."

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Thursday, July 9, 2015

Man and His Symbols

Filed under: General,Geometry — m759 @ 2:24 pm

(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.

The Galois field GF(3)

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.

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Tuesday, August 13, 2013

The Story of N

Filed under: General,Geometry — Tags: Story of N, The Coxeter Aleph — m759 @ 9:00 pm

(Continued from this morning)

The above stylized "N," based on
an 8-cycle in the 9-element Galois field
GF(9), may also be read as an Aleph.

Graphic designers may prefer a simpler,
bolder version:

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