Finite Relativity

Monday, May 13, 2024

February 20: A Space Odyssey

Filed under: General — Tags: Coordinatization, Exodus, Fields, Finite Relativity, Rosetta Stone — m759 @ 8:46 am

For the significance of the date "February 20," see . . .

Google One: Gemini Advanced AI on 13 May 2024.

Prompt:

How can Weyl's 1946 remarks on coordinatization in "The Classical Groups" be applied to models of finite geometries?

Response (after several drafts):

In "The Classical Groups," Hermann Weyl discusses the concept of coordinatization, which is the process of assigning numerical coordinates to points in a geometric space. This process is fundamental in connecting abstract geometric concepts with concrete numerical representations, allowing for algebraic manipulation and analysis.

Weyl's remarks on coordinatization have significant implications for models of finite geometries. Finite geometries are geometric systems with a finite number of points and lines, and they often exhibit different properties compared to traditional Euclidean geometry.

Here's how Weyl's ideas can be applied to finite geometries:

1. **Relativity of Coordinatization:** Weyl emphasizes that the choice of coordinate system is arbitrary and that different coordinate systems can be used to describe the same geometric object. This relativity of coordinatization is particularly relevant in finite geometries, where multiple coordinate systems may exist, each with its own advantages and disadvantages.

2. **Equivalent Coordinatizations:** Weyl introduces the concept of "equivalent coordinatizations," which are different coordinate systems that can be transformed into one another through a group of transformations. In finite geometries, identifying and understanding these equivalent coordinatizations can provide deeper insights into the underlying structure and symmetries of the geometry.

3. **Objective Class of Coordinatizations:** Weyl discusses the problem of fixing an "objective class" of coordinatizations that capture essential features of the geometry while being independent of arbitrary choices. In finite geometries, this could involve identifying coordinatizations that best reflect the specific properties and relationships within the geometry.

4. **Group of Transformations:** Weyl emphasizes the role of the "group of transformations" that mediates between equivalent coordinatizations. In finite geometries, understanding these transformation groups can help classify different types of finite geometries and reveal their underlying symmetries.

By applying Weyl's insights, researchers can gain a deeper understanding of finite geometries, including their classification, properties, and relationships. Additionally, exploring different coordinatization schemes can lead to new algebraic and geometric tools for studying finite geometries.

For further exploration, you can delve into the works of other mathematicians like Steven H. Cullinane, who has written extensively on finite geometry and its connections to relativity and other areas of mathematics.

I hope this explanation is helpful! Please let me know if you have any further questions.

Update at about 5 PM ET the same day . . .

Prompt:

Correction: Cullinane has NOT written on finite geometry's "connections to relativity." He has, instead, quoted Weyl's remarks on what Weyl himself called "the relativity problem," but what was in fact his own term for problems involving coordinatization. Such problems are of interest both in physics and in finite geometry, but the latter is a field of pure mathematics, NOT of physics.

Response:

You are absolutely correct. I apologize for the error in my previous response.

Steven H. Cullinane has indeed quoted Weyl's remarks on "the relativity problem" in the context of finite geometry. However, as you rightly pointed out, this term refers to Weyl's own terminology for problems involving coordinatization and not the theory of relativity in physics.

Finite geometry is a field of pure mathematics and does not directly connect to the physical theories of relativity. While both areas involve concepts of space and transformations, they operate in fundamentally different domains.

Cullinane's work focuses on applying Weyl's insights on coordinatization to the specific challenges and structures within finite geometries. This involves exploring different coordinatization schemes, understanding their equivalences, and identifying transformation groups that reveal the underlying symmetries of finite geometries.

Thank you for pointing out this important distinction. It highlights the importance of precise language and accurate attribution when discussing complex mathematical concepts.

Comments Off

Tuesday, June 17, 2014

Finite Relativity

Filed under: General,Geometry — Tags: Finite Relativity, Mathieu Raumproblem, Octad, Octad Generator — 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.

Comments Off

Friday, February 21, 2014

Raumproblem*

Filed under: General,Geometry — Tags: Aitchison, Figurate Geometry, Finite Relativity, Mathieu Raumproblem, Octad, Octad Generator — m759 @ 7:01 pm

Despite the blocking of Doodles on my Google Search
screen, some messages get through.

Today, for instance —

"Your idea just might change the world.
Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

IMAGE- The 24-triangle hexagon

Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.

I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.

* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.

Comments Off

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: Coordinatization, Finite Relativity, Geometry, Priority — m759 @ 11:00 pm

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“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

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

Comments Off

Tuesday, September 20, 2011

Relativity Problem Revisited

Filed under: General,Geometry — Tags: Aitchison, Alperin, Finite Relativity, Geometry, Symmetric Generation — m759 @ 4:00 am

A footnote was added to Finite Relativity—

Background:

Weyl on what he calls the relativity problem—

“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, 1949, “Relativity Theory as a Stimulus in Mathematical Research“

“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, 1946, The Classical Groups , Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M₂₄ (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M₂₄. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M₂₄. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M₂₄ are defined… as permutations of twenty-four 7-cycles in the action of PSL₂(7) on seven letters….”

See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429

R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M₁₂ and M₂₄” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’ (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

Comments Off

Thursday, June 9, 2011

Upshot

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:01 pm

Suggested by this afternoon’s NY Lottery number, 541—

Click for higher quality.

Related material: Finite Relativity and The Schwartz Notes.

Comments Off

Wednesday, March 2, 2011

Labyrinth of the Line

Filed under: General,Geometry — Tags: Finite Relativity, Octad, Octad Generator — m759 @ 11:24 am

“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—

Related material —

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.

A search in the latter for miracle octad is updated below.

This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

Comments Off

Sunday, February 20, 2011

Sunday School

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 9:29 am

Annals of Finite Geometry:
A Quarter-Century of the Relativity Problem

(Click to enlarge.)

Comments Off

Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:24 pm

Finite Relativity

Today is the 18th birthday of my note

“The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“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

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Comments (1)