Curtis Omega « Search Results

Thursday, August 22, 2024

Locke & Key: Alpha & Omega

Filed under: General — m759 @ 4:54 pm

A more sophisticated alpha and omega . . .

— R. T. Curtis, "A New Combinatorial Approach to M ₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42

A related key . . .

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A₈," in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

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Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: Octad, Octad Generator — m759 @ 5:00 am

The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem .

If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.

See the Miracle Octad Generator of R. T. Curtis.

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Sunday, May 6, 2018

The Osterman Omega

Filed under: General,Geometry — Tags: Fish Babel, The Curtis Omega — m759 @ 5:01 pm

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday —

Counting symmetries with the orbit-stabilizer theorem

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Monday, January 12, 2015

Points Omega*

Filed under: General,Geometry — Tags: Natural Diagram, Octad, Octad Generator, Points Omega — m759 @ 12:00 pm

The previous post displayed a set of
24 unit-square “points” within a rectangular array.
These are the points of the
Miracle Octad Generator of R. T. Curtis.

The array was labeled Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.

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Sunday, September 18, 2011

Alpha and Omega

Filed under: General,Geometry — Tags: Geometry, Octad — m759 @ 2:22 am

A transcription—

"Now suppose that α is an element of order 23 in M ₂₄; we number the points of Ω
as the projective line ∞, 0, 1, 2, … , 22 so that α : i → i + 1 (modulo 23) and fixes ∞. In
fact there is a full L₂(23) acting on this line and preserving the octads…."

— R. T. Curtis, "A New Combinatorial Approach to M ₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42

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Saturday, March 23, 2024

Alpha Bets

Filed under: General — Tags: Ghost Song, Mirror Worlds — m759 @ 6:31 am

"So you wanna play with magic?" — Katy Perry

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Friday, August 19, 2022

The Guy Embedding

Filed under: General — Tags: Octad, on220819, The Guy Embedding — m759 @ 3:43 pm

M. J. T. Guy discovered that the lexicographic version
of the Golay code contains, embedded within it, the
Miracle Octad Generator (MOG) of R. T. Curtis.

For 12 basis vectors of the lexicographic version, see below.