Log24

Wednesday, January 1, 2014

The 56 Spreads in PG(3,2)

Filed under: General,Geometry — m759 @ 11:07 PM

IMAGE- The 56 spreads in PG(3,2)

Click for a larger image

For a different pictorial approach, see Polster's
1998 Geometrical Picture Book , pp. 77-80.

Update:  Added to finitegeometry.org on Jan. 2, 2014.
(The source of the images of the 35 lines was the image
"Geometry of the Six-Element Set," with, in the final two
of the three projective-line parts, the bottom two rows
and the rightmost two columns interchanged.)

Friday, February 7, 2020

Correspondences

Filed under: General — Tags: , — m759 @ 1:05 PM

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)

Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.

See also Correspondences in this journal.

* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .

Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:

See as well the word correspondences  in 
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: — m759 @ 8:04 PM

There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.

Exercise:

It is well-known that

 There are 56 triangles in an 8-set.
 There are 56 spreads in PG(3,2).
 The alternating group An is generated by 3-cycles.
 The alternating group Ais isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M24.

Some background —

A Log24 post of May 19, 2013, cites

Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.

See also a Google search for "56 triangles" "56 spreads" Mathieu.

Update of October 31, 2019 — A related illustration — 

Update of November 2, 2019 —

See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel
  (Academic Press, 1991).
That page is from a paper published in 1970.

Update of December 20, 2019 —

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — m759 @ 7:11 PM

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell's heptads .

Update, morning of the following day (7:07 ET) — related material:

See also "56 spreads" in this  journal.

Saturday, February 15, 2014

Rosenhain and Göpel

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

(Continued)

See The Oslo Version in this journal and the New Year’s Day 2014 post.
The pictures of the 56 spreads in that post (shown below) are based on
the 20 Rosenhain and 15 Göpel tetrads that make up the 35 lines of
PG(3,2), the finite projective 3-space over the 2-element Galois field.

IMAGE- The 56 spreads in PG(3,2)

Click for a larger image.

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