Log24

Thursday, April 29, 2021

56 Three-Sets, 56 Spreads:

Filed under: General — Tags: , , — m759 @ 8:38 pm

The Steiner Quadruple System S(3,4,8)
underlies the Steiner System S(5,8,24).

A previous update to the Oct. 29, 2019, post Triangles, Spreads, Mathieu:

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.

That page, 284, contained an excerpt from

Bussemaker, F. C., & Seidel, J. J. (1970).
“Symmetric Hadamard matrices of order 36.”
(EUT report. WSK, Dept.of Mathematics and
Computing Science; Vol. 70-WSK-02).
Technische Hogeschool Eindhoven.

That paper is now available online:

https://pure.tue.nl/ws/files/1804543/252823.pdf .

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.)

Saturday, March 26, 2022

Box Geometry: Space, Group, Art  (Work in Progress)

Filed under: General — Tags: — m759 @ 2:06 am

Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").

Here is a draft for a table of related material, arranged
as internet URL labels.

Finite Geometry Notes — Summary Chart
 

Name Tag .Space .Group .Art
Box4

2×2 square representing the four-point finite affine geometry AG(2,2).

(Box4.space)

S4 = AGL(2,2)

(Box4.group)

 

(Box4.art)

Box6 3×2 (3-row, 2-column) rectangular array
representing the elements of an arbitrary 6-set. 		S6  
Box8 2x2x2 cube or  4×2 (4-row, 2-column) array. S8 or A8 or  AGL(3,2) of order 1344, or  GL(3,2) of order 168  
Box9 The 3×3 square. AGL(2,3) or  GL(2,3)  
Box12 The 12 edges of a cube, or  a 4×3  array for picturing the actions of the Mathieu group M12. Symmetries of the cube or  elements of the group M12  
Box13 The 13 symmetry axes of the cube. Symmetries of the cube.  
Box15 The 15 points of PG(3,2), the projective geometry
of 3 dimensions over the 2-element Galois field. 		Collineations of PG(3,2)  
Box16 The 16 points of AG(4,2), the affine geometry
of 4 dimensions over the 2-element Galois field.

AGL(4,2), the affine group of 
322,560 permutations of the parts
of a 4×4 array (a Galois tesseract)

 
Box20 The configuration representing Desargues's theorem.    
Box21 The 21 points and 21 lines of PG(2,4).    
Box24 The 24 points of the Steiner system S(5, 8, 24).    
Box25 A 5×5 array representing PG(2,5).    
Box27 The 3-dimensional Galois affine space over the
3-element Galois field GF(3). 		   
Box28 The 28 bitangents of a plane quartic curve.    
Box32 Pair of 4×4 arrays representing orthogonal 
Latin squares. 		Used to represent
elements of AGL(4,2) 		 
Box35 A 5-row-by-7-column array representing the 35
lines in the finite projective space PG(3,2) 		PGL(3,2), order 20,160  
Box36 Eurler's 36-officer problem.    
Box45 The 45 Pascal points of the Pascal configuration.    
Box48 The 48 elements of the group  AGL(2,3). AGL(2,3).  
Box56

The 56 three-sets within an 8-set or
56 triangles in a model of Klein's quartic surface or
the 56 spreads in PG(3,2).

   
Box60 The Klein configuration.    
Box64 Solomon's cube.    

— Steven H. Cullinane, March 26-27, 2022

Tuesday, October 6, 2020

Spreads via the Knight Cycle

Filed under: General — Tags: — m759 @ 2:10 am

A Graphic Construction of the 56 Spreads of PG(3,2)

(An error in Fig. 4 was corrected at about
10:25 AM ET on Tuesday, Oct. 6, 2020.)

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 A8 is 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 — Tags: , , , — 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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