Thursday, August 15, 2019

On Steiner Quadruple Systems of Order 16

Filed under: General — Tags: , — m759 @ 4:11 AM

An image from a Log24 post of March 5, 2019

Cullinane's 1978  square model of PG(3,2)

The following paragraph from the above image remains unchanged
as of this morning at Wikipedia:

"A 3-(16,4,1) block design has 140 blocks of size 4 on 16 points,
such that each triplet of points is covered exactly once. Pick any
single point, take only the 35 blocks containing that point, and
delete that point. The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points."

Exercise —

Prove or disprove the above assertion about a general "3-(16,4,1) 
block design," a structure also known as a Steiner quadruple system
(as I pointed out in the March 5 post).

Relevant literature —

A paper from Helsinki in 2005* says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2),
from which PG(3,2) can be derived using the well-known process
from finite geometry described in the above Wikipedia paragraph.

* "The Steiner quadruple systems of order 16," by Kaski et al.,
   Journal of Combinatorial Theory Series A  
Volume 113, Issue 8, 
   November 2006, pages 1764-1770.

Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: , — m759 @ 11:19 AM

A Midrash for Wikipedia 

Midrash —

Related material —


The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Sunday, August 25, 2019

Design Theory

Filed under: General — Tags: , , — m759 @ 7:58 PM

"Mein Führer Steiner"

See Hitler Plans and Quadruple System.

"There is  such a thing as a quadruple system."

— Saying adapted from a 1962 young-adult novel

Sunday, May 17, 2009

Sunday May 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM
Design Theory

Laura A. Smit, Calvin College, "Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams"–

"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."


Hans Ludwig de Vries, "On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16)," Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–

"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."

1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).

4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).

Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–

"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."

Olli Pottonen,  'Classification of Steiner Quadruple Systems'

"Simplify, simplify."

"Beauty is bound up
with symmetry."

Sixteen points in a 4x4 array

Pottonen's thesis is
 dated Nov. 16, 2005.

For some remarks on
images and theology,
see Log24 on that date.

Click on the above image
 for some further details.

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