Some mathematical background —
Saturday, May 10, 2025
The Bluesky Mystery Continues:
A Steiner Quadruple System
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A Steiner Quadruple System
A Steiner Quadruple System
Thursday, August 15, 2019
On Steiner Quadruple Systems of Order 16
An image from a Log24 post of March 5, 2019 —
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.
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Tuesday, March 5, 2019
A Block Design 3-(16,4,1) as a Steiner Quadruple System:
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
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Tuesday, March 28, 2023
Onestone Parable
“You’re literally looking for like a one in a million thing.
You filter out the 999,999 of the boring ones, then
you’ve got something that’s weird, and then that’s worth
further exploration.”
— Quote from a mathematics story today at Gizmodo
A different "one in a million" mathematics story —
On Steiner Quadruple Systems of Order 16.
See also Galois Tesseract.
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Thursday, April 29, 2021
56 Three-Sets, 56 Spreads:
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:
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Friday, January 8, 2021
Groups Act
"Somehow, a message had been lost on me. Groups act .
The elements of a group do not have to just sit there,
abstract and implacable; they can do things, they can
'produce changes.' In particular, groups arise
naturally as the symmetries of a set with structure."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry
in The American Mathematical Monthly , Vol. 94, No. 4
(April 1987), pp. 392-394.
"The concept of group actions is very useful in the study of
isomorphisms of combinatorial structures."
— Olli Pottonen, "Classification of Steiner Quadruple Systems"
(Master's thesis, Helsinki, 2005, p. 48).
“In a sense, we would see that change arises from
the structure of the object.”
— Nima Arkani-Hamed, quoted in "A Jewel at the Heart of
Quantum Physics," by Natalie Wolchover, Quanta Magazine ,
Sept. 17, 2013.
See as well "Change Arises" in this journal.
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From the Finland Station
The title refers to the Steiner quadruple systems in a 2005 thesis by
a Helsinki mathematician. See . . .
http://m759.net/wordpress/?tag=helsinki-math.
See as well “a million diamonds” and . . .
.
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Sunday, December 6, 2020
The Undoing
Today’s earlier post “Binary Coordinates” discussed a Dec. 6
revision to the Wikipedia article on PG(3,2), the projective
geometry of 3 dimensions over the 2-element field GF(2).
The revision, which improved the article, was undone later today
by a clueless retired academic, one William “Bill” Cherowitzo,
a professor emeritus of mathematics at U. of Colorado at Denver.
(See his article “Adventures of a Mathematician in Wikipedia-land,”
MAA Focus , December 2014/January 2015.)
See my earlier remarks on this topic . . . specifically, on this passage —
“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.”
As I noted on November 17, this is bullshit. Apparently Cherowitzo
never bothered to find out that an arbitrary “3-(16,4,1) block design”
(an example of a Steiner quadruple system ) does not yield a PG(3,2).
PG(3,2) is derived from the classical 3-(16,4,1) block design formed by the affine
space of 4 dimensions over GF(2). That design has 322,560 automorphisms.
In contrast, see a 3-(16,4,1) block design that is automorphism-free.
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“Binary Coordinates”
The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .
Related material from the Web —
Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —
A related anonymous change to Wikipedia today —
The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.
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Sunday, August 25, 2019
Design Theory
"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
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Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film "Triple Cross."
Related structures —
Greg Egan's animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
|
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
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Sunday, May 17, 2009
Sunday May 17, 2009
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."
"Simplify, simplify."
— Thoreau
"Beauty is bound up
with symmetry."
— Weyl
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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