Log24

Wednesday, December 7, 2016

Emch as a Forerunner of S(5, 8, 24)

Filed under: General,Geometry — m759 @ 1:00 pm

Commentary —

"The close relationships between group theory and structural combinatorics go back well over a century. Given a combinatorial object, it is natural to consider its automorphism group. Conversely, given a group, there may be a nice object upon which it acts. If the group is given as a group of permutations of some set, it is natural to try to regard the elements of that set as the points of some structure which can be at least partially visualized. For example, in 1861 Mathieu… discovered five multiply transitive permutation groups. These were constructed as groups of permutations of 11, 12, 22, 23 or 24 points, by means of detailed calculations. In a little-known 1931 paper of Carmichael [5], they were first observed to be automorphism groups of exquisite finite geometries. This fact was rediscovered soon afterwards by Witt [11], who provided direct constructions for the groups and then the geometries. It is now more customary to construct first the designs, and then the groups…."

  5.  R. D. Carmichael, Tactical configurations of rank two,
Amer. J. Math. 53 (1931), 217-240.

11.  E. Witt, Die 5-fach transitiven Gruppen von Mathieu,
Abh. Hamburg 12 (1938), 256-264. 

— William M. Kantor, book review (pdf), 
Bulletin of the American Mathematical Society, September 1981

Monday, May 23, 2016

23

Filed under: General,Geometry — m759 @ 8:25 am

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

Keywords: Golay, Mathieu, Steiner system, Carmichael, Harvard

Friday, February 6, 2015

The Annotated Spielraum

Filed under: General — Tags: — m759 @ 7:00 am

Comments on two sub-images from yesterday's
The Big Spielraum  (image, 1 MB) that may or
may not interest Emma Watson —

The Potter Sub-Image

This is from a link in a July 8, 2011, post:

The above "Childhood's End" link leads to
a midrash on the Harry Potter series:

"After pg. 759 in Harry Potter and the 
Deathly Hallows , my childhood ended."

The Carmichael Sub-Image

The number of the last page in the last Harry Potter
book is 759.  This number may, for those with
cabalistic tendencies, be interpreted as the
number 3*23*11 from a 1931 mathematics paper:

Monday, November 10, 2014

Narrative Line

Filed under: General,Geometry — Tags: , , — m759 @ 11:02 pm

"We live entirely, especially if we are writers, by the imposition
of a narrative line upon disparate images…." — Joan Didion

Narrative Line:

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

IMAGE- Harvard senior Jeremy Booher in 2010 discusses Carmichael's 1931 construction of S(5, 8, 24) without mentioning Carmichael.

Disparate images:

Exercise:

Can the above narrative line be imposed in any sensible way
upon the above disparate images?

Wednesday, December 25, 2013

Rotating the Facets

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 pm

Previous post—

“… her mind rotated the facts….”

Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:

IMAGE- Count rotational symmetries by rotating facets. Illustrated with 'Plato's Dice.'

“I’ve heard of affairs that are strictly Platonic”

— Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

 See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc.  31 (1929),
No. 1, 25–42.

 On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., three-dimensional cubes) to count
rotational symmetries —

See also Diamond Theory in 1937.

Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .

Sunday, July 7, 2013

Sunday School

Filed under: General,Geometry — m759 @ 7:59 am

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

IMAGE- Harvard senior Jeremy Booher in 2010 discusses Carmichael's 1931 construction of S(5, 8, 24) without mentioning Carmichael.

Saturday, October 8, 2011

An Ordinary Evening in Hartford

Filed under: General — Tags: , — m759 @ 7:59 am

From Rebecca Goldstein's Talks and Appearances page—

• "36 (Bad) Arguments for the Existence of God,"
   Annual Meeting of the Freedom from Religion Foundation,
   Marriot, Hartford, CT, Oct 7 [2011], 7 PM

From Wallace Stevens—

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

For those who prefer greater depth on Yom Kippur, yesterday's cinematic link suggests…

"Yo sé de un laberinto griego que es una línea única, recta."
 —Borges, "La Muerte y la Brújula " ("Death and the Compass")

See also Alpha and Omega (Sept. 18, 2011) and some context from 1931.

Sunday, September 18, 2011

Anatomy of a Cube

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 pm

R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—

"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955

http://www.log24.com/log/pix11B/110918-SprottAndCube.jpg

The figure by Cullinane included above shows a way to visualize Sprott's remarks.

For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."

Thursday, September 1, 2011

How It Works

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

“Design is how it works.” — Steven Jobs (See Symmetry and Design.)

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher’s thesis.  A book he does  cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937.  It should  be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010—

“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”

– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo  XXXIX)

See also “Do you like apples?“

Saturday, February 20, 2010

The Mathieu Relativity Problem

Filed under: General,Geometry — m759 @ 10:10 am

Weyl on what he calls the relativity problem—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, 1946, The Classical Groups, Princeton University Press, p. 16

Twenty-four years ago a note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M24.  That is, there is no obvious way to apply exactly 24 distinct transformable coordinates (or symbol-strings) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M24.

There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M24.

R.D. Carmichael in 1931 on his construction of the Steiner system S(5,8,24)–

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

— R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: General,Geometry — Tags: , , , — m759 @ 7:20 pm
Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."

— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

— William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.
 

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