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Wednesday, December 25, 2013

Rotating the Facets

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

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"… 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 .

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