Friday, November 18, 2011

Hypercube Rotations

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

The hypercube has 192 rotational symmetries.
Its full symmetry group, including reflections,
is of order 384.

See (for instance) Coxeter


Related material—

The rotational symmetry groups of the Platonic solids
(from April 25, 2011)—

Platonic solids' symmetry groups

— and the figure in yesterday evening's post on the hypercube


(Animation source: MIQEL.com)

Clearly hypercube rotations of this sort carry any
of the eight 3D subcubes to the central subcube
of a central projection of the hypercube


The 24 rotational symmeties of that subcube induce
24 rigid rotations of the entire hypercube. Hence,
as in the logic of the Platonic symmetry groups
illustrated above, the hypercube has 8 × 24 = 192
rotational symmetries.

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 .

Saturday, November 19, 2011

Star Wars (continued)

Filed under: General — m759 @ 1:14 PM

From Thursday's post All Things Fashion


From today's online New York Times


The nuclear symbol beneath the op-ed headline
is the most interesting part of this afternoon's front page—


Jung on projections

It is possible to project certain characteristics onto another person who does not possess them at all, but the one being projected upon may unconsciously encourage it.

"It frequently happens that the object offers a hook to the projection, and even lures it out. This is generally the case when the object himself (or herself) is not conscious of the quality in question: in that way it works directly upon the unconscious of the projicient. For all projections provoke counter-projections  when the object is unconscious of the quality projected upon it by the subject." ["General Aspects of Dream Psychology," CW 8, par. 519.]

For an object that "offers a hook to the projection," see yesterday's Hypercube Rotations


Central projection
of the hypercube

See also Stallion Gate  in this journal.

Saturday, January 4, 2014

For Phil Everly

Filed under: General — m759 @ 10:00 AM

A Souther song at YouTube.

See also the lyrics and, in this journal,
synchronicity on the uploading date.

Related art —

End of the Line Blues

IMAGE- YouTube statistics: 384 views, 3 thumbs up, 0 down.

and The Crosswicks Curse

IMAGE- The full group of hypercube symmetries, including both rotations and reflections, is of order 384.

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