Log24

Monday, May 7, 2012

More on Triality

Filed under: General,Geometry — Tags: — m759 @ 4:20 PM

John Baez wrote in 1996 ("Week 91") that

"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."

Baez seems to come close to doing this with a
somewhat different i , j , and kHurwitz
quaternions
— in his 2005 book review
quoted here yesterday.

See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

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