RE: CPS

Kami Rousseau wrote,

>This means that the CPS are a generalisation of the Euler Fokker
>generas.

That's one way of looking at it. Judging, however, from the preponderance of
hexanies, eikosanies, and hebdekotamies (whatever), I would venture that
having repeated factors goes againt the purest spirit of CPS.

However, there is another connection between Euler-Fokker genera and CPS.
The union of all

CPS i/(k+1) (1,a(1),a(2), . . . ,a(k)),

where the a's are mutually prime and i ranges from 1 to k, is

Eu (a(1),a(2), . . . ,a(k)).

In k-dimensional space, the "interior" of the Euler-Fokker genus is broken
into non-overlapping regions, the "interiors" of the CPSs. That is because
the only intersections in the above CPSs are between CPS iand CPS i1,
namely,

CPS (j+1)/k (a(1),a(2), . . . ,a(k)),

whic are mere "borders" of dimension k-1.

Now it is customary to view these Euler-Fokker genera as cubes, hypercubes,
etc., but if we view the CPSs as regular polygons, polyhedra, etc. (by
making the axes at 60-degree angles from each other), we obtain fascinating
results about regular tilings of k-dimensional space, since obliquely
stretched cubes tile just as well as cubes! Try it!


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