4-D and higher dimensional analogues of A065984

Jonathan Post jvospost3 at gmail.com
Mon Mar 12 07:19:49 CET 2007

Higher dimensional analogues of A065984  Volume (multiplied by 3) of
polyhedron formed by points (i,j,k) in Z^3 with i^2+j^2+k^2 = n.

These are relatively straightforward.  For example, in dimension 4,
a(n) = Volume (multiplied by 6) of polytope formed by points (h,i,j,k)
in Z^4 with h^2+i^2+j^2+k^2 = n.  Except for n = 0, these are always
nonzero, since each integer is the sum of four integer squares.  The
normalization factor of 6 is because the cross-polytope
(hyperoctahderon) such as the  4-D convex hull of (+-1,+-1,+-1,+-1)
as we need to calculate for a(4) has a factor of 6 needed since 1^2 +
1^2 + 1^2 + 1^2 = 4 has (assuming unit edge) hypervolume 1/6  and
"dichoral angles" (angles between regular polyhedra sharing a face)

For hyperoctahedra of dimension D in general, with unit edge,
D-hypervolume = (2^(D/2))/D! so we must have the D! normalization
factor for all values to be integers, just as we have the factor 3 for
3-D and 6 for 4-D.

Does Mathematica allow something like:

 polytopes = Flatten[ forms/@SumOfSquaresRepresentations[ 4, # ], 1
]&/@(Range[ 1, 36 ]); HullHyperVolume[ #, ConvexHull4D[ # ]

or is there a better way to calculate a(n) for the 4-D case to begin with?

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