4-D and higher dimensional analogues of A065984
Jonathan Post
jvospost3 at gmail.com
Thu Mar 15 20:14:06 CET 2007
The messy way is to pentatopalize (4-D version of 2-D trinagularize
and 3-D tetrahedralize) the convex hull into 4-simplexes, and then
calculate the hypervolume of each pentatope by the Cayley-Menger
determinant (generalization of 2-D Heron's formula). This invokes the
4th coefficient in -1, 2, -16, 288, -9216, 460800, ... (Sloane's
A055546).
If V_4 is the 4-D hypervolume of the general 4-simplex, then:
(V_4)^2 = ((-1)^4)/((2^4)(4!)^2) det B where B is the 6x6 matrix
determined by d(i,j) = the L2-norms of edges between vertex i and
vertex j.
B =
0 1 1 1 1 1
1 0 (d_12)^2 (d_13)^2 (d_14)^2 (d_15)^2
1 (d_21)^2 0 (d_23)^2 (d_24)^2 (d_25)^2
1 (d_31)^2 (d_32)^2 0 (d_34)^2 (d_35)^2
1 (d_41)^2 (d_42)^2 (d_43)^2 0 (d_45)^2
1 (d_51)^2 (d_52)^2 (d_53)^2 (d_54)^2 0
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