# 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

```