# volumes of integer polytopes n==i^2+j^2+k^2

vdmcc w.meeussen.vdmcc at vandemoortele.be
Thu May 6 15:58:33 CEST 1999

```hi,

as you know, the volume of a polytope is a sum of 1/6 of  v1.(v2 x v3)
with v1, v2 and v3 the vectors of vertices bounding the facets.
Since I'm into integer points on spherical shells, I calculated the
polytopes
defined by i^2+j^2+k^2 == n for n=1 .. 64 and got their volumes.
It turns out I never get a volume of Integer/6, just Integer/3.
I haven't prove that, although it feels easy?

the couples { n , volume of polytope i^2+j^2+k^2 == n } are:

{  {0,0},{1,4/3},{2,20/3},{3,8},{4,32/3},{5,32},{6,136/3},{7,0},{8,160/3},{9
,
272/3},{10,244/3},{11,344/3},{12,64},{13,404/3},{14,172},{15,0},{16,

256/3},{17,712/3},{18,248},{19,616/3},{20,256},{21,988/3},{22,644/3},{23,
0},{24,1088/3},{25,1156/3},{26,1484/3},{27,424},{28,0},{29,580},{30,

1628/3},{31,0},{32,1280/3},{33,664},{34,2092/3},{35,696},{36,2176/3},{37,
1360/3},{38,2608/3},{39,0},{40,1952/3},{41,3032/3},{42,2476/3},{43,
2408/3},{44,2752/3},{45,3344/3},{46,3136/3},{47,0},{48,512},{49,
3692/3},{50,1316},{51,3848/3},{52,3232/3},{53,4208/3},{54,1516},{55,0},{
56,1376},{57,4180/3},{58,3676/3},{59,5032/3},{60,0},{61,4528/3},{62,
1740},{63,0},{64,2048/3}}

triple volumes :
{0,4,20,24,32,96,136,0,160,272,244,344,192,404,516,0,256,712,744,616,768,988
,

644,0,1088,1156,1484,1272,0,1740,1628,0,1280,1992,2092,2088,2176,1360,2608,

0,1952,3032,2476,2408,2752,3344,3136,0,1536,3692,3948,3848,3232,4208,4548,0,
4128,4180,3676,5032,0,4528,5220,0,2048}

I am sorry, but the terms
4,20,24,32,96,136
do not match anything in the table

the polytopes with zero volume have n= {0,7,15,23,28,31,39,47,55,60,63} =
A004215
lower bound=4 n^(3/2)
upper bound=4 n^(3/2) Pi
-------------------------------------------------------------------------
for my archive:
Do[
n=Ceiling[Sqrt at sumofsq];
it=Flatten[
Table[If[x^2+y^2+z^2===sumofsq,{x,y,z},{}],{x,0,n},{y,x,n},{z,
Max[x,y],n}] ,2]~DeleteCases~{};
If[it==={},AppendTo[vols,{sumofsq,0}],vec=-1+2*IntegerDigits[Range[0,7],2,3];
p=Flatten[
Permutations/@Flatten[Outer[Times,it,vec,1],1],1];
p=Union[p];ch=ConvexHull3D[p]; AppendTo[vols,{sumofsq,HullVolume[p,ch]}];
HullShow[p,ch]],{sumofsq,37,48}]
vols

w.meeussen.vdmcc at vandemoortele.be
tel  +32 (0) 51 33 21 11
fax +32 (0) 51 33 21 75

```