Perfect Parallelepipeds Exist

by Jorge Sawyer and Cliff Reiter

12 Examples Confirmed

len2[X_]:=Factor[X.X]

len[X_]:=Sqrt[len2[X]]

lena[u_,v_,w_]:={{"u",len[u]},{"v",len[v]},{"w",len[w]},
				{"u+v",len[u+v]},{"u-v",len[u-v]},{"u+w",len[u+w]},{"u-w",len[u-w]},{"v+w",len[v+w]},{"v-w",len[v-w]},
{"u+v+w",len[u+v+w]},{"u+v-w",len[u+v-w]},{"u-v+w",len[u-v+w]},{"u-v-w",len[u-v-w]}}

Assemble[x1_,x2_,x3_,d23_,d13_,d12_]:=Block[{rho,c23,c13,c12,u,v,w},
c12=(x1^2+x2^2-d12^2)/(2 x1 x2);
c13=(x1^2+x3^2-d13^2)/(2 x1 x3);
c23=(x2^2+x3^2-d23^2)/(2 x2 x3);
rho=(c23-c12 c13)/(Sqrt[1-c12^2]Sqrt[1-c13^2]);
u=x1{1,0,0};
v=x2{c12,Sqrt[1-c12^2],0};
w=x3{c13,rho Sqrt[1-c13^2],Sqrt[1-rho^2]Sqrt[1-c13^2]};
{u,v,w}]

Example 1

{u,v,w}=Assemble[271,106,103,101,266,255]

lena[u,v,w]

Example 2

{u,v,w}=Assemble[630,365,335,444,595,385]

lena[u,v,w]

Example 3

{u,v,w}=Assemble[647,540,375,285,448,653]

lena[u,v,w]

Example 4

{u,v,w}=Assemble[739,595,342,463,661,774]

lena[u,v,w]

Example 5

{u,v,w}=Assemble[840,630,565,565,959,1050]

lena[u,v,w]

Example 6

{u,v,w}=Assemble[895,686,652,650,973,939]

lena[u,v,w]

Example 7

{u,v,w}=Assemble[925,579,422,431,577,776]

lena[u,v,w]

Example 8

{u,v,w}=Assemble[952,357,340,493,852,875]

lena[u,v,w]

Example 9

{u,v,w}=Assemble[1081,840,623,497,1206,1369]

lena[u,v,w]

Example 10

{u,v,w}=Assemble[1120,1035,840,969,1400,1525]

lena[u,v,w]

Example 11

{u,v,w}=Assemble[1381,1038,665,1087,1314,1583]

lena[u,v,w]

Example 12

{u,v,w}=Assemble[1535,1110,955,1015,1030,799]

lena[u,v,w]