OFFSET
1,1
COMMENTS
A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.
Primitive tetrahedron means 6 sides don't share a common factor.
EXAMPLE
see A272388
MATHEMATICA
aMax=360(*WARNING:takes a long time*);
heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
cayley=1/24Sqrt[2Det[{
{0, 1, 1, 1, 1},
{1, 0, #1^2, #2^2, #6^2},
{1, #1^2, 0, #3^2, #5^2},
{1, #2^2, #3^2, 0, #4^2},
{1, #6^2, #5^2, #4^2, 0}
}]]&;
Do[
S1=heron[a, b, c];
If[S1//IntegerQ//Not, Continue[]];
Do[
S2=heron[a, e, f];
If[S2//IntegerQ//Not, Continue[]];
Do[
If[GCD[a, b, c, d, e, f] > 1, Continue[]];
If[b==e&&c>f||b==f&&c>e, Continue[]];
S3=heron[b, d, f];
If[S3//IntegerQ//Not, Continue[]];
S4=heron[c, d, e];
If[S4//IntegerQ//Not, Continue[]];
V=cayley[a, b, c, d, e, f];
If[V//IntegerQ//Not, Continue[]];
If[V==0, Continue[]];
a//Sow(*{a, b, c, d, e, f, S1, S2, S3, S4, V}//Sow*);
, {d, Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1-S2)/a)^2]//Ceiling, Min[a, Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1+S2)/a)^2]]}];
, {e, a-b+1, b}, {f, a-e+1, b}];
, {a, 117, aMax}, {b, a/2//Ceiling, a}, {c, a-b+1, b}]//Reap//Last//Last
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Albert Lau, May 21 2016
STATUS
approved