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 4 edge lengths share no common factor.
Properties:
1. 3 pairs of opposite edge lengths are equal.
2. The perimeter must be an even number.
3. The faces are acute triangles, and cannot be isosceles triangle.
It is known that 5512,8484,11275,19695,32708,294175,683787 are in the sequence.
EXAMPLE
Below shows some example: (might contains gap)
a, b, c, S, V
203, 195, 148, 13650, 611520
888, 875, 533, 223860, 37608480
1804, 1479, 1183, 870870, 214582368
2431, 2296, 2175, 2277660, 1403038560
2873, 2748, 1825, 2419950, 1355172000
5512, 5215, 1887, 4919460, 1377448800
8484, 6625, 6409, 20980050, 30546952800
11275, 10136, 8619, 41861820, 103147524480
19695, 16448, 13073, 106675680, 323290060800
32708, 31493, 24525, 363332970, 2685757314240
MATHEMATICA
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}
}]]&;
aMin=203;
aMax=2000(*WARNING:runs very slow*);
Do[
If[GCD[a, b, c]>1, Continue[]];
S=heron[a, b, c];
If[S//IntegerQ//Not, Continue[]];
V=cayley[a, b, c, a, b, c];
If[V//IntegerQ//Not, Continue[]];
a(*{a, b, c, S, V}*)//Sow;
, {a, aMin, aMax}
, {b, a/Sqrt[2]//Ceiling, a-1}
, {c, Mod[a+b, 2, Floor[Sqrt[a^2-b^2]]+1], b-1, 2}
]//Reap//Last//Last(*//TableForm*)
{S, V}=.;
(*
(*this piece of code runs much faster but might contains gap*)
mMax=100;
Do[
{a, b, c}={n(m^2+k^2), m(n^2+k^2), (m+n)(m n-k^2)};
{a, b, c}={a, b, c}/GCD[a, b, c];
V=cayley[a, b, c, a, b, c];
If[V//IntegerQ//Not, Continue[]];
a(*{a, b, c, heron[a, b, c], V}*)//Sow
, {m, mMax}
, {n, m-1}
, {k, Floor[Sqrt[(m^2 n)/(2m+n)]+1], n-1}
]//Reap//Last//Last//Union(*TableForm*)
{a, b, c, V}=.;
*)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Albert Lau, May 26 2016
STATUS
approved