Integer Tetrahedra

Question

The points $\{(0, 0, 0), (12, 27, 44), (48, 0, 20), (48, 0, -64)\}$

have the property that

1. All vertices are on the integer grid,
2. All edge lengths are integers and different $\{51, 52, 53, 80, 84, 117\}$,
3. All face areas are integers and different $\{1170, 1800, 1890, 2016\}$

This is the only one I've found with these properties. Can someone find more examples that aren't similar to this tetrahedron?

Answer 1

Check out Heronian Tetrahedra are Lattice Tetrahedra, Marshall and Perlis, 2013, The American Mathematical Monthly, 120(2), 140-149.

Heronian tetrahedra satisfy the weaker constraints that all of their edge lengths and face areas are integral, plus the additional constraint that the volume is integral. Marshall and Perlis prove that they can all be positioned to satisfy your first constraint, so all Heronian tetrahedra which have distinct edge lengths and face areas will satisfy your three constraints.

The proof given is constructive, and they illustrate it with an example: edges $\{612, 480, 156, 185, 319, 455\}$ give vertices $(0, 0, 0)$, $(36, -432, 432)$, $(36, -144, 48)$, $(176, -264, 33)$. The face areas are $\{9570, 22464, 42000, 70686\}$.

Various other examples of Heronian tetrahedra are given in the corresponding MathWorld page. I haven't checked how many of them have distinct face areas.

Answer 2

There are many such tetrahedra. First note that the third follows from the first by the standard formula for the area (well, almost, since there is a factor $\dfrac 12$ but this cancels if enough coordinates are even). One can then consider terahedra with vertices of the form $(0,0,0)$, $(m,0,0)$, $(0,n,0)$ and $(0,0,p)$ for suitable integers $m$, $n$ and $p$ to get many examples.