Geodesic
Enumeration of integral tetrahedra
Journal of integer sequences, Tome 10 (2007) no. 9
We determine the number of integral tetrahedra with diameter $d$, up to isomorphism, for all $d \le 1000$, via computer enumeration. We give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^{5})$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula.
Classification : 33F05, 05A15
Keywords: implicit enumeration, integral tetrahedra, geometric probability, Euclidean metric, orderly generation, canonicity check 
@article{JIS_2007__10_9_a7,
     author = {Kurz,  Sascha},
     title = {Enumeration of integral tetrahedra},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {9},
     zbl = {1135.52003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_9_a7/}
}
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Kurz,  Sascha. Enumeration of integral tetrahedra. Journal of integer sequences, Tome 10 (2007) no. 9. http://geodesic.mathdoc.fr/item/JIS_2007__10_9_a7/