TITLE: Database of integer edge lengths that can form tetrahedrons
QUESTION [11 upvotes]: Is there a collection of lists of six integer edge lengths that form a tetrahedron?  Is there a computer program for generating such lists?  I need to find approximately thirty such tetrahedral combinations.

REPLY [2 votes]: It seems to me to be of considerable interest to explore the space of 6-tuples that realize tetrahedra in Euclidean 3-space with the tuples (a,b,c,d,e,f) when the points have coordinates with real numbers or more specifically positive integers. Because a set of 6 numbers as edge lengths can be realized by up to 30 incongruent tetrahedra it is indeed important to have a system to identify a sextuple with a particular tetrahedron - a "canonical" version that realizes a given 6-tuple. I have raised the question of finding sets of 6 real numbers (integers) that are such that for i taking on values from 0 to 30 can be realized with exactly i incongruent tetrahedra. See here:
http://www.math.illinois.edu/~ajh/ugresearch/Hildebrand-Calculus-Spring2015-report.pdf though it would be nice to see actual examples of integer 6-tuples that achieve the various values of i. It is also worth noting that there is a simpler determinant (which I refer to as the McCrea determinant) than the Cayley/Menger determinant to find the volume of a tetrahedron, see page 20, of the book  by William McCrea - Analytical Geometry: http://vignette3.wikia.nocookie.net/math/images/0/0d/Analytical_Geometry_of_Three_Dimensions.pdf/revision/latest?cb=20150228044133&path-prefix=ro For teachers it is worth pointing out one has to worry about numerical instability in determinant calculations, so one wants to be certain that the volume calculation for a determinant really yields a positive result.