I am looking for someone to discuss irregular pyramids. I have developed formulas and methods for finding solutions given only the length of edges.
For an irregular tetrahedron, six lengths assigned to specific edges gives up to one possible solution -- formulas for this have been around as early as Piero della Francesca's solution.
For an irregular quadrilateral pyramid, eight lengths assigned to specific edges gives up to four possible solutions. I have developed a formula for this (and don't know of any others). As an example of my "research," the series (14,15,16,17,18,19,20,21) is the smallest series to give a solution for every permutation (5040). Solutions include convex, concave and complex (bow-tie) constructions.
For an irregular hexagonal pyramid, eight lengths assigned to specific edges can give many solutions. This is what I'm working on now -- finding solutions, graphing and mapping the results. Unfortunately, with 12 edges, the permutations are too large (11! = nearly 40 million) to do an extensive study of any series.
If anyone has any interest in discussion, please feel free to respond.