The three dimensional Pareto front of the ammonoid dataset. (A) The RMS error for PCHA optimal polygons and polyhedra is computed for different numbers of possible vertices: line, triangle, tetrahedron, 5-vertex polyhedron, etc. Error decreases with increasing the number of archetypes up to 5. Increasing the number beyond 5 doesn't improve the fit by much (for more evidence for the pyramidal shape of the data, see Additional file 1). (B-D) The best fit 5-archetype polygon resembles a square pyramid. Blue points denote FF to DM ammonoids, red are DM to PT and green are post PT ammonoids. Archetypes are numbered, their morphology is shown, and the suggested tasks are listed in panel A.