All 17152 solutions to the Stomachion puzzle can be understood as a collection of 268 solutions in 64 symmetries. Of those 268 basic squares 24 are made from the same 4 triangular subsets of pieces but in different arrangements. They form a core set connecting 24 families of closely related squares. The study of these relationships is called graph theory in mathematics and provides surprising insights into the nature of the Stomachion.
In the Arabic version of Archimedes' writings, the first instructions for the construction of the puzzle are to divide the square in two rectangles and each of those in two triangles. This gives four identical rectangular triangles, each of which is subdivided in a different way to produce the final fourteen pieces. But disregarding the subdivisions, the four triangular groups of pieces can be rearranged among themselves in 24 arrangements, forming a core set. This number can be found empirically or perhaps by reasoning: the first can go in four positions, this leaves three options for the second and only two for the third, after which there is only one place left for the last segment, and 4 x 3 x 2 x 1 = 24. Each then evolves independently resulting in the 24 families. Some core arrangements offer more possibilities for further development than others, making some families larger than others.
The 24 core squares are those that consist of these four triangles, aptly named 1, 2, 3 and 4:
The core squares appear as numbers 2, 13, 16, 27, 30, 41, 44, 55, 70, 71, 98, 99, 122, 131, 142, 171, 178, 205, 216, 217, 236, 237, 256 and 257 in the series of named solutions for the puzzle. Core triangle 4 can mutate internally into four different configurations without changing its external shape. This explains the 96 squares containing exactly three of the core triangles. There are 50 squares with two triangles, and 81 with one. Only 17 squares (61, 62, 79, 80, 89, 90, 105, 106, 116, 117, 136, 137, 148, 149, 162, 163 and 164) contain none of these triangles.
The order in which the core squares have been arranged by default, forms a Hamiltonian path, allowing a trip along all members without missing out on any points or visiting any square more than once, and exchanging exactly two triangles at every step by mirroring a rectangle or a parallellogram. Drag the squares around to compare them one on one, or arrange them in one, two or three dimensions on a straight or circular line, in a wide or tall grid or in cyberspace, then step/back through the metamorphoses chain or play/stop an animation.