Stomachion

Stomachion

The stomachion, also known as the loculus or syntemachion, is a square dissected into fourteen pieces. The area of the individual elements is in rational proportion to the area of the square. Players can form as many shapes as they like with the stomachion. In doing so, all the pieces must be used without any overlapping.

Stomachion – history

The stomachion can be traced back to Archimedes, who is supposed to have set the task of dissecting the square into fourteen pieces and also solved it himself. The manuscript attributed to Archimedes in which the stomachion is described was discovered in 1899 in an Arabic translation and in 1906 in a Greek palimpsest (a parchment that has been erased and rewritten) from the tenth century.

Stomachion – mathematics

If the stomachion is placed on a grid of 12x12 squares, all the corners of the fragments lie on these grid points, which makes it easy to comprehend that the areas of the individual fragments are in rational proportion to the area of the square. Working out the areas is a nice application of George Pick's theorem to calculate the area of a polygon drawn on a grid (1899). 

In 2003 the American mathematician William Cutler used a computer programme to illustrate that there are 536 possibilities in all to assemble the fourteen pieces into a square. Patterns produced by rotating or reflecting another pattern or interchanging congruent pieces are counted only once.