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Pick’s Theorem

This description of the geometric construction of the Stomachion is somewhat different than that given in the Arabic manuscript, but results in the same figure.

Start with a square containing a uniform 12 x12 grid. The intersections of the grid lines are called lattice points. Take each small square to have unit area, so that the original square has area 144.

Draw lines through the indicated lattice points. The lines divide the square into 14 three-, four-, and five-sided polygons. These polygons are called lattice polygons because their vertices are at lattice points. The lattice polygons form the 14 pieces of the Stomachion.

The Arabic manuscript also contains computations for the areas of the pieces of the Stomachion. It shows that the areas of the pieces are all commensurate with the area of the square in the ratios 1:48 (2 pieces of area 3), 1:24 (4 pieces of area 6), 1:16 (1 piece of area 9), 1:12 (5 pieces of area 12), 7:48 (1 piece of area 21), and 1:6 (1 piece of area 24).

A contemporary approach to the computations of the areas is by the use of Pick’s theorem.