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Playing in the Sandreckoner's Box

By Eric W. Weisstein

November 19, 2003--The stomachion of Archimedes is a 14-piece dissection puzzle similar to the more familiar tangrams. It is described in fragmentary manuscripts attributed to Archimedes, the greatest mathematician of ancient times, by Magnus Ausonius (310-395 A.D.).

Archimedes may be known best for his alleged bathtub utterance "Eureka!" ("I have found it!"), but he also invented mathematics that was arguably at least a millennium ahead of its time. Archimedes not only invented the method of exhaustion, a technique that allows the exact computation of areas and volumes using ideas that sound very familiar to students of integral calculus, but also devised a system for representing very large numbers centuries before the place-based Hindu-Arabic system of numerals was introduced to Europe by Fibonacci in his 1202 treatise Liber Abaci.

Archimedes published this system of numerals in has manuscript The Sandreckoner, which was so titled because it attempted to compute the number of grains of sand it would take to fill the universe. Archimedes computed this number to be about 10⁶⁴ (Carroll), which is quite a feat for the third century B.C., even though a more modern estimate gives around 10⁹⁰ (Wolfram 2000).

Sadly and ironically, Archimedes met an untimely demise while again contemplating mathematics and sand. During the Roman siege of Syracuse, Archimedes is said to have single-handedly defended the city by constructing lenses to focus the sun's light on Roman ships and huge cranes to turn them upside down. When the Romans finally broke the siege, Archimedes was killed by a Roman soldier after snapping at him, "Don't disturb my circles," a reference to a geometric figure he had outlined on the sand.

Among his many other writings, Archimedes also described the puzzle he termed the "stomachion" (for reasons that I have not yet been able to determine), which that is variously known as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek , meaning "stomach." (Note that while the Roman Ausonius refers to the figure as the "ostomachion," this appears to be a distortion of the original Greek word.)

As illustrated above, the stomachion puzzle consists of 14 flat tiles of various shapes that can be arranged into a square, with the vertices of pieces occurring at the intersections of the lines of a 12 x 12 square grid. As can be seen, the configuration of the puzzle gives two pairs of duplicated pieces. Like tangrams, the object is to rearrange the pieces to form interesting shapes. Taking the bounding square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24, giving them relative areas 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 7, and 8. Interestingly, as noted by Coffin, it is always the case that all polygons formed by connecting points on a regular square grid must have areas in the ratios of whole numbers.