Equable Shape - Integer Dimensions

Integer Dimensions

Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10).

More generally, the problem of finding all equable triangles with integer sides (that is, equable Heronian triangles) was considered by B. Yates in 1858. As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).

The only equable rectangles with integer sides are the 4 × 4 square and the 3 × 6 rectangle. An integer rectangle is a special type of polyomino, and more generally there exist polyominoes with equal area and perimeter for any even integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.