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.
Read more about this topic: Equable Shape
Famous quotes containing the word dimensions:
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)