Euler Bricks and Perfect Polyhedra

There's something about integers that makes them perfectly irresistible to many mathematicians, both amateur and professional.

Number theorists have the advantage that they can indulge their pleasure without feeling overly guilty, whether it's in the connection between Fermat's last theorem and elliptic curves or the link between random matrices and the distribution of prime numbers (see "The Mark of Zeta").

The wide use of computers has also brought attention to the realm of the discrete. Indeed, that trend was noted more than 50 years ago. In his 1963 book Combinatorial Mathematics, H.J. Ryser remarked, "Our new technology with its vital concern with the discrete has given the recreational mathematics of the past a new seriousness of purpose."

The recreational aspect is alive and well, as seen, for example, in the continuing fascination with magic squares, magic cubes, and magic tesseracts.

In a 1999 article titled "Integer Antiprisms and Integer Octahedra," published in Mathematics Magazine, Blake E. Peterson and James H. Jordan drew attention to perfect boxes and polyhedra.

Their starting point was the problem of finding a rectangular box with integer dimensions and all diagonals of integer length. Such a figure is known as a perfect box. Whether it exists is an unsolved problem.

Leonhard Euler (1707-1783) described the smallest solution for the special case when the sides and face diagonals are all integers, but not the space diagonal passing through the box's center from one corner to its opposite. (Though Euler is often credited with its discovery, the German mathematician Paul Halcke mentioned this solution first in 1719.)

An Euler brick has integer dimensions.

Euler's "almost" perfect brick has the following dimensions: a = 240, b = 117, and c = 44. The face diagonals are 244, 125, and 267. The space diagonal is 5 times the square root of 2929.

Peterson and Jordan focused on other three-dimensional figures with integer edges and diagonals, particularly pyramids and prisms.

You can construct a pyramid by drawing a polygon (to serve as the base), then joining each vertex of the polygon to a point not in the plane of the polygon. A triangular pyramid, or tetrahedron, has a triangular base and four faces, counting the bottom.

An integer polyhedron is one in which the distance between each pair of vertices is an integer. Because the faces of an integer polyhedron must themselves be integer polygons, it's natural to use integer polygons as the building blocks of integer polyhedra, Peterson and Jordan remarked. Octahedral pyramids are a good starting point.

An octahedron has eight faces. In its most familiar form as one of the Platonic solids, each face is an equilateral triangle.

A regular octahedron, in which each face is an equilateral triangle.

An octahedral pyramid has a seven-sided heptagon as its base.

An octahedral pyramid with a heptagonal base.

To get an integer heptagon, adjacent vertices of the heptagon must lie on a circle and be separated by the following distances: 10, 16, 16, 10, 16, 16, and 16. In this case, all of the points lying along a line through this circle's center and perpendicular to the plane of the heptagon are equidistant from the polygon's vertices. You can then choose the lateral edges of the pyramid to be 17.

You can readily extend the same approach to other polyhedra, such as prisms and antiprisms. An antiprism consists of two identical polygons in parallel planes joined in such a way that all the other faces are isosceles triangles.

Example of an antiprism.

Peterson and Jordan went on to investigate interesting links between integer octahedra and integer antiprisms.

Originally posted October 25, 1999