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, June 21, 1999).

The wide use of computers has also brought attention to the realm of the discrete. Indeed, that trend was noted more than 30 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 serousness of purpose."

The recreational aspect is alive and well, as seen in the continuing fascination with magic squares, cubes, and tesseracts (see Magic Tesseracts, October 18, 1999).

In the October Mathematics Magazine, Blake E. Peterson of Brigham Young University in Provo, Utah, and James H. Jordan of Washington State University in Pullman draw attention to perfect boxes and polyhedra.

Their starting point is 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 boxÕs 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.)

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 focus 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 remark. 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.

An octahedral pyramid has a seven-sided heptagon as its base. To get an integer heptagon, adjacent vertices of the heptagon must be 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.

Peterson and Jordan go on to investigate interesting links between integer octahedra and integer antiprisms. There's no end to problems having to do with integers!

References

: Gardner, M. 1989. Magic stars and polyhedrons. In Mathematical Carnival. Washington, D.C.: Mathematical Association of America.

Peterson, B.E., and J.J. Jordan. 1999. Integer antiprisms and integer octahedra. Mathematics Magazine 72 (October):277.

______. 1995. Integer hexahedra equivalent to perfect boxes. American Mathematical Monthly 102 (January):41.

Ryser, H.J. 1963. Combinatorial Mathematics. Washington, D.C.: Mathematical Association of America.

You can learn more about Euler bricks at http://www.treasure-troves.com/math/EulerBrick.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.