Drawing Cubes, Finding Integral Cubes, and Solving Cubics

Abstract

Summary We present some variations on the theme of “cubes”. We start with right and wrong ways to draw cubes and orthogonal axes. From there we classify cubes in 3-space with integral coordinates. Further, we find that four complex numbers are the vertices of a drawing of a regular tetrahedron if their average of squares equals the square of their average. Tangentially, we find short proofs of the Siebeck-Marden theorem and a method for solving a cubic equation. Finally, we consider sets of complex numbers whose average of squares equals the square of their average.