The origins of most of the tiling problems we will explore go back just a short time. But Minkowski's problem, like many ideas in mathematics, can trace its roots to the Pythagorean theorem, a2 + b2 = c2. We shall trace these roots, and then go on to describe some of the problems that Minkowski's question, in turn, suggested. We do this in order to place the problem in its proper position in the intricate web that constitutes mathematics.

1. Introduction

The Greeks discovered all integer solutions of the equation x2 + y2 = z2: pick three integers d, m, and n, and let one of x and y be d(2mn) and the other be d(m2 − n2), and z = d(m2 + n2). Inspired by this, Diophantos, who lived some time between A.D. 150 and A.D. 350, described techniques for producing rational solutions of such equations as x2 + y2 = z3, x3 + y3 = z2, and x4 + y4 = z3. He does not refer to the equation x3 + y3 = z3, whose first recorded mention is in a letter written about the year A.D. 1000 by the mathematician Abu Djafar Mohammed Ben Alhocain [22]: