Foreword

Summary

In a famous address to the 1900 International Congress of Mathematicians, held in Paris, the great mathematician David Hilbert announced a list of 23 unsolved mathematical problems, many of which shaped the subsequent course of mathematics for the 20^th century. It would have been clear at the time that several of the problems concerned issues of profound mainstream mathematical interest. Some of the others may have seemed, then, more like curious mathematical side-issues; yet Hilbert showed a remarkable sensitivity in realising that within such problems were matters of genuine potential, mathematical subtlety and importance.

In this latter category was Problem 18, which raises issues of the filling of space with congruent shapes. Among other matters (such as the Kepler conjecture concerning the close-packing of spheres) was the question of whether there exists a polyhedron which tiles Euclidean 3-space, but only in a way that it is not the fundamental domain of any space group—that is to say, must every tiling by that polyhedron be necessarily isohedral, which would mean that every instance of the polyhedron is obtainable from every other, through a Euclidean motion of the entire tiling pattern into itself (i.e., all polyhedra in the tiling would thereby be on an ‘equal footing’ with respect to the pattern as a whole). Such shapes which tile space, but only in ways that are not isohedral, are now known as anisohedal prototiles (where the word ‘prototile’ simply means a tile shape, in current terminology).