Packing in two, three and more dimensions
How many equilateral triangles can be packed into the plane so that they all touch each other at a corner point without overlapping? This question is very easy to answer.
The same question can be asked for higher dimensions. Then, for example, in three-dimensional space the plane, equilateral triangles have to be replaced by spatial, equilateral tetrahedra. The medieval scholars Averroes (1126-1198) and Albertus Magnus (around 1200-1280) raised and studied this question when they commented on the writings of Aristotle. Their simple, geometric proposals for solutions initially seemed plausible at the time, but later proved to be wrong. In fact, the question is still unsolved. This tetrahedron packing problem is one of many geometric packing problems whose solution is not known, but which occur in mathematics and its applications in a very wide variety of ways. My field of work includes the search for mathematically correct approaches using modern techniques of mathematical optimization. These often require an immense
computational effort and can only be mastered with computer assistance.