ABSTRACT
The idea that space is flat, tenacious though it is, is more easily overcome than the idea that space is three-dimensional. The result is the four-dimensional arithmetic of quaternions. This system has most, though not all, the properties of real and complex numbers. It is called four-dimensional simply because quaternions have four coordinates, and one need not try to visualize the four perpendicular coordinate axes. In the first place, quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. William Rowan Hamilton's dream of three-dimensional numbers was indeed impossible, but the reality turned out to be more interesting. The known systems of numbers are exceptional structures, existing only in one and two dimensions, and the system of quaternions is even more exceptional. The regular polytopes in all dimensions were found by the Swiss mathematician Ludwig Schlafli in 1852.
