There is one
outstanding genius of the past who is still almost unknown. His name is Ludwig
Schlaefli and his achievement, unrecognised during his lifetime, defies belief.
He worked out the number of possible regular shapes which could exist in four
dimensions. That may not sound much, particularly if you start by considering
the equivalent problem in, say, two dimensions.
On a flat,
two-dimensional piece of paper, it is obvious that you can draw regular shapes
with three, four, five, any number of sides you like, triangles, squares,
pentagons, hexagons and so on, right up to infinity-gons. You might think the
same would be true in three dimensions. It isn't. As Lewis Carroll pointed out,
in 3D the regular shapes are "provokingly few in number", only five
of them. Plato and the ancient Greeks knew this and we call them the Platonic
solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron with respectively
4, 6, 8, 12 and 20 faces). You can't create any more, however hard you try.
It's fairly easy to
think in three dimensions and to convince yourself that the five Platonic
solids are the only ones possible. It's much harder to think in four
dimensions. In the 1800s, many very distinguished mathematicians tried to do
just that and to show which regular 4D shapes existed and what they looked
like. Schläfli, with a truly astonishing and genius-level feat of mental
gymnastics, beat the lot of them to it. Yet he died with this amazing
achievement completely unrecognised.
See the comment below for a brief biography.
