  An infinite type toric variety is a normal toric variety given by a fan with
infinitely many cones. We construct examples in this paper coming from
representation theory of loop groups. The fans that appear are cones on Voronoi
tilings on a vector space equipped with an inner product. We also explain the
affine analogue of the connection between a generic torus orbit closure in a
flag variety and the closure a maximal torus in the wonderful compactification.
