Section 10.3 Tiling the Klein Disk
¶The only difference between tiling the sphere and the Klein disk is that we must identify antipodal points of the sphere. All but one of the tilings of the sphere discussed in Section 10.2 have a reflection symmetry that allows them to be cut in half and used as a tiling of the Klein disk. The lone exception is the tetrahedron, which has no projective analog. The corresponding “hemi-polyhedra” are shown in Figure 10.3.1.
As on the sphere, there are also two infinite families of projective polyhedra, corresponding to the dihedron and hosohedron considered in Section 10.2.
Finally, we can use stereographic projection to convert the formulas given in Section 10.2 for the elliptic “radii” of the polyhedra to Euclidean distances from the center of the Klein disk, using the formula \(R=\tan\frac\theta2\) from Figure 7.2.1. This formula is useful when constructing models such as those shown in Figure 10.3.1.