By A B SosinskiiМ†

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The 12 two-sided ones are shown on the next page. We will not prove the second part of the classification theorem for regular plane tilings (it consists in finding the remaining 12 regular tilings, for which two-sided tiles are required). However, the reader can study some examples of these 12 tilings by doing some of the exercises. htm One can also visit the Escher website. 5. The 230 crystallographic groups The crystallographic groups are the analogs in R3 of the tiling groups in Euclidean space R2 .

The theorem is proved. 7. Let us denote by Dn the subgroup of SO(3) generated by the elements of Dn and the reflection ρ in the plane passing through the rotation axis of order n and one of the axes of order 2 in Dn . Obviously the subgroup Dn has 4n elements (because the compositions of ρ with different elements of Dn are all different from each other). Note also that the subgroup of SO(3) generated by the elements of Zn (interpreted as the motion group of the regular n-gon lying in the equatorial plane of the sphere and inscribed in it) and the reflection in a vertical plane passing through a vertex of the n-gon and the center of the sphere is Dn .

Iii) for g, h ∈ G the images g(T0 ), h(T0 ) of the fundamental tile coincide if and only if g = h. 37-38 of the French edition). The action of a tiling group G ⊂Sym(R2 ) on the plane R2 is, of course, a geometry in the sense of Klein that we call the tiling geometry (or Fedorov geometry) of the group G. 2. 4. ? ? ? ? ?? ?? ? ?? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?