Point groups in three dimensions the finite subgroups of SO(3), are: the cyclic groups "C" (the rotation group of a canonical pyramid), the dihedral groups "D" (the rotation group of a uniform prism, or canonical bipyramid), and the rotation groups "T", "O" and "I" of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron. In particular, the dihedral groups "D", "D" etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore, it is also called a "dihedron" (Greek: solid with two faces), which explains the name "dihedral group". The rotation group of
Smooth projective plane and if formula_30, then formula_7 is the classical real plane formula_1". All compact planes formula_7 with a 4-dimensional point space and formula_36 have been classified. Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane. According to , this shift plane is not smooth. Hence, the result on translation planes implies: Theorem. "A smooth 4-dimensional plane is isomorphic to the classical complex plane, or formula_37". Compact 8-dimensional "topological" planes formula_7 have been discussed in and, more recently, in . Put formula_39. Either formula_7 is the classical quaternion plane or formula_41. If formula_42,