  Retaining the combinatorial Euclidean structure of a regular icosahedron,
namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the
12 different vertices together with the incidence structure, we investigate
variations of the regular icosahedron admitting self-intersections of faces. We
determine all rigid equivalence classes of these icosahedra with non-trivial
automorphism group and find one curve of flexible icosahedra. Visualisations
and explicit data for this paper are available under
http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html.
