E The third generator E is distinguished - we will vary it to get the 480 multiplications. Start with a 3-dimensional tetrahedron of 4 facing mirrors. To get another, truncate the tetrahedron .
Truncation puts a mirror parallel to the floor, making a mirror roof.
For an exact icosahedral kaleidoscope, three of the triangles of the tetrahedron should be Golden isosceles triangles . The inexactness may be related to the aperiodicity of Penrose-Wang tilings of the plane. K ent Palmer has pointed out that the binary icosahedral group is isomorphic to the binary symmetry group of the 4-simplex, which he calls the pentahedron and which David Finkelstein and Ernesto Rodriguez have called the Quantum Pentacle . The pentahedron projected into 3 dimensions looks like a tetrahedron divided into 4 quarter-tetrahedra. A Dirichlet domain of a vertex in a packing is the set of points in the space in which the packing is embedded that are nearer to the given vertex than to any other.