Problem 113
Check that the numbers of vertices, edges, and faces of a cube are equal respectively to the numbers of faces, edges and vertices of an octahedron.
Problem 115
Prove that the polyhedron whose vertices are centers of faces of a tetrahedron is a tetrahedron again.
Problem 117
Which of the five Platonic solids have a center of symmetry?
Problem 118
Describe all ways to superimpose a regular tetrahedron onto itself by rotations, and show that there are 12 such rotations (including the trivial one).
Problem 119
Show that each of the 12 rotations of a regular tetrahedron permutes the four vertices, and that to different rotations there correspond different permutations of the set of vertices.
Problem 120
How many planes of symmetry does a regular tetrahedron have?
Problem 121
Realize all permutations of the four vertices of a regular tetrahedron by reflections in symmetry planes and rotations.
Problem 122
Prove that an octahedron has as many planes of symmetry and axes of symmetry of each order as a cube does.
Problem 124
Show that a cube has nine planes of symmetry.
Problem 126
Find all axes of symmetry (of any order) of an icosahedron, and show that there are in total 60 ways (including the trivial one) to superimpose the icosahedron onto itself by rotation.
