Math 445 Assignment Wed 1/30

Read these Facts about 3-space to help with the problems.

Symmetries and Cross-Sections of the simplest Regular Polyhedra

Tetrahedron

  1. Tell what are the planes of symmetry of a regular tetrahedron ABCD (and how many are there?)  Note that the half of ABCD on one side of such a plane is also a tetrahedron (but not regular).
  2. Construct an equilateral triangle ABC that is to be a face of a regular tetrahedron ABCD.  Then construct 3 other triangles so that the 4 triangle fit together to form the half of a regular tetrahedron described in 1.  In particular, if the side of ABC is s, what are the dimensions of the other triangles.
  3. Use 2 to prove what is the altitude of a regular tetrahedron resting on a base ABC.
  4. Prove that the altitudes AA', BB', CC', DD' of a regular tetrahedron ABCD are concurrent at a point M and show what is the ratio AM/AA'. Hint – use 2.
  5. Tell how to cut a tetrahedron by a plane so that the cross-section is a square.

Cube

  1. Given a cube, explain how you can choose 4 of the 8 vertices so that these 4 vertices form a regular tetrahedron.  Explain the relationship between the lengths of the edges of the cube and the length of the edges of the tetrahedron.
  2. Build a model of a cube that "boxes" your tetrahedron as in 1 i.e., the 4 vertices of the tetrahedron should be vertices of the cube.
  3. Tell what are the planes of symmetry of a cube.  How many are there?
  4. Tell how to cut a cube by a plane so that the cross-section is a regular hexagon.