Deltahedra

A deltahedron is a convex polyhedron, each of whose faces is an equilateral triangle. Each pair of adjacent triangles must ``bend outward'' (not be coplanar or bend inward).

1. Try to construct all of the deltahedra and draw a good sketch of each one.
2. How many of the deltahedra have an odd number of faces? An even number of faces? Find a proof for what you observe.
3. Assume that each triangle has a side length of 1 unit. Determine the volumes of the following deltahedra:
   1. Tetrahedron (4 triangles).
   2. Octahedron (8 triangles).
   3. The deltahedron with 10 triangles.
   4. The deltahedron with 14 triangles.
4. You can fit 4 tetrahedra around an octahedron to make a larger tetrahedron.
   1. Make a sketch to illustrate this.
   2. What is the relationship of the volume of the large tetrahedron to the volume of one of the small tetrahedra?
   3. Use this information to determine a relationship between the volume of the octahedron and one of the small tetrahedra.