# Convex Deltahedra

An interesting problem is to find the set of all convex polyhedra with equilateral triangular faces. This includes three Platonic solids: the tetrahedron, octahedron, and icosahedron. It also includes some other convex solids you can make by sticking together triangles in any haphazard manner which happens to close. Thus we are considering just those Johnson solids which happen to be made entirely of triangles.

How many sides might a deltahedron have ? Clearly four is the minimum number of faces, giving the tetrahedron. Because no vertex can have more than five equilateral triangles meeting, the largest convex equilateral deltahedron is the icosahedron, giving twenty as the maximum number of triangles. It is also not hard to show that any deltahedron has an even number of sides.

Exercise: Show that any deltahedron has an even number of faces.

The above arguments eliminate all numbers except the even numbers from 4 to 20. It turns out, when one examines the possibilities, that for all of these but one there is a convex deltahedron with that many faces. Various triangular construction toys are ideal for working on this problem, or you can search among the list of Johnson solids.

Exercise: For which of these numbers, n, is it impossible to find a convex equilateral deltahedron with n faces: 4, 6, 8, 10, 12, 14, 16, 18, 20 ?