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.
Answer: Answer.
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 ?
Answer: Answer.
The one with twelve sides is the least obvious. It was called the Siamese dodecahedron in the original reference by Freudenthal and van der Waerden. It can be thought of as a tetrahedron split into two wedges of two adjacent faces, which are then joined together by a band of eight triangles. Accordingly, Norman Johnson named it the snub disphenoid.
Here is a list of all eight.
Note that if we do not restrict ourselves to the convex case, there are an infinite number of deltahedra. For example, glue two icosahedra together face to face, or replace each pentagon of a dodecahedron with a dimple of five equilateral triangles. A nice example, based on an octahedron with its faces divided into 16 sub-triangles, is this 128-hedron sent to me by John Futhey.