Deltahedra

The 5 Platonic solids are the only convex polyhedra for which all the faces are identical regular polygons and all the vertices are identical. Let us now investigate what convex polyhedra result if we relax the requirement that every vertex be identical.

In the article on the Platonic Solids we realized that there is only one convex polyhedron with all square faces (the Cube), and only one convex polyhedron with all pentagonal faces (the Dodecahedron). Using equilateral triangles for the faces, however, we found 3 convex polyhedrons: the Tetrahedron, Octahedron and Icosahedron.

A polyhedron with faces that are all congruent equilateral triangles is called a Deltahedron. The question before us is whether equilateral triangles can be assembled to make any other convex polyhedrons, if we do not require that the vertices be all the same.

The answer is yes; there are several convex polyhedrons that can be constructed using equilateral triangles. Perhaps the most obvious example is obtained by connecting two tetrahedrons base-to-base, resulting in a convex polyhedron with 6 triangular faces:

Triangular Dipyramid (J12)

It turns out that there are still more Detahedra. Before we look at them, Let's figure out how many there could theoretically be. With only 4 faces, the Tetrahedron is the certainly the smallest deltahedron. Because there can be at most 5 equilateral triangles around a vertex, the Icosahedron must be the largest deltahedron (since it has 5 triangles surrounding every vertex). Therefore the number of faces of any convex deltahedron must be between 4 and 20. We now know that there can not be more than 17 convex Deltahedra.

With just a little more effort we can sharpen this result. For a deltahedron with F faces and E edges, it is always true that E=(3*F)/2. This is because every face has 3 edges, but the edges occur in pairs. This equation tells us that F must be even! We now know that there can be at most 9 deltahedra, and that the number of faces can only be 4, 6, 8, 10, 12, 14, 16, 18 or 20.

(Note that the requirement of convexity is pretty important here. If, for example, you start gluing tetrahedrons together, it becomes immediately apparent that there are an infinite number of nonconvex deltahedra!)

When we investigated the Platonic Solids we found that there could be at most 5 polyhedra that satisfied all the conditions, and that in fact there were 5. However, It turns out that there are not 9 convex deltahedra but only 8; there is no deltahedron with 18 faces!

The remaining 4 Deltahedra are shown below. Because they are not regular you will find it very helpful to view the VRML versions so you can spin them around.

Pentagonal Dipyramid (J13)

Gyroelongated Square Dipyramid (J17)

Triaugmented Triangular Prism (J51))

Snub Disphenoid (J84)


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