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:

There is no convex equilateral deltahedron with 18 faces. The only way one would be possible is if you removed two adjacent faces of an icosahedron and then closed the open wedge. However doing this creates a vertex with six triangles meeting, which can not be convex. (Try it with hinged triangles and see.) Here are the possible solutions:

• 4 sides: tetrahedron
• 6 sides: triangular dipyramid (J12) (glue two tetrahedra together)
• 8 sides: octahedron
• 10 sides: pentagonal dipyramid (J13) (glue two pentagonal pyramids together)
• 12 sides: snub disphenoid (J84) (see below)
• 14 sides: triaugmented triangular prism (J51) (attach three square pyramids to a triangular prism).
• 16 sides: gyroelongated square dipyramid (J17) (attach two square pyramids to a square antiprism)
• 18 sides: NONE
• 20 sides: icosahedron

Least obvious is the one with 12 sides. It was originally called the Siamese dodecahedron by Freudenthal and van der Waerden, who first described it.

Virtual Polyhedra, (c) 1996, George W. Hart