# Dodecahedra

No hard decode.  And in this case,
A solid answer you can claim.
It has (I'm told) a different face
for every letter of its name.
--Hubert Phillips*
There are a great many 12-sided polyhedra, all of which could be called dodecahedra. This page lists a number of the more important examples and their differences, organized according to what type of faces are present. At right is what minerologists call a tetartoid.  It is topologically equivalent to the regular dodecahedron (as it has twelve identical 5-sided faces, meeting three to a vertex) but the faces are irregular pentagons, and the overall symmetry is tetrahedral and chiral.

#### Irregular Pentagons

• The pyritohedron comprises twelve identical irregular pentagons. Each pentagon has mirror symmetry and the solid as a whole has tetrahedral symmetry (four 3-fold axes and three 2-fold axes) with three mirror planes. It is commonly given this name because the mineral pyrite can appear in crystals of this shape. So it is true that pentagonal dodecahedra are found in nature, but no mineral crystals are regular pentagonal dodecahedra. These minerals may have been an inspiration to the ancient Greeks who first wrote about the regular dodecahedron, but they still made a significant leap in going from the lesser symmetry of a pyritohedron to the greater symmetry of the regular pentagonal dodecahedron. (In pyrite, the pentagons are in the "0-1-2 planes" as crystallographers call it. An interesting relative, with slightly different angles, is a pyritohedron which is a stellation of the icosahedron, so the faces are in the planes of twelve of the regular icosahedron's twenty faces.)
• The tetartoid, shown at the top of this page, is another pentagonal dodecahedron of crystallographic importance. It is composed of twelve identical irregular pentagons, but has three different types of vertices. It is also called the tetrahedral pentagonal dodecahedron because of its chiral tetrahedral symmetry.
• If you take a regular pentagon made of paper and fold it on any diagonal, the five edges outline a concave pentagon. Twelve of these pentagons assemble into a concave dodecahedron with pyrite symmetry.  This form is particularly interesting because if you alternate it with regular dodecahedra you can pack space without gaps.  (You can easily make a very nice Zome model of the polyhedron and the packing, as described in Zome Geometry.)

#### Regular Triangles

• The snub disphenoid is the only convex polyhedron composed of twelve equilateral triangles. It is one of the eight regular convex deltathedra. It was given that name by Norman Johnson because you can construct it by separating a disphenoid into two parts (of two faces each; sphenoid is Greek for wedge) and then reconnecting them via a band of "snub" triangles. It was called a Siamese dodecahedron in the paper by Freudenthal and Waerden which first describes it.

#### Isosceles triangles

• Many different "dodecahedra" are composed of twelve isosceles triangles. The most important is probably the hexagonal dipyramid, but one would never call it a dodecahedron since that would obscure its important properties as a dipyramid.
• Another is the triakis tetrahedron, but it would also be odd to call it any kind of dodecahedron and obscure the fact that it is an Archimedean dual.
• Here's a nice isosceles dodecahedron. It comes from an infinite family which Joe Malkevitch showed me; it can be extended to any number of isosceles faces that is divisible by four if we make the faces pointy enough.

#### Scalene triangles

• There are a great many different "dodecahedra" composed of twelve non-regular triangles. One symmetric family is formed as follows: Pick three alternate vertices on the "equator" of a hexagonal dypyramid, and move them in or out slightly, by the same amount. The result is called a ditrigonal dipyramid by crystallographers.
• Another is called the hexagonal scalenohedron by crystallographers. It is formed by moving three of the equatorial vertices up and three down.  Both of these these have one degree of freedom in choosing the face shape. They are not of great interest to geometers, but minerologists use the terms for certain crystal forms.

#### Rhombi

• The rhombic dodecahedron has all the symmetry of a cube or octahedron. It is a zonohedron and the Archimedean dual to the quasi-regularcuboctahedron. The ratio of the lengths of the two diagonals of each rhombus is the square root of two.
• The rhombic dodecahedron of the second kind is another zonohedron, but with a different rhombus shape, and it only has 2-fold prism symmetry (three mutually orthogonal 2-fold axes and three symmetry planes). The ratio of the lengths of the two diagnals is the golden ratio (just like the rhombic triacontahedron to which it is related by the zone removal process.) Interestingly, both of these rhombic dodecahedra are space fillers. (The origin of this polyhedron is unclear to me. Coxeter [Regular Polytopes, p. 31] gives a 1960 reference for its first publication, but I have observed that it appears as a pop-up paper model in the 1787 Solid Geometry of John Lodge Cowley.)

#### Kites

• The hexagonal trapezohedron is the dual to the hexagonal antiprism.  It is called "hexagonal" because it is based on a hexagon and has a 6-fold axis of symmetry.  There are twelve "kite-shaped" faces---tetragons with a diagonal mirror, sometimes called deltoids, and called trapezoids in British English.
• The trapezoidal dodecahedron is a crystallographic polyhedron with twelve "kite-shaped" faces . It has tetrahedral symmetry with six mirror planes, and can be derived from the tetrahedron by dividing the triangles by thirds into kites and then puffing it out a bit. It is topologically equivalent to the (first) rhombic dodecahedron above, and can be derived from that by stretching it in four tetrahedral directions.

#### Irregular nonconvex faces

The poem above is in some references incorrectly attributed to Bishop Samuel "Soapy Sam" Wilberforce.  Notice how the anagram No hard decode is made from the twelve letters of dodecahedron.