Rhombic Dodecahedron

Links in the following text refer to MathWorld, the web’s most extensive mathematics resource. MathWorld usually has Java Applets for all linked polyhedra with which you can rotate them, and further informations.

A Rhombic Dodecahedron is a polyhedron consisting of twelve rhombs (thus the name), 24 edges and 14 vertices. It looks like this:

One of the most surprising properties of the Rhombic Dodecahedron is that it is possible to fill the space seamlessly with copies of it, glueing faces together:

The symmetries of the Rhombic Dodecahedron are closely related with Hexahedrons and Octahedrons. It is even possible to approximate a Hexahedron, Tetrahedron or Octahedron with lots of small Rhombic Dodecahedra (strictly speaking, it is possible to approximate any polyhedron with small Rhombic Dodecahedra, but not as nicely). The image below shows a “cube” consisting of 305 Rhombic Dodecahedra. Removing Rhombic Dodecahedra first gives a Half-Truncated Cube, then a Tetrahedron, then a Truncated Tetrahedron, then an Octahedron:

Changing the order how the Rhombic Dodecahedra are removed gives three different intermediate steps, as shown in the image below. All three intermediate steps, the Truncated Cube, the Cuboctahedron and the Truncated Octahedron, are Archimedean Solids.

(“Archimedean Solids” are polyhedron with indistinguishable vertices and regular polygons as faces; the Rhombic Dodecahedron, on the other hand, has two different kind of vertices and therefore can’t be, like an Archimedean Solid, isogonal — but it’s faces are indistinguishable, that means, it can be used as a fair “dice”, it is an Isohedron. And since it is the Dual of an Archimedean Solid, the Cuboctahedron shown below, it is furthermore a Catalan Solid.)

To make it easier to see all those solids, the image below shows just their skeletons:

It is of course also possible to approximate a Rhombic Dodecahedron with Rhombic Dodecahedra. Since yet again, the result might be a bit difficult to grasp, I added a second copy at the right side with the big Rhombic Dodecahedron superimposed:

That’s all. As a bonus, yet another Archimedean Solid, the Small Rhombicuboctahedron (well, admittedly, this version is more like a Truncated Icosihedron, since some of the rectangles are not square; or like something unnamed):

20.3.2005
Jan Thor
www.janthor.com
jan@janthor.de