### Mark Newbold's Rhombic Dodecahedron Page

This page discusses the Rhombic Dodecahedron and its relationship to the 4-dimensional hypercube.

3D glasses (red-blue) are required to view this page properly. The red filter should be on the left. The images with a green background are 3D images. So is the page background.

I am also experimenting with a Stellated Dodogahedron. Please check it out.

This is a screen-shot of the hypercube applet. The orientation of the hypercube has been adjusted (in 4-space) so that the 3-space image forms a perfect "rhombic dodecahedron" (plus some extra internal lines emanating from the center).

A "dodecahedron" is a polyhedron with 12 faces (do=2, deca=10, hedron=base). Most people associate the word "dodecahedron" with the pentagonal dodecahedron. Various other dodecahedra are known.

The faces of the rhombic dodecahedron are all identical rhombuses (a rhombus is a parallelogram with all edges equal, i.e. a squashed square). It is not considered a "regular" polyhedron because the vertices are not all the same. Some of the vertices join 3 faces and some join 4 faces.

## How to Make your own Rhombic Dodecahedron

You can make one of these for your very own by playing around with the hypercube applet. Once you have made it, you can admire it from all angles.

If you have read the user interface instructions for the hypercube applet, you know that you can rotate the object in 3-space by dragging with the mouse. You can rotate it in the 4th dimension by holding down the shift key and dragging with the mouse. By alternating between these two types of rotations, you can get the hypercube oriented as shown.

Your goal is to get two points to exactly coincide in the center (when viewed from all 3-space angles). The points that you want to coincide are opposite corners of the hypercube (the shortest path of edges from one to the other is 4 edges long).

## A 3-Dimensional Analogy

This screen shot is also taken from the hypercube applet. In this case, I adjusted the hypercube orientation so that its projection is a simple cube in 3-space.

You can do this too but you won't get exactly the same result. I cheated a little by modifying some hard-coded parameters in the applet (I multiplied the eye-to-screen distance and eye-to-nose distance by 100 so as to make the 3-space projection almost orthogonal.)

If you view this image with your 3D glasses, you see a 3-dimensional cube. (The 4-dimensional aspect is concealed and is irrelevant to this part of the discussion.)

Now close your left eye so you see only the blue image. Note that it is a perfect hexagon (plus some extra internal lines emanating from the center). By closing one eye, you have projected the 3-dimensional cube onto the 2-dimensional screen.

In summary, the analogy is:

• A rhombic dodecahedron is a projection of a 4-cube onto 3-space.
• A hexagon is a projection of the 3-cube onto 2-space.

You can do the same thing with all the higher-dimensional hypercubes. Projecting them onto 3 dimensions produces various increasingly complicated polyhedra, all having paralelograms for faces.

## Recipe for a Rhombic Dodecahedron

A solid rhombic dodecahedron doesn't photograph well. It is hard to visualize its shape from the faces which are visible. The following images are intended to give an intuitive understanding of the thing.

Recipe Ingredients: 2 identical cubes, greenish on the outside, purplish on the inside (preferably fresh, with no mushy spots)

Using a very sharp knife, slice one of the cubes into 6 identical pyramids as shown.

Line up the 6 pyramids with the 6 faces of the remaining cube. (You need 7 hands to do this.)

Smush the pyramids onto the cube so they stick. Smooth out the junctions so they are invisible. Chill before serving.

As I said, the rhombic dodecahedron doesn't photograph well.

It is helpful to compare the features of this image with those of the previous image.

You might think that this procedure would produce an object with 24 faces since each of the 6 pyramids has 4 exposed triangles. But in fact, pairs of triangles are co-planar so they join to form 12 rhombuses.

## A Question

On this page, I asked if anyone could provide a simple explanation of why the rhombic dodecahedron is a projection of the hypercube. Here are the responses I have received:
• Russell Towle has provided a discussion of this issue.
• David Christie has pointed out that H.S.M. Coxeter discusses this in section 13.8 of his book "Regular Polytopes" (Coxeter01). I haven't yet had a chance to try to understand it, or see whether his explanation can be translated into non-mathematical terms.

## Filling Space

Rhombic dodecahedra can be fitted together to fill space with no overlaps and no gaps (a spatial tesselation or "honeycomb"). This is illustrated by the background of this page, which shows some layers of a such a honeycomb. For other honeycomb backgrounds, see Mark's Page and my Java Page.

The following blank space is provided to allow unobstructed viewing of the background:

A spatial filling by rhombic dodecahedra occurs naturally in garnet crystals.

I know of two explanations of why rhombic dodecahedra can fill space:

1. Hypercubes can fill 4-space, just as cubes can be stacked to fill 3-space. Since you can project a hypercube onto 3-space as a rhombic dodecahedron, you should be able to project a 4-space tesselation of hypercubes onto 3-space to produce a 3-space tesselation of rhombic dodecahedra.

2. You can fill 3-space with alternate black and red cubes such that every black cube shares its faces only with red cubes and vice versa (making a 3-dimensional checkerboard). Slice up each of the red cubes into pyramids as in the recipe (above) and glue the bases of these pyramids onto the adjoining black-cube faces. It is apparent that this yields a 3-space tesselation by rhombic dodecahedra.

## My Paper Model

I made a paper model of the rhombic dodecahedron, which helped me to visualize the images on this page. I wouldn't use the recipe I described above for making an actual model. To make a paper model, you need to start with 12 identical paper rhombuses. The internal angles of the rhombuses are 70.53 degrees and 109.47 degrees. (70.53 degrees is ArcCos(1/3)). For more information about making paper models of polyhedra, see George Hart's Paper Polyhedra Page.

## VRML Models

If you have a VRML viewer, check out the VRML Rhombic Dodecahedron listed on George Hart's Zonohedra Page. George also has a VRML Compound of 5 Rhombic Dodecahedra listed on his "Uniform Compounds of Uniform Polyhedra" page under the heading of "Relatives of the 5 Cubes".

## Compound of 5 Rhombic Dodecahedra

I have a Stereoscopic 3D view of the compound of 5 Rhombic Dodecahedra.

September 19, 1996
Eric Swab says: "There is another (closely related) recipe for creating a Rhombic Dodecahedron. Begin with a single cube, slice off an edge with a plane that is parallel with that edge and passes through the mid-line of the 2 adjacent faces. Then do the same with all the remaining edges and the form you are left with is a Rhombic Dodecahedron."
January 28, 1997
Eric Swab says: "I was looking at Critchlow's "Order in Space" p. 55, this morning and discovered another recipe for the Rhombic Dodecahedron. Start with an octahedron, lay on each triangular face an irregular tetrahedron which is 1/4 of a regular tetrahedron. These irregular tetrahedron are formed by joining each vertex of the regular tetrahedron with its center. 4 irregular tets are formed in this way and 8 are required to cover all 8 faces of the octahedron."
February 10, 1997
Russell Towle says: "In Mathematica, you can apply the Stellate command to the built-in polyhedra. Using this command, and carefully choosing the parameter of stellation, you can create Kepler's rhombic dodecahedron by stellating either the cube or the octahedron. In each case it is as though we had "balanced planes" across the edges of these solids, letting them cut one another off. The analogous balancing, applied to a tetrahedron, gives a cube, and applied to the icosahedron and pentagonal dodecahedron, gives Kepler's rhombic triacontahedron, which is an isometric shadow of a 6-cube."

Russell has also pointed out that the Rhombic Dodecahedron under discussion here should be referred to as "Kepler's Rhombic Dodecahedron" since there are other Rhombic Dodecahedra. --MN

March 4, 1997
Matthew R. Kennedy says: "You may wish to see ours [web site] (under construction) at http://www.erols.com/mrkenn/index.html [this link no longer works] in which we emphasize the unity between the parallel and perspective projections of the hypercube in our "Inverted Projected Tesseract Resonating Chamber" loudspeaker design (patented)."
March 10, 1997
George Olshevsky says: "The RD also contains a 3-D projection of the regular icositetrachorema (or 24-cell), the regular 4D polytope (or polychorema) that consists of 24 octahedra. Twelve of the octahedra project flat into the 12 face-planes of the RD (the short diagonal of each face represents the square equator of the octahedron projected edgewise, so you need these lines to be drawn in addition to the edges of the RD), and the other twelve project into the somewhat flattened octahedra formed by a trivalent vertex, the four vertices edgewise adjacent to it, and the center of the RD. There are 6 such configurations within the RD, each representing a pair of coincident octahedra--one from in front, the other from behind."

I had already noticed that my 24-cell Java applet seems to produce a rhombic-dodecahedron. In fact, that is the start-up state of the animation. If you click the "Stop" button very quickly after the applet starts, you will see a rhombic dodecahedron. --MN

March 16, 1997
Stefan Scheller says: "How to make a Rhombic Dodecahedron? Simply intersect 3 cylinders orthogonally and take the common solid which belongs to all 3 cylinders. Well, it's a bit blown up..."

I tried it.
This is what I came up with (using a different color for each cylinder). --MN

September 6, 1997
September 28, 1997
Referring to George Olshevsky's note (above), Eric Swab says: "I was reminded that the cuboctahedron, the dual of the RD is also a "shadow" of the 24-cell.
November 9, 1997
Dr. Donald Carlson says: "An infinite lattice of spheres can be modeled as a single sphere inscribed in a unit cell with reflecting boundaries (i.e., a polyhedral box of mirrors)." He notes that the Rhombic Dodecahedron is the unit cell for the "face-centered-cubic" (fcc) close-packing of spheres. For the full contents of Dr. Carlson's email, click here.
December 26, 1998
Mark Newbold (me) says: "I note that the 1977 Sci-Fi movie Demon Seed features a large Rhombic Dodecahedron which unfolds into octahedra and attacks a nerd. The RD appears about 1 hour into the movie."
July 20, 2000
Ray Tomes says: "In your page ... you discuss why Rhombic Dodecahedra are space filling. My favorite way of thinking about how they do this is to start with the famous Kepler sphere packing problem, preferably by imagining oranges as usually stacked. Then apply pressure so that the oranges are squashed until all the air is squeezed out of the spaces between but the juice doesn't quite come out of the oranges. The resulting shape is the space packing of Rhombic Dodecahedra. Again, chill slightly before serving :)"
May 9, 2001
Ken Wauchope says: "As a garnet crystal enthusiast I enjoyed visiting your rhombic dodecahedron webpage. If you have a VRML2.0 browser you might be interested in a presentation on garnet geometry I put together recently for my rock and mineral club (also featuring the RD's close cousin the Trapezoidal Icositetrahedron) including animations of several of the geometric transformations your page discusses: http://home.comcast.net/~kwauchope/garnet/index.html.
June 9, 2013
Piet Ruhe made a jitterbug out of RD's: