Follow by Email

Friday, October 12, 2012

Cubic Nesting

For some reason humanity is obsessed with the cube. We build our skyscrapers based on it. We study its symmetries. We ship products in it (excepting, of course, Painter, which came in a cylindrical package).

Nature has cubic symmetry built right into the salt crystal. And, as we will presently see, many other shapes have cubic structure built right into them, by virtue of polyhedral nesting geometry.

If you snub the corners off a cube all the way to the midpoints of the edges, you get a cuboctahedron.

This shape, shown here, fits perfectly inside a cube and you can immediately see how the corners of the cube may be removed. As far as I know, this is one of the few examples of a 14-face polyhedron.

It is natural in the sense that the cuboctahedron has one face for each face and vertex of the cube.

It has been said that if you pack clay spheres into a space and press them down that each clay ball will have approximately 14 facets.

You could think of the spheres as mutually-avoiding points in space. The polyhedra made by the mid planes between the points would then be a three-dimensional Voronoi diagram.

Another naturally-occurring polyhedron is the rhombic dodecahedron. This is the natural shape of a garnet crystal. It is bounded by 12 rhomboids whose diagonals have the ratio 1::sqrt(2).

The cool thing about the rhombic dodecahedron is that it can tessellate space. So it makes a nice packing form. Honeybees use it to form the cells of their honeycomb.

If you look closely, a cube can nestle perfectly inside a rhombic dodecahedron. In particular, this shape has one face for each edge of the cube.

The rhombic dodecahedron is the dual of the cuboctahedron because you can construct each solid by putting a vertex in the center of each face of the dual. But these are not the only solids that nestle with a cube.

Perhaps the coolest solid to nestle with the cube is the dodecahedron itself. This shape is bounded by 12 pentagons.

This shape is used for the 12-sided die in Dungeons and Dragons because it is a regular polyhedron.

Here, if you look closely you can see the cube nestled inside. In fact, there are five distinct nestlings. This is because each cube edge travels along exactly one of the five diagonals of each pentagon.

This shape is ruled by the golden section: (1 + sqrt(5))/2 or 1.6180339.... This number is the limit of the ratio between successive Fibonacci numbers, defined by the recurrence relation Fn = Fn-1 + Fn-2.

The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... and it gets used for a lot of different purposes. For instance polyphase merge sorting and optimal one-dimensional searching.

If you connect the four vertices of a cube that don't neighbor each other, you get a tetrahedron. Each of the other four vertices caps a right tetrahedron.

Here I show it as an exploded view. The cube can be made up from five tetrahedra. One is of the platonic form, and nestles inside the cube perfectly (the center one).

This is one tessellation used for better cubic interpolation, it turns out. Diagonals become smooth when you interpolate in this way. Any tetrahedral tessellation converts more easily to barycentric interpolation, which is easier than trilinear, and offers less distortion on the diagonals.

Cubes can also be split into cubes, as everybody knows who has solved a soma cube or played with a Rubik's cube.

Or played Minecraft.

When you split cubes into cubes, you can irregularly cut a cube up into a paired set. This set consists of two pieces that are keyed to each other. The more sub-cubes you split a cube into, the more possibilities for keyed sets exist.

Here is a possibility, one of several, that exist when a cube is split into a 3X3X3 array of sub-cubes.

In all (but the trivial uninteresting) cases, a concavity on one side is met with a convexity on the other.

I have turned one of the pieces by 60 degrees so you can see that symmetry figures into how many of these keys there are.

For those people interested in splitting things up into pieces, you can see my Pieces post. For more secrets of three-dimensional thinking, my Three-Dimensional Thinking post, or my Three-Dimensional Design post.



  1. It looks like you should apply for a job to help out on Patterns! ( I was able to build your cuboctahedron but not the rhombic dodecahedron you mentioned above.

    1. Well, let that be a challenge! And of course the dodecahedron is the bigger win.

      I can assure you that drawing them freehand, like I did for the illustrations above, was not easy. No rulers no straight line tool. ;-)