Saturday, June 25, 2011

Investigations into Dodecahedral Space

Remember when I warned you that some of these posts may go into extravagant detail about some strange or obscure hobby that I enjoy? Well, this is one of them. So if you're not particularly interested in the interactions of certain polyhedra in three-dimensional space, you should probably leave now.

Today's topic is the interrelation of cubes and dodecahedra and the examination of said relations with regard to the potential of some kind of dodecahedral space.

For those of you who may not know, I really like polyhedra (especially Archimedian and Catlan Solids). I have models of several on my desk at work -- including an intersection of a snub dodecahedron and a pentagonal hexacontahedron, the patterns for which I calculated and designed myself -- and I thoroughly enjoy tinkering around with a Zome kit my parents got me several years ago. My current obsession is with the relation between dodecahedra and cubes. A dodecahedron has twelve regular pentagons for faces and is the shape for a 12-sided die.

I'm pretty sure you all know what a cube looks like.

Now, a dodecahedron doesn't seem to fit into 3D space very neatly: it's faces are pentagons, which aren't easily described in a square coordinate system and aren't as structurally fundamental as triangles, and neither pentagons nor dodecahedra tessellate (pack together without spaces in between). Cubes, on the other hand, represent the very basis of three-dimensional space. It is therefore fascinating to me that, by drawing a line between two vertices on each face, you can inscribe a cube on the surface of a dodecahedron.

(The struts used for this are unfortunately all the same color, so I covered the ones forming the cube in tin foil.)

The edges of the cube all fall on a unique face of the dodecahedron, and every face of the dodecahedron is occupied by an edge of the cube.

This inscribing of a cube on a dodecahedron allows for easier mapping of the latter into cubic space. And while dodecahedra cannot tessellate, they can be arranged without overlap in a repeating pattern along diagonal cubes:

I found this last bit worth further investigation, since the cube formed inside the dodecahedron is not the only one possible; the inscription process can be used to form 5 unique, rotated cubes.

Here are two, shown in black and silver.

So, if I were to arrange several dodecahedra according to the cubic space defined by the silver struts, then generated the black cubic space, would the black cubes intersect the other dodecahedra the same way as the one in which the first cube was generated? If this were true, I could see it giving rise to a new way of defining space according to the arrangement of dodecahedra rather than the cubic space we all know and love. While perhaps not practical, this would be kind of cool. However, preliminary construction showed that, at least one "layer" out from the original dodecahedron, the two cubic spaces did not converge. This didn't mean convergence was impossible, though; it was still possible that the two spaces converged at some more distant point. However, the increased distance would greatly increase the potential complexity of the "dodecahedral space," as every cube spacing the distance to convergence would itself be able to give rise to a dodecahedron and four other cubic spaces, which would in turn be able generate more cubic spaces before they converged, and so on. Further construction only got more and more flimsy the further I got from the original dodecahedron. This would have to be accomplished with math.

Several maths later...

It turns out the two cubic spaces never converge, leading to infinite complexity and the breakdown of "dodecahedral space." I arrived at this conclusion as follows: If I just took the two original cubes,

and then just looked at the top face of the silver cube, particularly the edges originating at the near-left corner,

If I were to figure out the the coordinates of the vertex of the black cube in silver space (given the cubes had sides of length 1), and then find some multiple of these coordinates that was an integer, it would mean that the two spaces would meet at that point. So taking the near-left corner to be (0,0,0),

An extra silver strut has been added to form the coordinate axes.

I calculated the unit vector of the black strut. If the cubes had edges of length 2 (a change from before, I know, but it makes the math easier), then the end of the black strut has coordinates (phi, 1, 1/phi), where phi is the golden ratio and an irrational number (my favorite irrational number, in case you were wondering). Since the coordinates contain irrational numbers, no multiple of them will ever be an integer, and the two spaces will never converge.

If you're disappointed that nothing came of this after such a long post, well... I kind of was, too. However, uniqueness is an important property in and of itself, so perhaps the knowledge that these two coordinate spaces never converge except in this one dodecahedron could be useful. The first thing that comes to mind is cryptography: is there a way to encode a message on the two cubic spaces so it can only be read at the point where they converge? I don't know.

I wish it were faster and easier to get these descriptions prepared and online; my mind moves too fast and sporadically. In the time that I've been preparing these models and writing this, I've been distracted by modeling the tessellation of rhombic dodecahedra

and experimenting with taking the volumes in a dodecahedra not filled by the inscribed cube and mirroring them on the inside of the cube's faces.

Interestingly, the points in the center mark the vertices of an icosahedron (20-sided die), the dual of a dodecahedron. Is this, combined with the planar golden rectangles of an icosahedron, a clue in the relation between cubes and dodecahedra? I must find out!

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