The Rhombicosidodecahedron: More than just a fancy name

The Rhombicosidodecahedron: More than just a fancy name

by Anneliese Slaton

MATH 493: Math Through 3D Printing

            If someone asked me if I wanted to see their rhombicosidodecahedron, I’d be skeptical. Did they have a dinosaur? A bug?? Some other terrifying mystery of nature??

http://ngm.nationalgeographic.com/2007/12/bizarre-dinosaurs/updike-text and https://6legs2many.files.wordpress.com/2011/01/lygaeidae_oncopeltus_fasciatus_female_milkweed_bug.jpg

Thankfully for my delicate sensibilities, a rhombicosidodecahedron is simply a sphere-ish thing.

http://mathworld.wolfram.com/SmallRhombicosidodecahedron.html

Let’s get a bit more technical. A rhombicosidodecahedron is an Archmiedean solid. The Archimedean solids are a group of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. This means that the polytope in question is nonprismatic, and has the property of being a convex set of points in n-dimensional real space. Additionally, for any two vertices there exists a symmetry of the polytope mapping the first vertex isometrically to the second (an isometry is a distance-preserving injection), and the polytope is constructed of two or more types of polygons that are both equiangular and equilateral.

https://en.wikipedia.org/wiki/Rhombicosidodecahedron and 

https://en.wikipedia.org/wiki/Rhombicosidodecahedron

Notice that the rhombicosidodecahedron is comprised of pentagons, squares, and equilateral triangles. 

I fashioned this fancy polytope into a 3D print file using Mathematica and OpenScad. Here, you can see the rhombicosidodecahedron in OpenScad:

The vertices of this polytope lie at all even permutations of

where phi is the golden ratio and edge length is 2.

During this project I also explored the morphing of the rhombicosidodecahedron to its dual, the deltoidal hexecontahedron.

	http://mathworld.wolfram.com/DeltoidalHexecontahedron.html and https://en.wikipedia.org/wiki/Deltoidal_hexecontahedron

The deltoidal hexecontahedron is comprised of a four-sided polytope fitted into pentagonal shapes. The deltoidal hexecontahedron is part of a group called the Catalan solids, which are a group of convex, face-transitive solids that are the duals of the Archimedean solids. (The Archimedian solids are vertex-transitive.)They were first described by the mathematician Eugene Catalan in 1865.

So what is a dual? Imagine a cube. If you replaced every vertex with a face, and every face with a vertex, you’d have the dual of that cube. 

http://www.software3d.com/PolyNav/PolyNavigator.php#morph

There are a five ways to find the dual of a solid, but we’ll focus on two of these techniques. The first is shown above. It is called morphing by size, and is as simple as placing the dual of the polytope inside the original solid, then slowly growing the dual so we can see its vertices poking out of the faces. Eventually, the dual just eats the original solid, and its done morphing.

http://www.software3d.com/PolyNav/PolyNavigator.php#morph

The morphing technique I employed for my project is called morphing by truncation. To find the dual this way, take your original solid and slice each vertex off like a serial polytope killer. Continue taking larger and larger slices at each vertex until the edges of the slices meet. You have created the dual. 

I created multiple truncated rhombicosidodecahedrons for my project to physically illustrate my morphing method. To do this, I used OpenScad to take the difference of my solid and its dual. Let me outline the morphing process of the rhombicosidodecahedron into the deltoidal hexecontahedron for you.

STEP 1: Find a trusting and naive rhombicosidodecahedron that’s just minding its own business.

STEP 2: Viciously lop of the unsuspecting polytope’s vertices. But just a bit.

STEP 3: Now that you have bloodlust flowing through your veins, lop of some more. Then even more. Then even more!! BWAHAHAHAHA.

 Congratulations. You have now created the deltoidal hexecontahedron. 

(No polytopes were actually harmed in the making of this project.)

Each of these solids were printed on George Mason’s 3D printer. As I mentioned before, I created the deltoidal hexecontahedron and the rhombicosidodecahedron in Mathematica, then imported them into OpenScad to create my mid-morph solids. Below, you can see an example of my code for the first morph.

The solids will hopefully be used to physically illustrate the morphing process of Archimedean solids to their duals for other students.

This is what the solids looked like when they were finished:

(left to right) rhombicosidodecahedron, dual 1, dual 2, deltoidal hexecontahedron