Morphing the Snub Dodecahedron

Morphing the Snub Dodecahedron

The following post is by Mae Markowski as a part of George Mason University

Math 493, Mathematics Through 3D Printing.

In this post we examine one of the 13 Archimedean solids, the snub dodecahedron,

and its dual, the pentagonal hexecontahedron. For starters, we

should define a few of the terms. Archimedean solids are 3-dimensional solids

made of regular polygonal faces, where we can have our faces consist of 2 or

more regular polygons.

We can also discuss an interesting feature of polyhedra: their duals. If we

replace each edge with a vertex in our polyhedra, what we see is that we give

rise to an entirely new 3-dimensional object. If we do this to the Archimedean

solids, we create the duals of these solids, known as the Catalan solids. Since

there are 13 Archimedean solids, there are 13 Catalan solids, as well. Additionally,

the dual operation is reversible, i.e. if we take the dual of the snub

dodecahedron to create the pentagonal hexecontahedron, we can take the

dual of the pentagonal hexecontahedron to obtain the snub dodecahedron.

The solid I chose was the snub dodecahedron. The snub dodecahedron contains

the most faces of the Archimedean solids, with 92, 12 of which are

pentagons, and the other 80 are equilateral triangles. Interestingly, you can

construct the snub dodecahedron from another Archimedean solid, the rhombicosidodecahedron,

by rotating the center of each face equally.

For this project, the goal was to start with the snub dodecahedron, which

can be created in Mathematica using the function "PolyhedronData", and

in the OpenSCAD software morph the polyhedron until its dual the pentagonal

hexecontahedron is obtained. For this I used the method of extrusion.

Essentially what occurs is starting with the snub dodecahedron, we place its

dual completely inside. Then we slowly scale up the dual until it begins to

poke through the other shape, as shown below in OpenSCAD:

 
 

 

The final dual prints of this transformation are shown below. Note that there

are only 5: the snub dodecahedron, a shape created at the start of the morph,

one a little before halfway through the morph, one right before the end, and

the final pentagonal hexecontahedron. I was hoping to print 6 polyhedra,

i.e. all six images shown above, but the print of the fourth figure was too

unstable and kept falling over, even with supports.