Wednesday, March 2, 2016

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.

No comments:

Post a Comment