Dual Solids, Morphing a Truncated Dodecahedron into a Triakis Icosahedron-Project 3 by nvanoort

Last crawled date: 3 years ago

Nicole Van Oort
02/29/16
George Mason University
Math 493-Mathematics Through 3D Printing
Shows the morphing from the Truncated Dodecahedron to its dual the Triakis Icosahedron.
The Truncated Dodecahedron is one of the 13 Archimedean Solids, meaning it is a convex polyhedron consisting of two or more regular polygons meeting at identical vertices. The Truncated Dodecahedron is composed of 20 triangles faces and 12 decagon faces, with 90 edges and 60 vertices. Polyhedra in general are paired with duels, meaning the faces of one polyhedron correspond to the vertices of another polyhedron. The group of duels to Archimedean solids are known as Catalan solids, first described in 1865 by Eugène Catalan. For the Truncated Dodecahedron, it duel or pair is the Triakis Icosahedron. This polyhedron has 60 isosceles triangle faces, with 90 edges and 32 vertices. Notice that the Truncated Dodecahedron has 32 faces, and the Triakis Icosahedron has 32 vertices. Also the Truncated Dodecahedron has 60 vertices while the Triakis Icosahedron has 60 faces. Thus the relationship between the faces and vertices of duels can be seen numerically.
The way one can obtain a polyhedron's duel from the polyhedral itself are various. The method I used is Truncation. This means that the vertices of the original polyhedron are continuously chopped or shortened towards the midpoint, until the original faces of the polyhedron disappear and the duel emerges. This is due to the fact, that a regular polygon uniformly truncated (truncated equally at each vertex) will become another regular polygon. The term truncation originates from the term "truncated" used by Johannes Kepler for the truncated Archimedean solids that he rediscovered in 1620, after originally being discussed by Archimedes and Pappus. (The Archimedean Solids were thought to be first described by Archimedes, although only heard through second hand sources. Kepler finally reconstructed the entire set in a book he wrote "The World Harmony.")
One benefit of choosing the truncation method is that mathematica has a truncation function under its Polyhedron Operations package. It also has multiple well known polyheda under its PolyhedronData package that allows you to produce 3D images of multiple polyhedrons with one command. Thus 3 of our 4 solids used to display the morphing of the Truncated Polyhedron to the Triakis Icosahedron were obtained readily through mathematica to be exported as an STL file into makerbot. The first step was to import the command "Needs[PolyhedronOperations]" into mathematica, calling in the further polyhedra commands we need to continue. Then to produce the first step of our morphing, or the original Truncated Dodecahedron we have two options. The first is to use the command "PolyhedronData["TruncatedDodecahedron"] to call the Truncated Dodecahedron directly. The other method, which can be seen in the mathematica image below uses the Truncate operation, brought in through the PolyhedronOperations package to truncate the Dodecahedron into the Truncated Dodecahedron. Secondly, to begin the initial truncation process, we can now truncate the Truncated Dodecahedron further to get the first step of morphing through truncation. This produces a polyhedron with circular faces instead of decagons. The other nice feature about the Truncate command is it allows you to specify the ratio of truncation you want from 0 to 0.5, where 0.5 means the polyhedron is halfway truncated towards its duel. I utilized 0.3 as my ratio to truncate about a third of the way to the duel. This code can be found below. Lastly I used mathematica and the Polyhedron Data command to obtain a 3D image of the Triakis Icosahedron duel, which is a stored polyhedron included in the database of mathematica. To get these three polyhedron into STL files that can be inserted into MakerBot, the Export command is used, specifying to export a .stl file.
The issue with the Truncate command in mathematica is that it doesn't allow truncation past the halfway point, thus to get to two-thirds of the way to my duel (to break the morphing process into 4 steps/solids) I needed to utilize OpenSCAD instead. To grasp an idea of the polyhedron I was looking for, I utilized the Stella4D Demo software and its truncate tool that allowed me to visualize and scroll from the original Truncated Dodecahedron to its duel and back. Finding 0.6 percent of truncation using the software I obtained the image of the polyhedron I was looking for in OpenSCAD. In order to obtain this polyhedron, I imported the STL files of the Truncated Dodecahedron and Triakis Icosahedron into OpenSCAD. The key is the idea that the faces of one are the edges of the other, thus the Triakis Icosahedron needed to be rotated until its vertices were in the middle of the Truncated Dodecahedron's faces in order for the truncation to work properly. Then using the intersection command, which correlates to truncation of polyhedron because you are cutting of the edges of the larger polyhedron (truncated dodecahedron) until it meets up with the edges of the smaller polyhedron (traikis icosahedron), I intersected the Triakis Icosahedron with the Truncated Dodecahedron to truncate the Truncated Dodecahedron. As you can see in the OpenSCAD image below, I had to play around with the proper numbers until I got a polyhedron that resembled the polyhedron I had generated in the Stella4D Demo tool, or that polyhedron two-thirds truncated. Thus when I observed this polyhedron, I also rendered it to an STL file.
Finally I combined all 4 STL files into one .thing file in MakerBot, equally scaled them and then exported it to be printed.
The code and equations for this project were straightforward thanks to the ease in use of the Polyhedron Operations package in Mathematica. If confusion ever arises about the Mathematica code to use, the Mathematica help window provides excellent instructions. All the STL files for each step of the morph, as well as the Mathematica and OpenSCAD files of the polyhedron too are found under thing files.
Citations:https://en.wikipedia.org/wiki/Archimedean_solidhttp://www.software3d.com/PolyNav/PolyNavigator.php#morph