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Hollow Polyhedra - Archimedean Solids

by pmoews, published

Hollow Polyhedra - Archimedean Solids by pmoews Feb 14, 2012
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Summary

I wanted to print the Archimedean Solids before doing the Catalans, thing:17137, but it was not possible at that time. Two lists are required to make an openSCAD polygon command: 1) a list of vertices and 2) a list of triangles which connect the vertices to make the surface. For many polyhedra the netlib library has a list of vertices and a list of faces; my program converts the list of faces to a list of triangles. However the netlib library does not have vertex or face information for the Archimedean Solids.

Archimedean vertices are available from Wikipedia and
I decided to revise my code so that only vertices are required. I changed my fortran program to work in two steps. In step one it produces a set of vertices for an Archimedean solid scaled to
mimic a cage molecule. The scaled vertices are input into a molecular modeling program. I used VMD from the University of Illinois. VMD shows the polyhedron as a wire frame model with labeled vertices. I then made, by inspection, a list of faces to be input into the second step of the program. Output is a polyhedron command. It looks possible to program the whole process but as only 13 solids were involved I chose to use my existing code and identify the faces by hand.

The time spent looking at wire frame models made me realize how nice they look and how little material would be necessary to print them. Then I came across the truncated icosahedron - ah the Buckyball - see thing 12675. The 60 atoms of the Buckyball fit onto the 60 vertices of the truncated icosahedron. All of the edges of an Archimedean polyhedron have the same length so one could use the Buckyball program to make ball and stick and wire frame models of all of the Archimedean solids.

The complete set of hollow Archimedean stl files is in hollow_archimedean_stl.zip and the scad files used to produce them are in hollow_archimedean_scad.zip. Note that errors are always possible.

All 13 files produced attractive models. The maximum dimension is about 50 mm so as to print quickly but scaling up should be possible. Six of my favorites are included as stl files (see images) :
hollow_cuboctohedron.stl
hollow_trunc_octahedron.stl
hollow_rhombi_cuboctohedron.stl
hollow_icosidodecahedron.stl
hollow_trunc_dodecahedron.stl
hollow_trunc_icosahedron.stl

Instructions

The files were printed on a Thing-O-Matic using a smooth heated surface. Kapton tape, cleaned with acetone, covered the surface. No external support was required as all of the Archimedeans have large faces available. I used rafts.

The files provided all print in under 2 hours,

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thanks

Over the past year, there has been an explosion of all things polyhedral. We have libraries to create vertices, and with your sets, we must have all the most interesting solids known to man.

I think this is a great teaching tool. In a classroom environment, you can go from the math book, the the whiteboard, to the physical object. If I had this ability when I was a child, I might have understood the maths all the more better earlier.

Great work!

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