This page demonstrates some ringed models, each formed using a single polyhedral component which is duplicated numerous times but always placed in the same orientation, and used as the "vertices" and "edges" that make up the "skeleton" of large polyhedral arrangement. We have already seen this approach before, in Chapter 7 of Stewart's book, in which he created a rhombic dodecahedral arrangement composed of 38 octahedra. 14 of those octahedra served as the "vertices", which were connected to each other by 24 additional octahedra that acted as "edges" of this arrangement. Similarly, he used dodecahedra to create the skeleton of the rhombic triacontahedron.
On this page, we are trying to create the skeletons of the platonic and archimedean solids. Originally, I could not think of a simple way to do this, but then I realized that if octahedra can be combined to create a rhombic dodecahedron skeleton, maybe a rhombic dodecahedron can be used as the component of the octahedral platonic and archimedean skeletons. I was able to create some of them, but others looked like they would also need the planes of the platonic solids in order to work. Then I remembered that the rhombicuboctahedron and truncated cuboctahedron both had square faces in the planes of the rhombic dodecahedron. Now, I figured, I would be able to create any of the models using either of these two base components. Likewise, the icosahedral models could be composed of rhombicosidodecahedra or truncated icosidodecahedra.
The models below were all done with edge length of 3 (each edge is 3 components long). We could increase the number of components along each edge if we wanted to open things up. In the table below, Where there are empty spaces in the left column, the component figure did not have the required face planes to construct the skeleton. In the middle and right columns, you will see that the component pieces are joined on faces other than the squares from the rhombic dodecahedral planes.
Click on the images below to see a VRML model of the figure, or click on the name below the images to get the .Stel file (for users of "Great Stella"). Note to Great Stella users - images below were taken using "Orthogonal View" to keep them neat and clean. Also note that the last few Icosahedral models require Stella version 3.0 or later to open (I could not create them in version 2.x due to their level of complexity). So make sure you get the latest version! (VRML files for these are also not yet available, pending a fix from Robert Webb).
Disclaimer: This page is dedicated to B.M. Stewart. The ideas presented here are new figures I came up with that are based in part on Stewarts original models. His book has tons of information that have been inspirational to many polyhedronists, and I highly encourage anyone interested in this subject to buy the book. If you like my illustrations, please let me know. And if you have any new models based on Stewart's work and would like to see them here, please let me know and I will do what I can.
Thanks also to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).
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Link to this page as http://Polyhedra.Doskey.com/Rings.html
Alexander's Polyhedra, (c) 1998-2006, Alex Doskey