A Polyhedral Journey, Beginning With an Expansion of the Rhombic Triacontahedron

The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.

Rhombic Triaconta

I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.

Rhombic Triaconta augmented

After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.

Convex hull

Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.

ch after ttmfr

For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.

ch after ttmfr dual

Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.

aug ch after ttmfr dual

Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.

Convex hull again

 

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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