Here is a collection of polyhedra that can be generated by "Prism Expansions" of other polyhedra. The basic concept here is a variation on the technique discussed on the Parallel Expansions page. In this case, the starting figure is a decomposed polyhedron, and the different component polyhedra are going to be expanded away from each other. As discussed on the Parallel Expansions page, the common edges will be replaced with squares. In this case, however, the squares can be seen as the faces of a prism that are inserted between the original polyhedral components that shared a face.
As a first example, we can see how an octahedron can be decomposed to form two square pyramids joined by a common square. If we expand those components apart, forming a square prism between them, we will get an elongated square dipyramid (Johnson Solid J15).
Prism Expansions of Dissected Polyhedra
All dissected polyhedra can be expanded by placing prisms between their shared faces. However, not all of these prisms are right-prisms with regular faces. If we are only interested in regular faced polyhedra, here are some interesting expandable dissections:
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B4 ≈ 6Y4 ⊕ 8Y3 -
Cuboctahedron - decomposes into square and triangular pyramids. Expand with triangular prisms between the square pyramids and triangular pyramids.
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T4 ≈ 6Q4 ⊕ 8Y3 ⊕ (P4) -
Truncated Cube - decomposes into square cupolas and triangular pyramids around a central cube. Expand with triangular prisms between the square cupolas and triangular pyramids (remove the central cube).
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K3 ≈ 6Y4 ⊕ 8Q3 ⊕ (S3) -
Truncatd Octahedron - decomposes into square pyramids and triangular cupolas around a central octahedron. Expand with triangular prisms between the square pyramids and triangular cupolas (remove the central octahedron).
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K4 ≈ 6Q4 ⊕ 8Q3 ⊕ 12P4 ⊕ (E4) -
Truncated Cuboctahedron - decomposes into square and triangular cupolae, and cubes around a central rhombicuboctahedron. Expand with triangular prisms between the square cupolas and the triangular cupolas (remove the cubes and the central rhombicuboctahedron).
For those polyhedronists who have read B.M. Stewart's Adventures Among the Toroids, these four figures are particularly interesting because they satisfy all of his conditions (Q)(A)(R)(T)(D) but they have convex hulls that include irregular polygons. It does not appear that Stewart was aware of these figures. (For those who have not read this great book, the criteria he was most interested in were (Q)-All edges of the convex hull of the figure are also edges of the figure, (A)-No consecutive faces are coplanar, (R)-All faces are regular polygons, (T)-The overall figure is tunnelled - genus > 0 and all excavations from the hull go toward increasing the genus, (D)-The figure cannot be self-intersecting.)
Since these figures are my own discovery, and I don't have any obscure collection of polyhedra named after me, I propose that all (Q)(A)(R)(T)(D) toroids with convex hulls that include irregular faces henceforth be referred to as "Doskey Toroids".
Note: All polyhedra on this page were discovered by me, and most of them were modeled using my Jovo polyhedron toys. To my knowledge none of these figures had been published prior to my discovery. Generation of VRML models was expedited by the use of Robert Webb's Stella application. If you have this program, here are the .Stel files:
6Y4+8Y3+12P3.stel,
6Q4+8Y3+24P3.stel,
6Y4+8Q3+24P3.stel,
6Q4+8Q3+24P3.stel,