Here is a collection of polyhedra that can be generated by "Parallel Expansions" of other polyhedra. The basic concept is to "expand" components of a polyhedron - either faces or edges - outward to form a new figure, while keeping those components parallel to the originals. In doing so, new faces may be formed, and/or old faces may be transformed from n-gons to 2n-gons. In particular, if two faces that share an edge are expanded outward (and away from each other), then the edge that they shared will be replaced with a square. If k faces that meet at a vertex are expanded outward, then the vertex they shared will be replaced by a regular k-gon. And if the n edges of a single face are expanded outward from each other (which I will refer to as exploding), then the n-gon will become a regular 2n-gon. The reverse process of "expansion" can be referred to as "contraction".One particular use of this parallel expansion is Prism Expansion which I have moved to another page to make room for some new models on this page.
As a first example, we can see how a triangular pyramid can be expanded by moving three of the triangles outward, and exploding the fourth triangle into a hexagon. The edges between the expanded triangles become squares, and the top vertex becomes a new triangle. The base triangle of the pyramid becomes a hexagon, and the result is a triangular cupola (one of the Johnson Solids - J3).
This operation can also be performed on square pyramids and pentagonal pyramids to produce square cupolas pentagonal cupolas. There are also numerous Johnson Solids that can be arrived at by expanding edges into squares. They are generally referred to as "elongated" polyhedra.
This parallel expansion method can also be seen as a means of creating some of the archimedean solids, either from a platonic solid or another archimedean solid. Below are a few examples. Click on the links to see an animation of the expansion process.
| Animation | Base Polyhedron | Expanded Polyhedron | Expanded | Exploded | New from Edges | New from Vertices |
| Y3-to-B4 | tetrahedron | cuboctahedron | 4 triangles | 6 squares | 4 triangles | |
| S3-to-T3 | octahedron | truncated tetrahedron | 4 triangles | 4 triangles | ||
| T3-to-K3 | truncated tetrahedron | truncated octahedron | 4 hexagons | 4 triangles | 6 squares | |
| P4-to-E4 | cube | small rhombicuboctahedron | 6 squares | 12 squares | 8 triangles | |
| P4-to-T4 | cube | truncated cube | 12 edges | 6 squares | 8 triangles | |
| S3-to-E4 | octahedron | small rhombicuboctahedron | 8 triangles | 12 squares | 6 squares | |
| S3-to-K3 | octahedron | truncated octahedron | 12 edges | 8 triangles | 6 squares | |
| B4-to-T4 | cuboctahedron | truncated cube | 8 triangles | 6 squares | ||
| B4-to-K3 | cuboctahedron | truncated octahedron | 6 squares | 8 triangles | ||
| B4-to-K4 | cuboctahedron | great rhombicuboctahedron | 24 edges | 6 squares + 8 triangles | 12 squares | |
| T4-to-K4 | truncated cube | great rhombicuboctahedron | 6 octagons | 8 triangles | 12 squares | |
| K3-to-K4 | truncated octahedron | great rhombicuboctahedron | 8 hexagons | 6 squares | 12 squares | |
| E4-to-K4 | small rhombicuboctahedron | great rhombicuboctahedron | 12 squares | 6 squares + 8 triangles | ||
| D5-to-E5 | dodecahedron | small rhombicosidodecahedron | 12 pentagons | 30 squares | 20 triangles | |
| D5-to-T5 | dodecahedron | truncated dodecahedron | 30 edges | 12 pentagons | 20 triangles | |
| I5-to-E5 | icosahedron | small rhombicosidodecahedron | 20 triangles | 30 squares | 12 pentagons | |
| I5-to-C5 | icosahedron | truncated icosahedron | 30 edges | 20 triangles | 12 pentagons | |
| B5-to-T5 | icosidodecahedron | truncated dodecahedron | 20 triangles | 12 pentagons | ||
| B5-to-C5 | icosidodecahedron | truncated icosahedron | 12 pentagons | 20 triangles | ||
| B5-to-K5 | icosidodecahedron | great rhombicosidodecahedron | 60 edges | 12 pentagons + 20 triangles | 30 squares | |
| T5-to-K5 | truncated dodecahedron | great rhombicosidodecahedron | 12 decagons | 20 triangles | 30 squares | |
| C5-to-K5 | truncated icosahedron | great rhombicosidodecahedron | 20 hexagons | 12 pentagons | 30 squares | |
| E5-to-K5 | small rhombicosidodecahedron | great rhombicosidodecahedron | 30 squares | 12 pentagons + 20 triangles |
Keep in mind that if two base polyhedra can both be expanded to form the same new polyhedron, then we have a new way of transforming between the two base polyhedra. For example, the cube and octahedron can both be expanded to form the rhombicuboctahedron. So if we want to transform from cube to octahedron, we can do so by expanding the cube to form the rhombicuboctahedron, and then contracting the rhombicuboctahedron to form the octahedron:
P4-to-E4-to-S3
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Link to this page as http://Polyhedra.Doskey.com/ParallelExpansions.html