This page was inspired by the works of Bonnie Madison Stewart in his book Adventures Among the Toroids. In this book he mentions a few examples of regular-faced convex polyhedra that can be decomposed into numerous smaller regular-faced polhyedra. These decomposable figures are often the source for some interesting Toroidal polyhedra.
The most interesting ones are the ones that the decomposition is of higher symmetry than prisms. These are few in count, and I am working on coming up with a nice collection of them. I will also have a few of prism symmetry, but these are not nearly as interesting.
The sample model shown here is the most fascinating dissection I have seen yet. Stewart describes it in his notation as:
- S3 ≈ 2Y4 - Octahedra (triangular antiprisms) can be split into two square pyramids.
- I5 ≈ Y5 ⊕ S5 ⊕ Y5 - Icosahedra can be split into a pentagonal antiprism augmented with two pentagonal pyramids.
- There are also 4 Archimedean solids that will soon be listed here.
- There are also 2/3 of the Johson solids that will soon be listed on a separate page.
- P6 ≈ 6P3 - Hexagonal prisms can be split into six triangular prisms.
- P12 ≈ 6P3 ⊕ 6P4 ⊕ (P6) - Dodecagonal prisms can be split into 6 cubes and 6 tri prisms, around a central hex prism.
- P12 ≈ 6P3 ⊕ 6P4 ⊕ (6P3) - Same as above, but the central hexagonal prism can be further split.
- There are also several of the Johson solids that could be listed in this section, and will also soon be listed on a separate page.
- B4 ≈ 6Y4 ⊕ 8Y3 - Cuboctahedron - decomposes into square and triangular pyramids.
- T4 ≈ 6Q4 ⊕ 8Y3 ⊕ (P4) - Truncated Cube - decomposes into cupolae and pyramids around a cube.
- K3 ≈ 6Y4 ⊕ 8Q3 ⊕ (S3) - Truncatd Octahedron - decomposes into cupolae and pyramids around an octahedron.
- K4 ≈ 6Q4 ⊕ 8Q3 ⊕ 12P4 ⊕ (E4) - Truncated Cuboctahedron - decomposes into cupolae and cubes around a rhombicuboctahedron.
- E5 ≈ 6J91 ⊕ 8G3 ⊕ 12Y5 ⊕ (P4) - Rhombicosidodecahedron - decomposes into several interesting figures.
- D5 ≈ 2G3 ⊕ 4Y5 ⊕ 5-4-3-acrohedron (Mode C) - Dodecahedron - decomposes into several interesting figures. This dissection is not a highly symmetrical one, but it was rather interesting to tie it in with the acrohedra from Jim McNeill's page
- Remember, this is a hobby of mine, not my day job. There will eventually be several more in this category as well.
E5 ≈ 6J91 ⊕ 8G3 ⊕ 12Y5 ⊕ (P4)
The Rhombicosidodecahedron (E5) can be dissected into six Bilunabirotundas (J91), eight unnamed figures simply referred to as G3 (which is discussed on my Stewart G3 page), and twelve pentagonal pyramids (Y5) all around a central cube (P4).
I have broken the dissections up into three categories of interest - linear, planar and spherical. Linear models are decomposed into a linear collection of polyhedra, that can be stacked upon each other (the centers of the stacked polyhedra are linear). This is generally only interesting if the resulting conglomeration is a uniform solid. Planar models correspond to conglomerations that can be obtained by sliding several coplanar components together to form a larger figure (here the centers of the component polyhedra are coplanar instead of colinear). Primarily this will consist of prisms sliding together to form larger prisms, or augmentations being added to a central figure in prism symmetry fashion. The third category are the ones that interest me most, and the components connect together with a symmetry greater than that of a prism.
Linear Dissections.
Planar Dissections.
Spherical Dissections
Each one will eventually link to its own page to show various perspectives.
Back to the main Polyhedron Page.
Link to this page as http://Polyhedra.Doskey.com/DecomposablePolyhedra.html