by Marek14 » Wed Jan 11, 2012 2:26 am
And just for fun, how it works in hyperbolic space (convex and concave borders might be the other way around sometimes):
There are 3 kinds of C-D diagrams, 3 linear ones, 1 branching and 5 cyclical.
Linears are 353, 435 and 535, corresponding to 4 regular tilings of H3, icosahedral, cubical and two kinds of dodecahedral tilings.
353:
Regular: Made from icosahedra, 12 to vertex. You can cut pentagonal pyramids off 12 icosahedra and fit a dodecahedral cell inside. You can also border it with pseudohedra, in three different ways:
1: {5,4} pseudohedron. Both convex and concave border is formed by alternating pentagonal pyramids and gyroelongated pentagonal pyramids, but concave border has more icosahedra fit between them, while on convex border they fit together snugly.
2: {5,5} pseudohedron. The convex border is made purely of gyroelongated pentagonal pyramids, while the concave border is made of pentagonal pyramids.
3: (3,5,3,5,3,5) pseudohedron. The convex border is made of icosahedra and pentagonal pyramids, the concave border is made of icosahedra and gyroelongated pentagonal pyramids.
Rectified: Made from dodecahedra and icosidodecahedra. Can be bordered by a (5,10,10) pseudohedron: convex border is made of dodecahedra and pentagonal rotundas, concave border is made of icosidodecahedra and pentagonal rotundas.
Truncated: Made from dodecahedra and truncated icosahedra. Can't be diminished.
Small rhombated: Made from rhombicosidodecahedra, triangular prisms and icosidodecahedra. An icosidodecahedral cell can be enlarged to truncated dodecahedron by swallowing 20 nearby triangular prisms and cutting pentagonal cupolas from 12 nearby rhombicosidodecahedra. In addition, the tiling can be bordered by a (4,10,10) pseudohedron, convex border is made of triangular prisms, diminished rhombicosidodecahedra and pentagonal rotundas, while concave border is bordered by rhombicosidodecahedra, pentagonal cupolas and pentagonal rotundas.
Small prismated: Made from icosahedra and triangular prisms. An icosahedral cell can be enlarged to rhombicosidodecahedron by cutting pentagonal pyramids from 12 nearby icosahedra and swallowing 20+30 nearby triangular prisms. It can be bordered by a (4,5,4,5) plane, the border is made of triangular prisms, gyroelongated pentagonal pyramids and pentagonal pyramids.
Bitruncated: Made from truncated dodecahedra. Can't be diminished.
Great rhombated: Made from truncated icosidodecahedra, triangular prisms and truncated dodecahedra. Can't be diminished.
Prismatotruncated: Made from truncated icosahedra, hexagonal prisms, triangular prisms and rhombicosidodecahedra. A truncated icosahedral cell can be enlarged to truncated icosidodecahedron by cutting pentagonal cupolas from 12 nearby rhombicosidodecahedra and swallowing 20 nearby hexagonal prisms and 30 nearby triangula prisms.
Great prismated: Made from truncated icosidodecahedra and hexagonal prisms. Can't be diminished.
435 group:
Regular 435: Made from cubes. Can be bordered by a {4,5} pseudohedron, both convex and concave border are formed by cubes, but concave border has extra cubes fit to the edges.
Regular 534: Made from dodecahedra. Can be bordered by a {5,4} plane, border formed by dodecahedra.
Rectified 435: Made from cuboctahedra and icosahedra. Icosahedral cell can be enlarged to truncated icosahedron by cutting pentagonal pyramid from 12 nearby icosahedra and cutting 20 nearby cuboctahedra in half. In addition, the tiling can be bordered by a (3,6,5,6) pseudohedron; convex border consists of cuboctahedra, gyroelongated pentagonal pyramids and triangular cupolas, concave border consists of icosahedra, pentagonal pyramids and triangular cupolas. It can be also bordered by a (4,5,4,5) pseudohedron; convex border consists of cuboctahedra and pentagonal pyramids, concave border consists of cuboctahedra and gyroelongated pentagonal pyramids.
Rectified 534: Made from icosidodecahedra and octahedra. Can be bordered by a (4,10,10) pseudohedron, convex border consists of square pyramids and pentagonal rotundas, concave border consists of those as well, but with additional icosidodecahedra fit to edges. Can be also bordered by a (3,10,3,10) plane, border consisting of octahedra, icosidodecahedra and pentagonal rotundas, or by a (4,5,4,5) plane, border consisting of icosidodecahedra and square pyramids.
Truncated 435: Made from truncated cubes and icosahedra. Can be bordered by a (5,8,8) pseudohedron, convex border consisting of pentagonal pyramids and truncated cubes, concave border consisting of gyroelongated pentagonal pyramids and truncated cubes.
Truncated 534: Made from truncated dodecahedra and octahedra. Can be bordered by a (4,10,10) plane, border consisting of square pyramids and truncated dodecahedra.
Small rhombated 435: Made from rhombicuboctaherda, pentagonal prisms and icosidodecahedra. Can be bordered by a (5,8,8) pseudohedron, convex border consisting of pentagonal prisms and gyroelongated square cupolas, concave border consisting of icosidodecahedra and square cupolas. Can be also bordered by a (4,8,10) pseudohedron, convex border consisting of pentagonal prisms, square cupolas and pentagonal rotundas, convave border consisting of rhombicuboctahedra, elongated square cupolas and pentagonal rotundas.
Small rhombated 534: Made from rhombicosidodecahedra, cubes and cuboctahedra. A rhombicosidodecahedral cell can be enlarged to truncated icosidodecahedron by cutting pentagonal cupolas from 12 nearby rhombicosidodecahedra, cutting 20 nearby cuboctahedra in half and swallowing 30 nearby cubes. The tiling can be also bordered by a (4,10,10) pseudohedron, convex border consisting of cubes and diminished rhombicosidodecahedra, concave border consisting of cuboctahedra and pentagonal cupolas.
Small prismated 435: Made from cubes of two kinds, pentagonal prisms and dodecahedra. Can be bordered by a (4,4,4,5) pseudohedron, convex border consisting of cubes and pentagonal prisms, concave border consisting of cubes, pentagonal prisms and dodecahedra.
Bitruncated 435: Made from truncated octahedra and truncated icosahedra. Can't be diminished.
Great rhombated 435: Made from truncated cuboctahedra, pentagonal prisms and truncated icosahedra. Can't be diminished.
Great rhombated 534: Made from truncated icosidodecahedra, cubes and truncated octahedra. Can't be diminished.
Prismatotruncated 435: Made from truncated cubes, octagonal prisms, pentagonal prisms and rhombicosidodecahedra. Can be bordered by a (4,8,10) pseudohedron, convex border consisting of octagonal prisms, truncated cubes and pentagonal cupolas, concave border consisting of pentagonal prisms, octagonal prisms and diminished rhombicosidodecahedra.
Prismatotruncated 534: Made from truncated dodecahedra, decagonal prisms, cubes and rhombicuboctahedra. Can be bordered by a (4,8,10) pseudohedron, convex border consisting of decagonal prisms, truncated dodecahedra and square cupolas, concave border consisting of cubes, decagonal prisms and elongated square cupolas.
Great prismated 435: Made from truncated cuboctahedra, octagonal prisms, decagonal prisms and truncated icosidodecahedra. Can't be diminished.
535 group:
Regular: Made from dodecahedra. Can be bordered by a {5,5} pseudohedron, convex border consisting of dodecahedra, concave border as well, but with extra dodecahedra fit to edges.
Rectified: Made from icosidodecahedra and icosahedra. Can be bordered by a (5,10,10) pseudohedron, convex border consisting of pentagonal pyramids and pentagonal rotundas, concave border consisting of gyroelongated pentagonal pyramids and pentagonal rotundas. Can be also bordered by a {5,4} pseudohedron, convex border consisting of pentagonal pyramids and icosidodecahedra, concave border consisting of gyroelongated pentagonal pyramids and icosidodecahedra. Can be also bordered by a (3,10,5,10) pseudohedron, convex border consisting of icosidodecahedra, gyroelongated pentagonal pyramids and pentagonal rotundas, concave border consisting of icosahedra, pentagonal pyramids and pentagonal rotundas.
Truncated: Made from truncated dodecahedra and icosahedra. Can be bordered by a (5,10,10) pseudohedron, convex border consisting of pentagonal pyramids and truncated dodecahedra, concave border consisting of gyroelongated pentagonal pyramids and truncated dodecahedra.
Small rhombated: Made from rhombicosidodecahedra, pentagonal prisms and icosidodecahedra. Can be bordered by a (4,10,10) pseudohedron, convex border consisting of cubes, pentagonal cupolas and pentagonal rotundas, concave border consisting of rhombicosidodecahedra, diminished rhombicosidodecahedra and pentagonal rotundas. Can be also bordered by a (5,10,10) pseudohedron, convex border consisting of pentagonal prisms and diminished rhombicosidodecahedra, concave border consisting of icosidodecahedra and pentagonal cupolas.
Small prismated: Made from dodecahedra and pentagonal prisms. Can be bordered by a (4,5,4,5) plane, border consisting of pentagonal prisms and dodecahedra.
Bitruncated: Made from truncated icosahedra. Can't be diminished.
Great rhombated: Made from truncated icosidodecahedra, pentagonal prisms and truncated icosahedra. Can't be diminished.
Prismatotruncated: Made from truncated dodecahedra, decagonal prisms, pentagonal prisms and rhombicosidodecahedra. Can be bordered by a (4,10,10) pseudoherdon, convex border consisting of decagonal prisms, truncated dodecahedra and pentagonal cupolas, concave border consisting of pentagonal prisms, decagonal prisms and diminished rhombicosidodecahedra.
Great prismated: Made from truncated icosidodecahedra and decagonal prisms. Can't be diminished.
Branching {3,3,5} group:
Semiregular (alternating): Made from tetrahedra and icosahedra. A vertex can be enlarged to icosidodecahedral cell, cutting pentagonal pyramid from 12 nearby icosahedra and swallowing 20 nearby tetrahedra. The tiling can be also bordered by a (3,5,3,5,3,5) pseudohedron, convex border consisting of tetrahedra and gyroelongated pentagonal pyramids, concave border consisting of icosahedra and pentagonal pyramids. It can be also bordered by a {5,5} pseudohedron, convex border consisting of pentagonal pyramids, concave border consisting of gyroelongated pentagonal pyramids. It can be also bordered by a {5,6} plane, border consisting of pentagonal pyramids and gyroelongated pentagonal pyramids. It can be also bordered by a {3,10} plane, border consisting of tetrahedra and icosahedra.
Truncated: Made from truncated tetrahedra, truncated icosahedra and icosidodecahedra. It can be bordered by a (6,6,10) plane, border consisting of truncated tetrahedra, truncated icosahedra and pentagonal rotundas.
Small rhombated: Made from rhombicosidodecahedra, tetrahedra and dodecahedra. A dodecahedral cell can be enlarged to truncated dodecahedron by cutting a pentagonal cupola from 12 nearby rhombicosidodecahedra and swallowing 20 nearby tetrahedra. The tiling can be also bordered by a (5,10,10) pseudohedron, convex border consisting of dodecahedra and diminished rhombicosidodecahedra, concave border consisting of rhombicosidodecahedra and pentagonal cupolas.
Great rhombated: Made from truncated icosidodecahedra, truncated tetrahedra and truncated dodecahedra. Can't be diminished.
Cyclical (3,3,3,4):
(3,3)-Semiregular: Made of tetrahedra and two kinds of octahedra. A vertex can be enlarged to a rhombicuboctahedral cell by cutting 6+12 nearby octahedra in half and swallowing 8 nearby tetrahedra. The tiling can be also bordered by a (3,4,3,4,3,4) pseudohedron, convex border consisting of octahedra and square pyramids, concave border consisting of tetrahedra and square pyramids. It can be also bordered by a {3,8} pseudohedron, convex border consisting of tetrahedra and octahedra, concave border consisting of octahedra only.
(3,4)-Semiregular: Made of tetrahedra, cubes and cuboctahedra. Can be bordered by a (3,6,4,6) pseudohedron, convex border consisting of tetrahedra, cubes and triangular cupolas, concave border consisting of cuboctahedra and triangular cupolas. Can be also bordered by a (3,4,3,4,3,4) pseudohedron, convex border consisting of tetrahedra and cuboctahedra, concave border consisting of cuboctahedra and cubes.
Truncated (343): Made of tetrahedra and truncated cubes. Can be bordered by a (3,8,3,8) plane, border consisting of tetrahedra and truncated cubes.
Truncated (334): Made of tetrahedra, cubes, truncated tetrahedra and truncated octahedra. Can be bordered by a (3,6,4,6) pseudohedron, convex border consisting of tetrahedra, truncated octahedra and truncated tetrahedra, concave border consisting of truncated tetrahedra, cubes and truncated octahedra.
Truncated (333): Made of octahedra and truncated tetrahedra. Can be bordered by a (3,6,4,6) pseudohedron, convex border consisting of truncated tetrahedra and square pyramids, concave border consisting of octahedra, square pyramids and truncated tetrahedra.
Small rhombated: Made of octahedra, two kinds of cuboctahedra and rhombicuboctahedra. A cuboctahedral cell can be enlarged to a truncated cuboctahedron by cutting square cupolas from 6 nearby rhombicuboctahedra, cutting 8 nearby cuboctahedra in half and cutting 12 nearby octahedra in half. The tiling can be also bordered by a (6,6,8) pseudohedron, convex border consisting of triangular cupolas and elongated square cupolas, concave border consisting of triangular cupolas and square cupolas. It can also be bordered by a (3,8,3,8) pseudohedron, convex border consisting of cuboctahedra, octahedra and square cupolas, concave border consisting of cuboctahedra and elongated square cupolas. It can also be bordered by a (3,6,4,6) pseudohedron, convex border consisting of octahedra, rhombicuboctahedra and triangular cupolas, concave border consisting of rhombicuboctahedra, cuboctahedra and triangular cupolas. It can also be bordered by a (4,4,4,6) plane, border consisting of square pyramids, cuboctahedra, rhombicuboctahedra and triangular cupolas.
Great rhombated (33): Made from two kinds of truncated octahedra, truncated tetrahedra and rhombicuboctahedra. Can be bordered by a (6,6,8) pseudohedron, convex border consisting of truncated tetrahedra, truncated octahedra and square cupolas, concave border consisting of truncated octahedra and elongated square cupolas.
Great rhombated (34): Made from truncated tetrahedra, truncated cubes, truncated cuboctahedra and cuboctahedra. Can be bordered by a (6,6,8) pseudohedron, convex border consisting of truncated tetrahedra, triangular cupolas and truncated octahedra, concave border consisting of truncated cuboctahedra, triangular cupolas and truncated cubes.
Great prismated: Made from truncated octahedra and truncated cuboctahedra. Can't be diminished.
Cyclical (3,3,3,5):
(3,3)-Semiregular: Made of tetrahedra, octahedra and icosahedra. A vertex can be enlarged to a rhombicosidodecahedral cell, cutting pentagonal pyramids from 12 nearby icosahedra, cutting 30 nearby octahedra in half and swallowing 20 nearby tetrahedra. Can be bordered by a (3,5,3,5,3,5) pseudohedron, convex border consisting of octahedra and pentagonal pyramids, concave border consisting of tetrahedra and gyroelongated pentagonal pyramids. Can be also bordered by a {3,10} pseudohedron, convex border consisting of octahedra and tetrahedra, concave border consisting of icosahedra and octahedra. Can be also bordered by a (3,5,3,5,3,5,3,5,3,5) pseudohedron, convex border consisting of octahedra and gyroelongated pentagonal pyramids, concave border consisting of tetrahedra and pentagonal pyramids. Can be also bordered by a (3,4,4,3,4,4,3,4,4) pseudohedron, convex border consisting of icosahedra and square pyramids, concave border consisting of octahedra and square pyramids.
(3,5)-Semiregular: Made of tetrahedra, dodecahedra and icosidodecahedra. Can be bordered by a (3,10,5,10) pseudohedron, convex border consisting of tetrahedra, dodecahedra and pentagonal rotundas, concave border consisting of icosidodecahedra and pentagonal rotundas. Can be also bordered by a (3,5,3,5,3,5) pseudohedron, convex border consisting of tetrahedra and icosidodecahedra, concave border consisting of icosidodecahedra and dodecahedra.
Truncated (353): Made of tetrahedra and truncated dodecahedra. Can be bordered by a (3,10,3,10) plane, border consisting of tetrahedra and truncated dodecahedra.
Truncated (335): Made of tetrahedra, dodecahedra, truncated tetrahedra and truncated icosahedra. Can be bordered by a (3,6,10,6) pseudohedron, convex border consisting of tetrahedra, truncated icosahedra and truncated tetrahedra, concave border consisting of truncated tetrahedra, dodecahedra and truncated icosahedra.
Truncated (333): Made of icosahedra and truncated tetrahedra. Can be bordered by a (3,6,5,6) pseudohedron, convex border consisting of truncated tetrahedra and pentagonal pyramids, concave border consisting of icosahedra, gyroelongated pentagonal pyramids and truncated tetrahedra. Can be also bordered by a (5,6,5,6) plane, border consisting of pentagonal pyramids, gyroelongated pentagonal pyramids and truncated tetrahedra.
Small rhombated: Made of octahedra, icosidodecahedra, cuboctahedra and rhombicosidodecahedra. An icosidodecahedral cell can be enlarged to a truncated icosidodecahedron by cutting pentagonal cupolas from 12 nearby rhombicosidodecahedra, cutting 20 nearby cuboctahedra in half and cutting 30 nearby octahedra in half. The tiling can be also bordered by a (6,10,10) pseudohedron, convex border consisting of triangular cupolas, diminished rhombicosidodecahedra and pentagonal rotundas, concave border consisting of triangular cupolas, pentagonal cupolas and pentagonal rotundas. It can also be bordered by a (3,10,3,10) pseudohedron, convex border consisting ofcuboctahedra, octahedra and pentagonal cupolas, concave border consisting of icosidodecahedra, cuboctahedra and diminished icosidodecahedra. It can also be bordered by a (3,6,5,6) pseudohedron, convex border consisting of octahedra, rhombicosidodecahedra and triangular cupolas, concave border consisting of rhombicosidodecahedra,icosidodecahedra and triangular cupolas. It can also be bordered by a (4,4,4,10) plane, border consisting of square pyramids, cuboctahedra, rhombicosidodecahedra and pentagonal rotundas.
Great rhombated (3,3): Made of truncated tetrahedra, truncated octahedra, truncated icosahedra and rhombicosidodecahedra. Can be bordered by a (6,6,10) pseudohedron, convex border consisting of truncated tetrahedra, truncated octahedra and pentagonal cupolas, concave border consisting of truncated octahedra, truncated icosahedra and diminished rhombicosidodecahedra.
Great rhombated (3,5): Made of truncated tetrahedra, truncated dodecahedra, truncated icosidodecahedra and cuboctahedra. Can be bordered by a (6,6,10) pseudohedron, convex border consisting of truncated tetrahedra, triangular cupolas and truncated icosidodecahedra, concave border consisting of truncated octahedra, triangular cupolas and truncated dodecahedra.
Great prismated: Made from truncated octahedra and truncated icosidodecahedra. Can't be diminished.
Cyclical (3,4,3,4):
Semiregular: Made of cubes, octahedra and cuboctahedra. Can be bordered by a (4,6,4,6) pseudohedron, convex border consisting of cubes, square pyramids and triangular cupolas, concave border consisting of cuboctahedra, square pyramids and triangular cupolas. Can be also bordered by a {4,6} pseudohedron, convex border consisting of cuboctahedra and square pyramids, concave borded consisting of cubes and square pyramids. Can be also bordered by a (3,4,3,4,3,4,3,4) pseudohedron, convex border consisting of cuboctahedra and cubes, concave border consisting of octahedra and cuboctahedra.
Truncated (434): Made of cubes and truncated octahedra. Can be bordered by a (4,6,4,6) plane, border consisting of cubes and truncated octahedra.
Truncated (343): Made of octahedra and truncated cubes. Can be bordered by a (3,8,4,8) pseudohedron, convex border consisting of truncated cubes and square pyramids, concave border consisting of octahedra, square pyramids and truncatedcubes.
Small rhombated: Made of cuboctahedra and rhombicuboctahedra. Can be bordered by a (6,8,8) pseudohedron, convex border consisting of triangular cupolas, square cupolas and elongated square cupolas, concave border consisting of triangular cupolas, elongated square cupolas and square cupolas, but with more polyhedra fit to edges. Can also be bordered by a (3,8,4,8) pseudohedron, convex border consisting of cuboctahedra, rhombicuboctahedra and square cupolas, concave border consisting of cuboctahedra, rhombicuboctahedra and elongated square cupolas. Can also be bordered by a (4,6,4,6) plane, border consisting of rhombicuboctahedra and triangular cupolas.
Great rhombated: Made of truncated cubes, truncated octahedra, truncated cuboctahedra and rhombicuboctahedra. Can be bordered by a (6,8,8) pseudohedron, convex border consisting of truncated octahedra, truncated cuboctahedra and elongated square cupolas, concave border consisting of truncated cuboctahedra, truncated cubes and square cupolas.
Great prismated: Made from truncated cuboctahedra. Can't be diminished.
Cyclical (3,4,3,5):
(3,4)-Semiregular: Made of cubes, cuboctahedra and icosahedra. Can be bordered by a (4,6,5,6) pseudohedron, convex border consisting of cubes, pentagonal pyramids and triangular cupolas, concave border consisting of cuboctahedra, gyroelongated pentagonal pyramids and triangular cupolas. Can be also bordered by a (4,5,4,5,4,5) pseudohedron, convex border consisting of cuboctahedra and pentagonal pyramids, concave border consisting of cubes and gyroelongated pentagonal pyramids. Can be also bordered by a (3,4,3,4,3,4,3,4,3,4) pseudohedron, convex border consisting of cuboctahedra and cubes, concave border consisting of icosahedra and cuboctahedra. Can be also bordered by a (4,5,4,5,4,5,4,5,4,5) pseudohedron, convex border consisting of cuboctahedra and gyroelongated pentagonal pyramids, concave border consisting of cubes and pentagonal pyramids.
(3,5)-Semiregular: Made of dodecahedra, octahedra and icosidodecahedra. Can be bordered by a (4,10,5,10) pseudohedron, convex border consisting of square pyramids, pentagonal rotundas and dodecahedra, concave border consisting of square pyramids, pentagonal rotundas and icosidodecahedra. Can be also bordered by a (4,5,4,5,4,5) pseudohedron, convex border consisting of square pyramids and icosidodecahedra, concave border consisting of square pyramids and dodecahedra. Can be also bordered by a (3,5,3,5,3,5,3,5) pseudohedron, convex border consisting of icosidodecahedra and dodecahedra, concave border consisting of octahedra and icosidodecahedra.
Truncated (435): Made of cubes, dodecahedra, truncated octahedra and truncated icosahedra. Can be bordered by a (4,6,5,6) pseudohedron, convex border consisting of cubes, truncated icosahedra and truncated octahedra, concave border consisting of truncated octahedra, dodecahedra and truncated icosahedra.
Truncated (353): Made of octahedra and truncated dodecahedra. Can be bordered by a (3,10,4,10) pseudohedron, convex border consisting of truncated dodecahedrahedra and square pyramids, concave border consisting of octahedra, square pyramids and truncated dodecahedra.
Truncated (343): Made of icosahedra and truncated cubes. Can be bordered by a (3,8,5,8) pseudohedron, convex border consisting of truncated cubes and pentagonal pyramids, concave border consisting of icosahedra, gyroelongated pentagonal pyramids and truncated cubes. Can be also bordered by a (5,8,5,8) plane, border consisting of pentagonal pyramids, gyroelongated pentagonal pyramids and truncated cubes.
Small rhombated: Made of cuboctahedra, icosidodecahedra, rhombicuboctahedra and rhombicosidodecahedra. Can be bordered by a (6,8,10) pseudohedron, convex border consisting of triangular cupolas, elongated square cupolas and pentagonal cupolas, concave border consisting of triangular cupolas, square cupolas and diminished rhombicosidodecahedra. Can be also bordered by a (8,10,10) pseudohedron, convex border consisting of square cupolas, diminished rhombicosidodecahedra and pentagonal rotundas, concave border consisting of elongated square cupolas, pentagonal cupolas and pentagonal rotundas. Can also be bordered by a (3,10,4,10) pseudohedron, convex border consisting of rhombicuboctahedra, cuboctahedra and pentagonal cupolas, concave border consisting of icosidodecahedra, rhombicuboctahedra and diminished rhombicosidodecahedra. Can also be bordered by a (3,8,5,8) pseudohedron, convex border consisting of cuboctahedra, rhombicosidodecahedra and square cupolas, concave border consisting of rhombicosidodecahedra,icosidodecahedra and elongated square cupolas. Can also be bordered by a (4,6,4,10) plane, border consisting of rhombicuboctahedra, rhombicosidodecahedra, triangular cupolas and pentagonal rotundas.
Great rhombated (3,4): Made of truncated cubes, truncated icosahedra, truncated cuboctahedra and rhombicosidodecahedra. Can be bordered by a (6,8,10) pseudohedron, convex border consisting of truncated icosahedra, truncated cuboctahedra and diminished rhombicosidodecahedra, concave border consisting of truncated cuboctahedra, truncated cubes and pentagonal cupolas.
Great rhombated (3,5): Made of truncated octahedra, truncated dodecahedra, truncated icosidodecahedra and rhombicuboctahedra. Can be bordered by a (6,8,10) pseudohedron, convex border consisting of truncated octahedra, elongated square cupolas and truncated icosidodecahedra, concave border consisting of truncated icosidodecahedra, square cupolas and truncated dodecahedra.
Great prismated: Made from truncated cuboctahedra and truncated icosidodecahedra. Can't be diminished.
Cyclical (3,5,3,5):
Semiregular: Made of dodecahedra, icosidodecahedra and icosahedra. Can be bordered by a (5,10,5,10) pseudohedron, convex border consisting of dodecahedra, pentagonal pyramids and pentagonal rotundas, concave border consisting of icosidodecahedra, gyroelongated pentagonal pyramids and pentagonal rotundas. Can be also bordered by a {5,6} pseudohedron, convex border consisting of icosidodecahedra and pentagonal pyramids, concave border consisting of dodecahedra and gyroelongated pentagonal pyramids. Can be also bordered by a (3,5,3,5,3,5,3,5,3,5) pseudohedron, convex border consisting of icosidodecahedra and dodecahedra, concave border consisting of icosahedra and icosidodecahedra. Can be also bordered by a {5,10} pseudohedron, convex border consisting of icosidodecahedra and gyroelongated pentagonal pyramids, concave border consisting of dodecahedra and pentagonal pyramids.
Truncated (535): Made of dodecahedra and truncated icosahedra. Can be bordered by a (5,6,5,6) plane, border consisting of dodecahedra and truncated icosahedra.
Truncated (353): Made of icosahedra and truncated dodecahedra. Can be bordered by a (3,10,5,10) pseudohedron, convex border consisting of truncated dodecahedra and pentagonal pyramids, concave border consisting of icosahedra, gyroelongated pentagonal pyramids and truncated dodecahedra. Can be also bordered by a (5,10,5,10) plane, border consisting of pentagonal pyramids, gyroelongated pentagonal pyramids and truncated dodecahedra.
Small rhombated: Made of icosidodecahedra and rhombicosidodecahedra. Can be bordered by a {10,3} pseudohedron, convex border consisting of pentagonal rotundas, pentagonal cupolas and diminished rhombicosidodecahedra, concave border consisting of pentagonal rotundas, diminished rhombicosidodecahedra and pentagonal cupolas, but with more polyhedra fit to edges. Can also be bordered by a (3,10,5,10) pseudohedron, convex border consisting of rhombicosidodecahedra, icosidodecahedra and pentagonal cupolas, concave border consisting of icosidodecahedra, rhombicosidodecahedra and diminished rhombicosidodecahedra. Can also be bordered by a (4,10,4,10) plane, border consisting of rhombicosidodecahedra and pentagonal rotundas.
Great rhombated: Made of truncated dodecahedra, truncated icosahedra, truncated icosidodecahedra and rhombicosidodecahedra. Can be bordered by a (6,10,10) pseudohedron, convex border consisting of truncated icosahedra, truncated icosidodecahedra and diminished rhombicosidodecahedra, concave border consisting of truncated icosidodecahedra, truncated dodecahedra and pentagonal cupolas.
Great prismated: Made from truncated icosidodecahedra. Can't be diminished.