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  2. Snub disphenoid - Wikipedia

    en.wikipedia.org/wiki/Snub_disphenoid

    It can be constructed in different approaches. This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron. The snub disphenoid can be visualized as an atom cluster surrounding a central atom, that is the dodecahedral molecular geometry.

  3. Rhombicosidodecahedron - Wikipedia

    en.wikipedia.org/wiki/Rhombicosidodecahedron

    Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron ...

  4. Dodecahedron - Wikipedia

    en.wikipedia.org/wiki/Dodecahedron

    A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular anticupola connected base-to-base, called a triangular gyrobianticupola. It has D 3d symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which ...

  5. Geodesic polyhedron - Wikipedia

    en.wikipedia.org/wiki/Geodesic_polyhedron

    In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+} b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valence vertices. The + symbol indicates the valence of the vertices being increased. b,c represent a subdivision description, with 1,0 representing the ...

  6. Dodecahedral molecular geometry - Wikipedia

    en.wikipedia.org/wiki/Dodecahedral_molecular...

    In chemistry, the dodecahedral molecular geometry describes the shape of compounds where eight atoms or groups of atoms or ligands are arranged around a central atom defining the vertices of a snub disphenoid (also known as a trigonal dodecahedron). This shape has D 2d symmetry and is one of the three common shapes for octacoordinate transition ...

  7. Icosidodecahedron - Wikipedia

    en.wikipedia.org/wiki/Icosidodecahedron

    The dihedral angle of an icosidodecahedron between pentagon-to-triangle is ⁡ (+), determined by calculating the angle of a pentagonal rotunda. [ 4 ] An icosidodecahedron has icosahedral symmetry , and its first stellation is the compound of a dodecahedron and its dual icosahedron , with the vertices of the icosidodecahedron located at the ...

  8. Polyhedron - Wikipedia

    en.wikipedia.org/wiki/Polyhedron

    It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem, flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as ...

  9. Pentagonal hexecontahedron - Wikipedia

    en.wikipedia.org/wiki/Pentagonal_hexecontahedron

    Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal ...