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1. Catalan solid – In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, the Catalan solids are all convex. They are face-transitive but not vertex-transitive and this is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons, however, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra, additionally, two of the Catalan solids are edge-transitive, the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids, just as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral, the pentagonal icositetrahedron and these each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids, the Catalan solids, along with their dual Archimedean solids, can be grouped by their symmetry, tetrahedral, octahedral, and icosahedral. There are 6 forms per symmetry, while the self-symmetric tetrahedral group only has three forms and two of those are duplicated with octahedral symmetry. J. lÉcole Polytechnique 41, 1-71,1865, alan Holden Shapes, Space, and Symmetry. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Williams, the Geometrical Foundation of Natural Structure, A Source Book of Design. California, University of California Press Berkeley, chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Catalan Solids. Archived from the original on 4 February 2007, Archimedean duals – at Virtual Reality Polyhedra Interactive Catalan Solid in Java

2. Triakis tetrahedron – In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron and it can be seen as a tetrahedron with triangular pyramids added to each face, that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name, the length of the shorter edges is 3/5 that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area 5/3√11, a triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will be coplanar and this can be seen by adding the 6 edges of tetrahedron inside of a cube. This chiral figure is one of thirteen stellations allowed by Millers rules, the triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design

3. Triakis octahedron – In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube and it can be seen as an octahedron with triangular pyramids added to each face, that is, it is the Kleetope of the octahedron. It is also called a trisoctahedron, or, more fully. Both names reflect the fact that it has three triangular faces for every face of an octahedron, the tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron. This convex polyhedron is topologically similar to the concave stellated octahedron and they have the same face connectivity, but the vertices are in different relative distances from the center. The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube, the triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design. Triakis Octahedron – Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, dtC

4. Tetrakis hexahedron – In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the octahedron, an Archimedean solid. It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron, the tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. Naturally occurring formations of tetrahexahedra are observed in copper and fluorite systems, polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers. The tetrakis hexahedron appears as one of the simplest examples in building theory, consider the Riemannian symmetric space associated to the group SL4. Its Tits boundary has the structure of a building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices can be obtained by taking the radial projection of a tetrakis hexahedron, with Td, tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere and it can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point. The edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles and these edges can also be seen as a compound of 3 orthogonal square hosohedrons. If we denote the length of the base cube by a. The inclination of each face of the pyramid versus the cube face is arctan. One edge of the triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of √5a/4 in the triangle and its area is √5a/8, and the internal angles are arccos and the complementary 180° −2 arccos. The volume of the pyramid is a3/12, so the volume of the six pyramids. It can be seen as a cube with square pyramids covering each square face and it is a polyhedra in a sequence defined by the face configuration V4.6. 2n. With an even number of faces at every vertex, these polyhedra, each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face Catalan solid, the Geometrical Foundation of Natural Structure, A Source Book of Design

5. Triakis icosahedron – In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron, the last two correspond to the A2 and H2 Coxeter planes. It can be seen as an icosahedron with triangular pyramids augmented to each face and this interpretation is expressed in the name, triakis. The triakis icosahedron is a part of a sequence of polyhedra and tilings and these face-transitive figures have reflectional symmetry. Triakis triangular tiling for other triakis polyhedral forms, the Geometrical Foundation of Natural Structure, A Source Book of Design. The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Triakis Icosahedron – Interactive Polyhedron Model

6. Pentakis dodecahedron – In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face, that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name, there are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include, The usual Catalan pentakis dodecahedron, a convex hexecontahedron with sixty isosceles triangular faces illustrated in the sidebar figure and it is a Catalan solid, dual to the truncated icosahedron, an Archimedean solid. As the heights of the pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi. As the height is raised further, the shape becomes non-convex, other more non-convex geometric variants include, The small stellated dodecahedron. Great pentakis dodecahedron Wenningers third stellation of icosahedron, if one affixes pentagrammic pyramids into Wenningers third stellation of icosahedron one obtains the great icosahedron. The pentakis dodecahedron in a model of buckminsterfullerene, each surface segment represents a carbon atom, equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom. The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses and these have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron. The pentakis dodecahedron has three positions, two on vertices, and one on a midedge, The Spaceship Earth structure at Walt Disney Worlds Epcot is a derivative of a pentakis dodecahedron. The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https, in Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron and these Domes also appear whenever the player transforms on a dome in the Hypno Ray level. Some Geodomes in which play on are Pentakis Dodecahedra. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Pentakis dodecahedron at MathWorld. Pentakis Dodecahedron – Interactive Polyhedron Model