# Studies into Polyhedra - Regular

Look around the Graphics Lab

## Regular and Semi-Regular Solids

On this first page I present the Regular (Platonic), Semi-Regular (Archimedian) and their mathematical Duals. So as to just present the results I have move the details of my study to a separate page (See link above). It is recomended reading, as it explains a lot of the results below, including how images are generated.

### Platonic Solids (5) -- Regular Polyhedra

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 ```Name V F E F-Type Truncation Dual tetrahedron 4 4 6 triangles truncated tetrahedron tetrahedron cube 8 6 12 squares truncated cube octahedron octahedron 6 8 12 triangles truncated octahedron cube dodecahedron 20 12 30 pentagons truncated dodecahedron icosahedron icosahedron 12 20 30 triangles truncated icosahedron dodecahedron ``` tetrahedron off   vrml wikipedia cube off   vrml wikipedia octahedron off   vrml wikipedia dodecahedron off   vrml wikipedia icosahedron off   vrml wikipedia All vertices, edge mid-points and face mid-points lie on concentric spheres All faces are the same shape and are all regular polygons Thus all edges are equal in length and face corners equal in angle. Duals are also all Plationic Solids. The cube is also called a hexahedron

### Archimedean Solids (13) Semi-Regular Polyhedra

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 ```tetrahedral symmetry Name V F E Truncation Generates truncated tetrahedron 12 8 18 ``` truncated_tetrahedron off   vrml wikipedia ```cub-octahedral symmetry Name V F E Truncation Generates truncated cube 24 14 36 truncated octahedron 24 14 36 cuboctahedron 12 14 24 rhombicuboctahedron rhombicuboctahedron 24 26 48 great rhombicuboctahedron great rhombicuboctahedron 48 26 72 snub cuboctahedron 24 38 60 ``` truncated_cube off   vrml wikipedia truncated_octahedron off   vrml wikipedia cuboctahedron off   vrml wikipedia rhombicuboctahedron off   vrml wikipedia truncated_cuboctahedron off   vrml wikipedia snub_cuboctahedron off   vrml wikipedia ```icosi-dodecahedral symmetry Name V F E Truncation Generates truncated dodecahedron 60 32 90 truncated icosahedron 60 32 90 icosidodecahedron 30 32 60 rhomb-icosidodecahedron rhombicosidodecahedron 60 62 120 great rhomb-icosidodecahedron great rhombicosidodecahedron 120 62 180 snub icosidodecahedron 60 92 150 ``` truncated_dodecahedron off   vrml wikipedia truncated_icosahedron off   vrml wikipedia icosidodecahedron off   vrml wikipedia rhombicosidodecahedron off   vrml wikipedia truncated_icosidodecahedron off   vrml wikipedia snub_icosidodecahedron off   vrml wikipedia All faces are regular polygons but not of same shape. All edges are equal in length. All vertices have the same number of edges attached (usually 3) All vertices and edges are co-spherical. BUT NOT face mid points! The "truncated icosahedron" is more commonly known as a soccer ball The "cuboctahedron" is produced if the truncation of the "cube" and/or "octahedron" continues in sequence. Similarly for the "icosi-dodecahedron" for the "dodecahedron" and/or "icosahedron". Truncation is the term commonly used but as true truncation can produce rectangular faces. The correct term is rombic, where the rectangles are modified to more regular squares. By doing this the resulting Archimedean Solid retains the regular polygonal faces, and the "all edges of equal length" fact. The "snub cuboctahedron" (sometimes called "snub cube") has a left and right-handed version. As does the "snub icosidodecahedron" (or "snub dodecahedron") A "irregular rhombicuboctahedron" (also called a "elongated square gyrobicupola" - J37), also exists, where one section is given a 45 degree twist (caution is advised is building). Similarly an "irregular cuboctahedron" (or "triangular orthobicupola - J28), and "irregular icosidodecahedron" (or "pentagonal orthobirotunda" - J34), also exist by twisting two equal halves, of these objects.

### Catalan solid or Archimedean Duals (13)

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 ```Archimedean Dual of the Archimedean F Type triakis tetrahedron truncated tetrahedron 12 triangles triakis octahedron truncated cube 24 triangles tetrakis hexahedron truncated octahedron 24 trianlges triakis icosahedron truncated dodecahedron 60 triangles pentakis dodecahedron truncated icosahedron 60 triangles ``` triakis_tetrahedron off   vrml wikipedia triakis_octahedron off   vrml wikipedia tetrakis_hexahedron off   vrml wikipedia triakis_icosahedron off   vrml wikipedia pentakis_dodecahedron off   vrml wikipedia ```rhombic dodecahedron cuboctahedron 12 rhombus kite icositetrahedron rhombicuboctahedron 24 kites disdyakis dodecahedron great rhombicuboctahedron 48 triangles pentagonal icositetrahedron snub cuboctahedron 24 tears ``` rhombic_dodecahedron off   vrml wikipedia kite_icositetrahedron off   vrml wikipedia disdyakis_dodecahedron off   vrml wikipedia pentagonal_icositetrahedron off   vrml wikipedia ```rhombic tricontahedron icosidodecahedron 30 rhombus kite hexacontahedron small rhombicosidodecahedron 60 kites disdyakis triacontahedron great rhombicosidodecahedron 120 triangles pentagonal hexacontahedron snub dodecahedron 60 tears ``` rhombic_triacontahedron off   vrml wikipedia kite_hexecontahedron off   vrml wikipedia disdyakis_triacontahedron off   vrml wikipedia pentagonal_hexecontahedron off   vrml wikipedia ```Face descriptions: triangles are isosceles triangles rhombus are equal sided parallelogram or diamond shaped quadrilaterals. kites are diagonally mirror symmetrical quadrilaterals, or kite shaped. tears are roughly hexagonal shaped with two sides extended to remove one point, EG: a tear shaped symmetrical pentagon. ``` Faces are NOT regular polygons, but are all symmetrical in some way. EG: isosceles triangles, rather than equilateral triangles. All faces in an object are the same size, shape, and co-spherical. IE: they make good, unusual looking dice! Edge mid-points and vertices are NOT co-spherical. As the two "snub" archmidean solids have left and right versions, their duals, also have left and right versions. EG: left and right versions of the "pentagonal icositetrahedron" and "pentagonal hexacontahedron". The phrase xx-aconta-xx means ten, EG tri-aconta means 30. The phrase xx-akis-xx means a face has a low pyramid built on it, EG: tri-akis means 3 triangles forming a low pyramid on a triangular face George Hart uses the phase disdyakis-xx, which is a low four sided pyramid (like a tetrakis-xx) but on a rombic or diamond shaped face. Naming objects by pyramid building on simpler objects is not without problems. A "disdyakis-dodecahedron" (pyramids on 12 rombic faces) is often called a "hexakis-octahedron" (octahedron based), or even a "octakis-hexahedron" (cube based). It all depends on how you look at the object. The same goes for the "disdyakis-triacontahedron", which can be called a "hexakis-icosahedron" or just as easily, "decakis-dodecahedron". I named the "kite_xx" objects based on the shape of the face, though George Hart names them "trapezoidal_xx" (which is confusing) and Wikipedia names them "deltoidal_yy" where "yy" is a different base object, and not the count of the number of faces.

Created: 30 April 2001
Updated: 28 January 2009
Author: Anthony Thyssen, <anthony@cit.gu.edu.au>
URL: `http://www.cit.gu.edu.au/~anthony/graphics/polyhedra/`