Studies into Polyhedra - Regular

Look around the Graphics Lab

Regular, Semi-Regular Polyhedra, and thier Duals (first page)
Prisms, Anti-prisms, Pryamids, and related Polyhedra
Miscellanous Polyhedra: Deltahedra
Johnson Solids -- The other convex polyhedra with regular faces

Why I studied polyhedra, and Image Generation Techniques
Known Polyhedral Mathematical Formula
Data Sources and links for Polyhedral Data

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

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
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tetrahedron
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cube
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octahedron
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dodecahedron
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icosahedron
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  • 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

tetrahedral symmetry
Name                            V   F   E    Truncation Generates
truncated tetrahedron          12  8   18
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truncated_tetrahedron
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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
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truncated_cube
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truncated_octahedron
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cuboctahedron
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rhombicuboctahedron
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truncated_cuboctahedron
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snub_cuboctahedron
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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
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truncated_dodecahedron
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truncated_icosahedron
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icosidodecahedron
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rhombicosidodecahedron
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truncated_icosidodecahedron
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snub_icosidodecahedron
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  • 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)

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
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triakis_tetrahedron
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triakis_octahedron
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tetrakis_hexahedron
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triakis_icosahedron
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pentakis_dodecahedron
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rhombic dodecahedron          cuboctahedron                  12  rhombus
kite icositetrahedron         rhombicuboctahedron            24  kites
disdyakis dodecahedron        great rhombicuboctahedron      48  triangles
pentagonal icositetrahedron   snub cuboctahedron             24  tears
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rhombic_dodecahedron
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kite_icositetrahedron
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disdyakis_dodecahedron
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pentagonal_icositetrahedron
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rhombic tricontahedron        icosidodecahedron              30  rhombus
kite hexacontahedron          small rhombicosidodecahedron   60  kites
disdyakis triacontahedron     great rhombicosidodecahedron  120  triangles
pentagonal hexacontahedron    snub dodecahedron              60  tears
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rhombic_triacontahedron
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kite_hexecontahedron
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disdyakis_triacontahedron
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pentagonal_hexecontahedron
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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/