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Wythoff's Construction

Notion

Wythoff's construction by mirror-edges creates figures by placing the vertex somewhere in the fundemental region, and dropping half-edges to each mirror of the region.

           o               I
           |               |\
           |               | \
   ========+========       |  \
           |           [3]tI   rr [2]
           |               | ot \
           e               |     \
                          ID--tD--D
                              [5]
     A Mirror Edge     The Icosahedral
                        group (2,3,5)

In the figure above, there is a mirror-egde o-e. To the right, there is a triangle representing the fundemental region of the icosahedral group. A vertex may be placed at any of the seven points shown in the triangle, to yield one of the seven uniform figures of icosahedral symmetry.

I: icosahedron: A regular figure, having five triangles at each vertex.
D: dodecahedron: A figure with twelve pentagonal faces.
ID: icosadodecahedron: A figure having alternating faces of triangles and pentagons.
tI: truncated icosahedron: This figure has pentagons and hexagons as faces. It is sometimes called the Buckyball after the inventor of the geodesic dome, Buckminster Fuller, or the Soccer ball, which often employ 32 panels of this type.
tD: truncated dodecahedron: This figure has triangles and decagons as faces.
rr: rhombi-icosadodecahedron: This figure has triangles, squares and pentagons as faces.
ot: omnitruncated dodecahedron: This figure has squares, hexagons and decagons as faces.

Notation

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