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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

Copyright 2002 Wendy Krieger

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