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## Polyhedra

This category displays the five Platonic
solids (tetrahedron, hexahedron (or cube), octahedron, dodecahedron,
and icosahedron) both in their regular (Platonic) forms, and in their
truncated (Archimedean) versions.

Truncation cuts off the same portion of the
facet centered at each vertex by a plane orthogonal to the line from
the center to that vertex. The degree of truncation is controlled by
the parameter aa. When aa is 1 (or greater) there is no truncation
and the polyhedron is regular. When aa is 1/2 (or less) every facet
is completely cut off. The value aa=2/3 gives the standard
Archimedean truncation, characterized by the fact that all its edges
have equal length.

Perhaps the most famous of the Archimedean
polyhedra is the Buckyball, which is the truncated icosahedron. For
this reason it is given a special entry on the Polyhedron
menu.

The choice of a polyhedron is made in the
Regular submenu of the Polyhedron menu. When one of the five regular
solids is chosen, it is immediately displayed. Choosing "Create
Truncated" from the Action menu will display the Archimedean
truncation of the solid. Of course, choosing "Set Parameters..." from
the Settings menu and giving aa a value different from its default
value of 2/3 will produce a different degree of truncation. Choosing
Create from the Action menu will go back to the Platonic version of
the solid. You can also choose Create Stellated and Create Subdivided
to see other variations of these polyhedra.

Choosing Morph from the Animation menu will
produce a filmstrip of the solid being truncated from its Platonic to
its Archimedean version. If you choose Set Morphing... from the
Settings menu and give a1 the value 1/2 instead of 2/3, then morphing
will produce a filmstrip of the facets being cut off
completely.

While developing the algorithms for
truncation I received advice, help, and encouragement from Fan Chung,
Han Sah, Darko Babic, and Igor Rivin, and I would like to thank them
all.

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