This Geometer's Corner stems from work I am doing on
a book that focuses on how to use GeoGebra to construct
the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids.
I found one unifying principle for my work in using Jeff
Weeks' wonderful application KaleidoTile , which allows
you to construct the Cube-Octa and Icosa-Dodeca families
(minus the Snub Cube and the Snub Dodecahedron, which
are discussed in Consortium 120's Geometer's Corner: Snub
Polyhedra) in a way that shows how the solids are related to
each other. All the colored polyhedral images in Figures 1
and 2 were created with KaleidoTile.
In the case of the Cube Octa family, KaleidoTile transforms
a Cube into a Truncated Cube that is transformed into a Cuboctahedron that is transformed into a Truncated Octahedron that is transformed into an Octahedron that is
transformed into a Small Rhombicuboctahedron that is
transformed into either a Large Rhombicuboctahedron or a
Cube. In the case of the Icosa Dodeca family, a Dodecahedron is transformed into a truncated Dodecahedron that is
transformed into an Icosadodecahedron that is transformed
into a Truncated Icosahedron that is transformed into an
Icosahedron that is transformed into a Small Rhombicosadodecahedron that is transformed into either a Large
Rhombi cosadodecahedron or a Dodecahedron. The progression fascinated me, and I did some research to find out
how it worked.
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