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 Discussion: All Topics Topic: How to calculate the ratio of Platonic solid's duals? Related Item: http://mathforum.org/mathtools/tool/500/

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 Subject: ratio of platonic solid's duals Author: Dennis Date: Apr 8 2003
It is a matter of scale and depends on how you define the dual.  The dual of any
planar graph is obtained by replacing each face with a vertex.  Faces that were
adjacent become vertices connected by a line.  this tells us nothing about how
big each graph is.

In the case of the platonic solids you need to decide what the physical
relationship is between a solid and its dual.  For example, when you consider
the octahedron as the dual of a cube, do you mean the octahedron inscribed in
the cube with a vertex at the center of each face of the cube, an octahedron
containing the cube with each vertex of the cube at the center of a face of the
octahedron, both inscribed in the same shpere, or some other relationship.  Once
you have decided on their physical relationship, their relative volumes can be
calculated.

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