Monday, August 28, 2017

[qllbwysh] Truncating or extending to a sphere

Given a regular polyhedron and inscribed sphere (insphere), truncate its vertices creating new faces that are tangent to the original inscribed sphere.  This process can be done on any polyhedron, and the successively truncated polyhedron approaches a sphere.  The new faces do not need to be constructed where old vertices were: the grinding stone could be applied anywhere, grinding to tangency to the insphere.

Dually, consider a starting polyhedron and sphere it is inscribed within (exsphere).  Above a face construct a vertex on the exphere then add edges joining the new vertex to the vertices of the face it is above.  These new faces will typically be triangular.  The successively truncated polyhedron also again approaches the sphere.  There is freedom of where above each face to construct the new vertices (perhaps one of the many triangle centers).  Sometimes edges of the starting polyhedron disappear as adjacent faces are coincident.  I suspect this happens when a cube is grown into a rhombic dodecahedron.

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