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Midsphere
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In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, includin
The centers of four pairwise tangent spheres form the vertices of a Crelle's tetrahedron. Here, four equal spheres form a regular tetrahedron. The mid
The centers of four pairwise tangent spheres form the vertices of a Crelle's tetrahedron. Here, four equal spheres form a regular tetrahedron. The midsphere passes through the six points of tangency of these spheres, which in this case form a regular octahedron.
Dual polyhedron
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In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to
Midsphere
…polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals (Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed…
The dual of a Platonic solid can be constructed by connecting the face centers. In general this creates only a topological dual. Images from Kepler's
The dual of a Platonic solid can be constructed by connecting the face centers. In general this creates only a topological dual. Images from Kepler's Harmonice Mundi (1619)