Midsphere

An opaque white polyhedron with four triangular faces and four quadrilateral faces is crossed by a transparent blue sphere of approximately the same size, tangent to each edge of the polyhedron. The visible portions of the sphere, outside the polyhedron, form circular caps on each face of the polyhedron, of two sizes: smaller in the triangular faces, and larger in the quadrilateral faces. Red circles on the surface of the sphere, passing through these caps, mark the horizons visible from each polyhedron vertex. The red circles have the same two sizes as the circular caps: smaller circles surround the polyhedron vertices where three faces meet, and larger circles surround the vertices where four faces meet. — A polyhedron and its midsphere. The red circles are the boundaries of spherical caps within which the surface of the sphere can be seen from each vertex.

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, 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 about this sphere.^[1]

When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing.

Every convex polyhedron has a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere, centered at the centroid of the points of tangency of its edges. Numerical approximation algorithms can construct the canonical polyhedron, but its coordinates cannot be represented exactly as a closed-form expression. Any canonical polyhedron and its polar dual can be used to form two opposite faces of a four-dimensional antiprism.

^ Grünbaum (2005).

[FOOTNOTEGrünbaum2005-1] Grünbaum (2005).

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