Midsphere

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.

Equivalently, it is a sphere that contains the inscribed circle of every face of the polyhedron.

Not every convex polyhedron has a midsphere; to have a midsphere, every face must have an inscribed circle (that is, it must be a tangential polygon), and all of these inscribed circles must belong to a single sphere.

For example, a rectangular cuboid has a midsphere only when it is a cube, because otherwise it has non-square rectangles as faces, and these do not have inscribed circles.

[3] For a unit cube centered at the origin of the Cartesian coordinate system, with vertices at the eight points

In this case, the six edge lengths of the tetrahedron are the pairwise sums of the four radii of these spheres.

)[1] Dually, if v is a vertex of P, then there is a cone that has its apex at v and that is tangent to O in a circle; this circle forms the boundary of a spherical cap within which the sphere's surface is visible from the vertex.

The circles formed in this way are tangent to each other exactly when the vertices they correspond to are connected by an edge.

The face planes of the polar polyhedron pass through the circles on O that are tangent to cones having the vertices of P as their apexes.

[2] The edges of the polar polyhedron have the same points of tangency with the midsphere, at which they are perpendicular to the edges of P.[10] For a polyhedron with a midsphere, it is possible to assign a real number to each vertex (the power of the vertex with respect to the midsphere) that equals the distance from that vertex to the point of tangency of each edge that touches it.

For instance, Crelle's tetrahedra can be parameterized by the four numbers assigned in this way to their four vertices, showing that they form a four-dimensional family.

[12] One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by the vertices and edges of a polyhedron with a midsphere.

[15] Any polyhedron with a midsphere, scaled so that the midsphere is the unit sphere, can be transformed in this way into a polyhedron for which the centroid of the points of tangency is at the center of the sphere.

The result of this transformation is an equivalent form of the given polyhedron, called the canonical polyhedron, with the property that all combinatorially equivalent polyhedra will produce the same canonical polyhedra as each other, up to congruence.

[17] For polyhedra with a non-cyclic group of orientation-preserving symmetries, the two choices of transformation coincide.

[18] For example, the canonical polyhedron of a cuboid, defined in either of these two ways, is a cube, with the distance from its centroid to its edge midpoints equal to one and its edge length equal to

[19] A numerical approximation to the canonical polyhedron for a given polyhedral graph can be constructed by representing the graph and its dual graph as perpendicular circle packings in the Euclidean plane,[20] applying a stereographic projection to transform it into a pair of circle packings on a sphere, searching numerically for a Möbius transformation that brings the centroid of the crossing points to the center of the sphere, and placing the vertices of the polyhedron at points in space having the dual circles of the transformed packing as their horizons.

[21] Alternatively, a simpler numerical method for constructing the canonical polyhedron proposed by George W. Hart works directly with the coordinates of the polyhedron vertices, adjusting their positions in an attempt to make the edges have equal distance from the origin, to make the points of minimum distance from the origin have the origin as their centroid, and to make the faces of the polyhedron remain planar.

Unlike the circle packing method, this has not been proven to converge to the canonical polyhedron, and it is not even guaranteed to produce a polyhedron combinatorially equivalent to the given one, but it appears to work well on small examples.

[1] The midsphere in the construction of the canonical polyhedron can be replaced by any smooth convex body.

[22] Moreover, fixing three edges of the cage to have three specified points of tangency on the egg causes this realization to become unique.

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.
Four white spheres of equal sizes, centered on the vertices of a regular tetrahedron, touch each other.
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.
An outlined magenta cube and green octahedron, arranged so that each cube edge crosses an octahedron edge at the midpoint of both edges. A translucent sphere, concentric with the cube and octahedron, passes through all of the crossing points.
Cube and dual octahedron with common midsphere
Six blue circles, each tangent to four other circles, arranged in two triangles of three large outer circles and three small inner circles. Three more red circles cross each other and the blue circles at right angles. Each of the six red-red crossings is inside one of the blue circles, and each red-blue crossing is at a point where two blue circles touch each other. The red-red crossings are highlighted by small yellow circles.
A circle packing in the plane (blue) obtained by stereographically projecting the horizon circles on the midsphere of an octahedron. The yellow vertices and red edges represent the octahedron itself, centrally projected onto the midsphere and then stereographically projected onto the plane.