midsphere

Short definition from Wiktionary

Noun

1. 1
   A middle sphere or region.
2. 2
   The sphere which, if it exists, is tangent to every edge of a given polyhedron and with respect to which the polyhedron's vertices are the inversion poles of the planes of the faces of its dual polyhedron (and vice versa).
3. 3
   The sphere, between two given spheres, by which each of the latter may be inverted into the other.
4. 4
   A supposed realm between earth and heaven.