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MIDSPHERE

Midsphere

Sphere tangent to every edge of a polyhedron

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

Midsphere

Dual polyhedron

Polyhedron associated with another by swapping vertices for faces

that, for a polyhedron with a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents), this can be used. However, it is possible

Dual polyhedron

Dual_polyhedron

Platonic solid

Any of the five regular polyhedra

^{\ast }\rho .} Dualizing with respect to the midsphere (d = ρ) is often convenient because the midsphere has the same relationship to both polyhedra.

Platonic solid

Platonic_solid

Circle packing theorem

On tangency patterns of circles

polyhedron that has the given graph as its vertices and edges and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Each vertex of

Circle packing theorem

Circle_packing_theorem

Steinitz's theorem

Graph-theoretic description of polyhedra

graph of a convex polyhedron all of whose edges are tangent to a common midsphere. An undirected graph is a system of vertices and edges, each edge connecting

Steinitz's theorem

Steinitz's_theorem

Polyhedron

Flat-sided three-dimensional shape

the triangular prism are elementaries. Some convex polyhedra possess a midsphere, a sphere tangent to each of their edges, which is intermediate in radius

Polyhedron

Inscribed sphere

Sphere tangent to every face of a polyhedron

'inspheres' of their polyhedra. Circumscribed sphere Inscribed circle Midsphere Sphere packing Coxeter, H.S.M. Regular Polytopes 3rd Edn. Dover (1973)

Inscribed sphere

Inscribed_sphere

Concentric objects

Geometric objects with a common centre

polygon#Regular polygons. The same can be said of a regular polyhedron's insphere, midsphere and circumsphere. The region of the plane between two concentric circles

Concentric objects

Concentric_objects

Circumscribed sphere

Sphere touching all of a polyhedron's vertices

time. Other spheres defined for some but not all polyhedra include a midsphere, a sphere tangent to all edges of a polyhedron, and an inscribed sphere

Circumscribed sphere

Circumscribed_sphere

Regular tetrahedron

Solid with four equal triangular faces

r_{\text{in}}} (a sphere within a regular tetrahedron and touches to its faces), midsphere r m i {\displaystyle r_{\mathrm {mi} }} (a sphere that touches its edges)

Regular tetrahedron

Regular_tetrahedron

Regular icosahedron

Solid with twenty equal triangular faces

is a sphere that contains the polyhedron and touches every vertex. The midsphere of a convex polyhedron is a sphere tangent to every edge. Given that the

Regular icosahedron

Regular_icosahedron

Rhombic dodecahedron

Catalan solid with 12 faces

r_{\mathrm {i} }={\frac {\sqrt {6}}{3}}a\approx 0.817a,} the radius of its midsphere is: OEIS: A179587) r m = 2 2 3 a ≈ 0.943 a , {\displaystyle r_{\mathrm

Rhombic dodecahedron

Rhombic_dodecahedron

Cube

Solid with six equal square faces

r_{i}} is a sphere tangent to the faces of a cube at their centroids. Its midsphere r m {\displaystyle r_{m}} is a sphere tangent to the edges of a cube.

Cube

Stellated octahedron

Polyhedral compound

called "compound of two tetrahedra". The two tetrahedra share a common midsphere, making the compound self-dual. Other compounds of two tetrahedra can

Stellated octahedron

Stellated_octahedron

Paul Koebe

German mathematician (1882–1945)

mehrfach zusammenhängender Bereiche". Jahresbericht DMV. Koebe groups Midsphere Riemann mapping theorem Koebe 1907a. Koebe 1907b. Koebe 1907c. Koebe 1910a

Paul Koebe

Paul_Koebe

Regular polyhedron

Polyhedron with regular congruent polygons as faces

share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices. The regular

Regular polyhedron

Regular_polyhedron

Petrie polygon

Skew polygon derived from a polytope

rectangular intersections in the points where the edges touch the common midsphere. The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6}

Petrie polygon

Petrie_polygon

Polytope compound

3D shape made of polyhedra sharing a common center

composed of a polyhedron and its dual, arranged reciprocally about a common midsphere, such that the edge of one polyhedron intersects the dual edge of the

Polytope compound

Polytope_compound

Deltoidal icositetrahedron

Catalan solid with 24 kite faces

{\sqrt {2}}{2}}}};} so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius r a = 1 = ρ 2 R = 2 (

Deltoidal icositetrahedron

Deltoidal_icositetrahedron

Regular octahedron

Solid with eight equal triangular faces

that tangent to each of the octahedron's faces), and the radius of a midsphere r m {\displaystyle r_{m}} (one that touches the middle of each edge),

Regular octahedron

Regular_octahedron

600-cell

Four-dimensional analog of the icosahedron

the unit-radius 24-cell, the first step is to reciprocate it around its midsphere to construct its outer canonical dual: a larger 24-cell, since the 24-cell

600-cell