Midsphere

In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere.

It is so-called because it is between the inscribed sphere (touches every face) and the circumscribed sphere (touches every vertex).

The radius of this sphere is called the midradius.

Important classes of polyhedra which have interspheres include:

• Canonical polyhedra.

  These have the unit sphere for their midsphere, i.e. midradius = 1.

• The Uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals.

Where the dual polyhedron is also considered, for example in constructing a dual compound, the intersphere is commonly used as the reciprocating sphere (or inversion sphere) for polar reciprocation. When a canonical polyhedron is dualised in this way, the canonical dual is obtained.

See also

Other spheres

• Inscribed sphere
• Circumscribed sphere

Applications

• Dual polyhedron

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• Cundy, H.M. and Rollett, A.P. Mathematical Models, OUP (Second Edition 1961).
• Hart, G. Calculating canonical polyhedra, Mathematica in Education and Research 6, Issue 3 (1997), pp 5–10.

External links

• Weisstein, Eric W., "Midsphere" from MathWorld.

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