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.

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. When a canonical polyhedron is dualised in this way, the canonical dual is obtained.

It can also be convenient to use it as an inversion sphere.

See also

• Inscribed sphere
• Circumscribed sphere

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

• Eric W. Weisstein, Midsphere at MathWorld.

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