# Midsphere

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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

### Applications

## 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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