Editing Platonic solid (section)

=== Dual polyhedra ===
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Every polyhedron has a [[dual polyhedron|dual (or "polar") polyhedron]] '''with faces and vertices interchanged'''. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.

* The tetrahedron is [[self-dual polyhedron|self-dual]] (i.e. its dual is another tetrahedron).
* The cube and the octahedron form a dual pair.
* The dodecahedron and the icosahedron form a dual pair.

If a polyhedron has Schläfli symbol {''p'',&nbsp;''q''}, then its dual has the symbol {''q'',&nbsp;''p''}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.

One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.

More generally, one can dualize a Platonic solid with respect to a sphere of radius ''d'' concentric with the solid. The radii (''R'',&nbsp;''ρ'',&nbsp;''r'') of a solid and those of its dual (''R''*,&nbsp;''ρ''*,&nbsp;''r''*) are related by

<math display="block">d^2 = R^\ast r = r^\ast R = \rho^\ast\rho.</math>

Dualizing with respect to the midsphere (''d''&nbsp;=&nbsp;''ρ'') is often convenient because the midsphere has the same relationship to both polyhedra. Taking ''d''<sup>2</sup>&nbsp;=&nbsp;''Rr'' yields a dual solid with the same circumradius and inradius (i.e. ''R''*&nbsp;=&nbsp;''R'' and ''r''*&nbsp;=&nbsp;''r'').