Semi-regular polyhedron

A semi-regular polyhedral is a Polyèdre with regular faces and a group of symmetry which is transitive on its tops. Or at least, it is what rises from the definition from 1900 from Gossett on the most general Polytope semi-regular. These polyhedrons include:

thirteen solid of Archimedes .
the infinite series of the convex prisms .
the infinite series of the convex Antiprisme S (their semi-regular nature was observed in first by Kepler).

These solid semi-regular can be entirely specified by a Configuration of top, a list of the faces by the number of with dimensions in the order where they appear around a top. For example 3.5.3.5 , represents the Icosidodécaèdre where two alternate Triangle S and two pentagons around each top. 3.3.3.5 on the contrary is a pentagonal Antiprisme. These polyhedrons are sometimes described like uniform Sommet.

Since Gossett, other authors used the semi-regular term in various ways. Elte gave a definition that Coxeter found too artificial. Coxeter itself doubled the uniform figures of Gossett, with only one completely restricted subset classified like semi-regular.

Others still, took the opposed way, categorizing more polyhedrons like semi-regular. Those include:

Three spangled polyhedral whole of which coincides with the definition of Gossett, similar to the three convex units listed above.
the duaux of the semi-regular solids above, pointing out that since the duaux polyhedrons share same symmetries that the originals, they should also be looked like semi-regular. These duaux includes the solid of Catalan , the convex Diamant S and the antidiamants or Trapèzoèdre S , and their analogues not-convex.

An additional source of confusion is given in the way in which the solid of Archimedes are defined, with the appearance of new different interpretations.

The definition of Gossett of the semi-regularity includes figures of higher symmetry, the regular polyhedrons and quasi-regular. Certain later authors prefer to say than these polyhedrons are not semi-regular, because they are more regular than that - the polyhedral uniforms are then known as to include the regular ones, theregular ones and theregular ones. This nomenclature goes well and reconciles much (but not all) confusions.

In practice, even the most eminent authorities can itself be muddled, defining a given whole of polyhedrons like semi-regular and/or archimédien, and then supposing (or even establishing) a different whole during the following discussions. To suppose that the definition established on applies only to the convex polyhedrons is probably the most common fault. Coxeter, Cromwell and Cundy & Rollett are all guilty such slips.

External references

George Binder: polyhedrons of semi-regular Archimedes
David Darling: semi-regular polyhedron
polyhedra.mathmos.net: semi-regular polyhedron
Encyclopedia of mathematics: semi-regular, polyhedral polyhedrons uniform, solids of Archimedes

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