Exceptional object

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions that don't fit into any series. These are known as exceptional objects.

Frequently these exceptional objects play a further and important role in the subject. Surprisingly or not, the exceptional objects in one branch of mathematics are often related to the exceptional objects in others.

A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values.

The prototypical examples of exceptional objects arise in the classification of regular polytopes. In two dimensions there is a series of regular n-gons for n ≥ 3. In every dimension above 2 we find analogues of the cube, tetrahedron and octahedron. In three dimensions we find two more regular polyhedra – the dodecahedron (12-hedron) and the icosahedron (20-hedron) – making five Platonic solids. In four dimensions we have a total of six regular polytopes including the 120-cell, the 600-cell and the 24-cell. There are no other regular polytopes; in higher dimensions the only regular polytopes are of the hypercube, simplex, orthoplex series. In all dimensions combined, there are therefore three series and five exceptional polytopes.

The pattern is similar if non-convex polytopes are included. In two dimensions there is a regular star polygon for every rational number p/q > 2. In three dimensions there are four Kepler–Poinsot polyhedra, and in four dimensions ten Schläfli–Hess polychora; in higher dimensions there are no non-convex regular figures.

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