Regular polytope

Regular polytopes are classified primarily according to their dimensionality.

In one dimension, the Line Segment simoltaneously serves as all of these polytopes, and in two dimensions, the square can act as both the Measure Polytope and Cross Polytope at the same time.

The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.

A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

If the Schläfli symbol is palindromic, i.e. reads the same forwards and backwards, then the polyhedron is self-dual. The self-dual regular polytopes are:

A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. The rotation takes place in the xw plane.

At the start of the 20th century, the definition of a regular polytope was as follows.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.

The Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.

This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.

The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace.

A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the geometric. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.

The theory has since been further developed, largely by McMullen & Schulte (2002), but other researchers have also made contributions.

Regularity has a related, though different meaning for abstract polytopes, since angles and lengths of edges have no meaning.

The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.

Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes don't care about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope.

All polygons are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world.

Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures may or may not be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons.

The classical vertex figure will, however, be a realisation of the abstract one.

Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.

The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface.

In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of an object having 4 or more dimensions. There are several approaches normally taken to overcome this matter.

Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner.