Movatterモバイル変換

Regular Polytopes

• regular polygons, polygrams, or compounds ofdimension 2
• regular polyhedra or compounds ofdimension 3
• regular polychora or compounds ofdimension 4

This is the title of a famous book of H.S.M. Coxeter, and accordingly lots of the following can be found there too.(The below provided tables by dimensions had been assembled by D. de Winters, but his webpage ceased to exist after his death.They are thus reproduced here, in a further enriched form.)

The high degree of symmetry of regular polytopes induces lots of inter-relations onto the set of regular polytopes.There are not only dual pairings (which, according to regularity, both have to be regular polytopes!), but even theset of vertices (up to radial scalings) often are commonly used for large subsets each.Polytopes can be grouped accordingly to the dimensional size of shared sub-elements. The following systematics of Olshevsky does not only apply to regulars, but for the above reasons clearly has lotsof applications here:

In more detail, the army general of some regular (or even uniform) polytope always is itsconvex hull. Conversely, that polytope is a (full symmetrical) faceting of its army general.Alike, any polytope can be considered as an edge faceting of its regiment colonel, i.e. a faceting which respects the givenedge skeleton.Thereby its vertex figure obviously belongs to the army of thevertex figure of its colonel. Note that this doesnot imply the other way round, that the colonel always will have a convex vertex figure(which also is known aslocal convexity of the polytope itself).Sure, most often it does, but there will be cases, where a coating of the edge skeleton in the sense of a convex vertexfigure, demanded from one end of the edges, would ask for additional edges at the opposite end, which do not belongto the actual edge skeleton. Then the colonel would be chosen as the one with a vertex figure beingas convex as possible.For instance this takes place for the 10-tet-compound (e), where there are only 2 provided edges within each potential locally coating face plane.

(Johnson extended this nomenclatura slightly, by calling 2 or more different regiments (of the same army), using thesame edge length, abrigade. Further, a subset of polytopes of a brigade, which is closed under the operation ofblending,is called acohort. – Even close relatives of colonels deserve a special name, attributed by Bowers:lieutenants he calls the conjugates of colonels, which not themselves are colonels.)

Compounds too can fall into the same army as some regular polytope. This is what Coxeter once calledvertex regularity.

Just as thechords of polygons are defined as its vertex-to-vertex line-segments, generally any polytope shows up similarilychords of different lengths. Those might be grouped into ones of the same size, and further the classes might be ordered (and thereby named) by increasing size.A 0-chord then would be the vertex adjoined to itself, and for uniform convex polytopes the 1-chords would bethe set of edges.

In the following, compounds (esp. the polygonal ones), if not stated otherwise, are understood to be (fully) regular ones.

----2D----

----3D----

©

----4D----

The 10 regular polychora, which are obtained from theex  ©

1st :ex,fix,gohi,gahi	2nd :sishi,gaghi	5th :gax,gofix,gashi,gishi

©