Abstract Regular Polytopes (Encyclopedia of Mathematics and by Peter McMullen, Egon Schulte PDF

By Peter McMullen, Egon Schulte

ISBN-10: 0511065000

ISBN-13: 9780511065002

ISBN-10: 0521814960

ISBN-13: 9780521814966

Summary average polytopes stand on the finish of greater than millennia of geometrical study, which started with average polygons and polyhedra. The fast improvement of the topic long ago 20 years has ended in a wealthy new concept that includes an enticing interaction of mathematical components, together with geometry, combinatorics, workforce conception and topology. this can be the 1st entire, updated account of the topic and its ramifications. It meets a severe desire for any such textual content, simply because no publication has been released during this region given that Coxeter's "Regular Polytopes" (1948) and "Regular advanced Polytopes" (1974).

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A simple example is the icosahedron and the hemi-icosahedron (constructed from the former by identifying antipodes); both are of type {3, 5}. In using the Schl¨afli symbol { p1 , . . , pn−1 }, we repeat the convention introduced in Section 1B. If a symbol contains a string of k (say) equal integers p (say), then we shorten the symbol by replacing the string by a single p with a superscript k. For example, the symbol for the ordinary n-cube would be {4, 3n−2 }, standing for {4, 3, 3, . . , 3} with n − 2 entries 3.

20 1 Classical Regular Polytopes (if only to dismiss) a generalization of regularity for maps formulated in [310]. A map is called equivelar of type { p, q} if each of its faces is a p-gon, and these faces meet q at each vertex. A regular map must, of course, be equivelar, but the converse is far from true. Large classes of equivelar maps were constructed in [310, 311], where the interest was in whether they admitted polyhedral embeddings in E3 ; some of these maps are regular, particularly the simplest cases of those of type {4, q} and { p, 4} (see [50, 51, 309]).

21 22 2 Regular Polytopes In this chapter, we introduce the basic notation and concepts of the theory of abstract regular polytopes. Our notation will generally be patterned after the traditional theory, and will provide a convenient framework to study combinatorially regular structures resembling the classical regular polytopes. The chapter is organized as follows. In Section 2A, we discuss the concept of an abstract polytope without reference to regularity. Regular abstract polytopes, or as we shall say abstract regular polytopes, are then the subject of Section 2B.

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Abstract Regular Polytopes (Encyclopedia of Mathematics and its Applications 92) by Peter McMullen, Egon Schulte


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