ISSN:
1432-0444
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract. The study of the polyhedra (in Euclidean 3-space) in which faces may be self-intersecting polygons, and distinct faces may intersect in various ways, was quite fashionable about a century ago. The Kepler—Poinsot regular polyhedra, and several of their generalizations, were investigated about that time by Cayley, Wiener, Badoureau, Fedorov, Hess, Pitsch, and others; the accumulated wisdom was presented in Max Brückner's well-known book Vielecke und Vielflache in 1900. Despite the intrinsic interest of the topic, and its relations to various other disciplines, there have been very few additional investigations during the intervening century, except for discussions of uniform polyhedra. In particular, there has been no mention or clarification of the many errors and other shortcomings of Brückner's book. One of our aims is to point out the extent of these inadequacies; they are illustrated by a discussion of isogonal prismatoids, the investigation of which is our main objective. A prismatoid is a polyhedron having all its vertices in two parallel planes. Familiar examples are prisms and antiprisms. A polyhedron P is isogonal if all its vertices form one transitivity class under isometric symmetries of P. Although these restrictions appear very severe, there exist many different kinds of isogonal prismatoids. Some general concepts concerning polyhedra with possible self-intersections are presented, and several classes of isogonal prismatoids are discussed in some detail.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00009307
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