52B: Polytopes and polyhedra
Here are a few files concerning geometric objects made from straight pieces: polygons, polyhedra, and generalizations.
Questions regarding the underlying spaces (and algebraic invariants) are included on the topology pages. Classic questions regarding polygons and regular solids are included on the geometry pages. There is a separate section on constructibility of polygons (and other things) with ruler and compass.
Many of these questions are related to themes arising in geometric visualization, a topic covered reasonably well on the net. In particular, the newsgroup comp.graphics.algorithms considers such themes from time to time. Some such topics are here, others on the page for computational geometry.
Parent field: Convex and discrete geometry
Browse all (old) classifications for this area at the AMS.
Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; Petrie, J. F.: "The fifty-nine icosahedra", Springer-Verlag, New York-Berlin, 1982. 26 pp. ISBN 0-387-90770-X
"Regular Polytopes", H.S.M Coxeter (Dover reprint) -- lots of formulas for polyhedra and so on.
Schreiber, Peter: "What is the true number of semiregular (Archimedean) solids?", Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt. Beiträge Algebra Geom. 35 (1994), no. 1, 91--94. MR95e:52020
"Polyhedron Models", by Magnus J. Wenninger (Cambridge University Press, London-New York, 1971): has models of all 52 uniform polyhedra and some stellations.
"Space, Shapes, and Symmetry" by Holden -- lots of pictures of models, not limited to paper.
A few basic questions keep arising with regard to polygons:
A couple of odds and ends of pointers to software (not tested here!):
Then a few comments about polyhedral surfaces in 3-space.
A chance encounter with polyhedral tori led to the chance to try some models. There are ways to build these with few polygonal pieces. We find some information when looking up piecewise-linear embeddings of n-holed tori into R^3. (n=0 includes the tetrahedron, for example.)
This in turn led to a discussion of just how few vertices you need to create g-holed tori.
General questions on polyhedra: