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# 52B: Polytopes and polyhedra

## Introduction

Here are a few files concerning geometric objects made from straight pieces: polygons, polyhedra, and generalizations.

## Applications and related fields

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.

## Subfields

• 52B05: Combinatorial properties (number of faces, shortest paths, etc.), See also 05Cxx
• 52B10: Three-dimensional polytopes
• 52B11: n-dimensional polytopes
• 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
• 52B15: Symmetry properties of polytopes
• 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also 06A08, 13F20, 13Hxx
• 52B22: Shellability [new in 2000]
• 52B30: Arrangements of hyperplanes [to be dropped in MSC2000]
• 52B35: Gale and other diagrams
• 52B40: Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), See Also 05B35
• 52B45: Dissections and valuations (Hilbert's third problem, etc.)
• 52B55: Computational aspects related to convexity, For computational geometry and algorithms, See 68Q20, 68Q25, 68U05; for numerical algorithms, See 65Yxx
• 52B60: Isoperimetric problems for polytopes
• 52B70: Polyhedral manifolds
• 52B99: None of the above but in this section

Parent field: Convex and discrete geometry

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

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 Weinbaum: 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.

## Selected topics at this site

A few basic questions keep arising with regard to polygons:

A couple of odds and ends of pointers to software (not tested here!):