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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 52B: Polytopes and polyhedra

Introduction

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

History

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]
• 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 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.

Software and tables

Other web sites with this focus

• virtual polyhedra

Selected topics at this site

A few basic questions keep arising with regard to polygons:

• How do you compute the area enclosed by a polygon?
• How do you find its center of mass of a polygon (program included)
• How do you find the centroid of a polygon (pointer)
• How do you decide if a point is interior
• How to decide if you're inside a polygon? (pointer, citation)
• How can you check a corner for concavity.
• A combinatorial question: how many regions result when connecting all the vertices of a regular polygon?
• Number of regions formed by diagonals in a polygon
• Unbounded number of regions as intersection of convex polygons

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

• A pointer to code for Delaunay triangulation
• How to plot circular motion on a display?
• How to compute the convex hull of some points in the plane?
• (An interesting variant on computing areas of cyclic polygons is also included.
• For variety, here's a sample of a trigonometric approach to determining the area of a pentagon.
• For triangles, you may wish to use Heron's formula

Then a few comments about polyhedral surfaces in 3-space.

• Using Cayley-Menger determinant to determine radius of circumscribed sphere around a tetrahedron
• What shape is a soccer ball?
• Regular, semiregular polyhedra and the disphenoid
• Can one always unwrap the surface of a polyhedron to get something flat and nonoverlapping? (open)
• Here's a long spiel (with short punchline) on evaluating volumes of polyhedra.
• Another post computing volumes of polyhedra.
• How to compute the volume of a polyhedron? Pointers, citations, summary
• Volume of a tetrahedron (in terms of sides)
• How to compute the volume of a simplex in R^n in terms of its sides.
• Relating volumes of simplices to vertices, edges, or lengths.
• Finding volumes of an n-dimensional polyhedron
• Proving the analogue of the Pythagorean theorem in higher dimensions.
• realizability of polyhedral surfaces.
• Pointers to hexaflexagons
• Flexible polyhedra.
• Bellows theorem: flexible polyhedra maintain their volume
• Instructions for making a kaleidocycle (flexible polyhedron)

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.)

• Read about Polyhedral versions of 1- and 2-holed tori which have a small number of vertices
• Pasting information for 1- and 2-holed tori with few cells.
• Image of a one-holed torus made only with triangles, in which all pairs of the (seven) vertices are joined by edges.

This in turn led to a discussion of just how few vertices you need to create g-holed tori.

• A summary of what the questions are regarding polyhedral tori.
• Program and literature review (both long) for g-holed tori with few vertices
• Another program for polyhedral g-holed tori.
• A short summary of some basic data for polyhedral tori.
• A related post on polyhedral tori.
• Triangularizations of tori -- how nice can they be?
• Is there a polyhedral torus made of equilateral triangles?
• [Offsite]Models of minimal polyhedral models of genus-6 tori (evidently cannot be linearly embedded into R^3)

General questions on polyhedra:

• Just what is a polytope and how does it differ from a polygon or polyhedron? (opinions vary!)
• Open question: can every convex polyhedron be cut along edges, then laid flat without self-overlap?
• If you want to really build these things, here are a couple of construction tips
• The rhombic dodecahedron, use as space filler.
• Pointer: virtual polyhedra (pretty pictures).
• References on Euler's formula for polyhedra.
• The four regular nonconvex polyhedra (Kepler-Poinsot)
• Numerical data for many polyhedra -- pointer
• Pointer to numerical data on 4-dimensional polytopes
• Coordinates of a dodecahedron
• Some information about the vertices of the dodecahedron
• Coordinates of an icosahedron and "rhombicubeoctahedron".
• Some information about the edges of the dodecahedron.
• Description of the truncated octahedron.
• Pointer to gallery of Archimedean solids
• Edge-transitive polyhedra in R^3
• Decomposing polyhedra into convex or tetrahedral pieces
• Under what circumstances can we decompose a polyhedron into pieces which reassemble into another given polyhedron? (The Dehn invariant)
• Can a 3-dimensional polyhedron be decomposed into tetrahedra? (Not without adding interior points in general)
• Tiling S^3 with 600 congruent spherical tetrahedra
• Visualizing the 16-cell (solid in R^4)
• Construction of the 120-cell and 600-cell (solids in R^4)
• That four-dimensional polytope with no three-dimensional analogue.
• Unusual four-dimensional polyhedra: the 24-cell, 120-cell, and 600-cell.
• Generalizations of the Platonic solids to dimensions 4 and up.
• From the sci.math FAQ: How can you chop up a ball and reassemble the parts (the Banach Tarski paradox, and related issues).
• Decomposing a square and a circle into congruent (nonmeasurable!) parts
• Two polygons of equal area may be decomposed into congruent triangles.
• A dissection problem: how to dissect a square into pieces with minimal perimeter.
• Dissecting a rectangle into squares -- ratio of sides rational
• Dissecting an equilateral triangle into incongruent equilateral triangles (can't)
• Dissecting a cube into distinct smaller cubes
• Dissecting n-cubes into (minimal numbers of) simplices
• Using Kirchhoff's laws to solve geometric combinatorial problems
• Divide a square into acute triangles
• Decomposing a square into incongruent squares: [No link here, in an attempt to avoid the ASCII art. See Guy's "Unsolved Problems in Geometry", section C2.]
• The maximum surface-area tetrahedron inscribed in a sphere is the regular one.
• A "tetrahedral inequality": under what circumstances can six line segments be joined into a tetrahedron?
• Pointer to KALEIDO (program for regular polyhedra)