The word polyhedron has slightly different meanings in geometry and algebraic geometry. In
geometry, a polyhedron is simply a three-dimensional solid which consists of a collection
of polygons, usually joined at their
edges. The word derives from the Greek poly (many) plus the Indo-European
hedron (seat). A polyhedron is the three-dimensional version of the more general
polytope (in the geometric sense),
which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra"
(or sometimes "polyhedrons").
The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can
be built from such "building blocks" as line segments, triangles, tetrahedra,
and their higher dimensional analogs by "gluing them together" along their
faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space of a simplicial
complex (with the additional constraint sometimes imposed that the complex be
finite; Munkres 1993, p. 9). In the usual definition, a polyhedron can be viewed
as an intersection of half-spaces, while a polytope
is a bounded polyhedron.
A convex polyhedron can be
formally defined as the set of solutions to a system of linear inequalities
where  is a real  matrix and  is a real  -vector. Although
usage varies, most authors additionally require that a solution be bounded for it
to define a convex polyhedron.
An example of a convex polyhedron is illustrated above.
The following table lists the name given to a polyhedron having  faces for small
 . When used without qualification for polyhedron
for which symmetric forms exist, the term may mean this particular polyhedron or
may mean a arbitrary  -faced polyhedron, depending on context.
A polyhedron is said to be regular if its faces and vertex figures are regular
(not necessarily convex) polygons (Coxeter
1973, p. 16). Using this definition, there are a total of nine regular polyhedra, five being the convex Platonic solids and four being the
concave (stellated) Kepler-Poinsot solids. However, the term "regular polyhedra"
is sometimes used to refer exclusively to the Platonic
solids (Cromwell 1997, p. 53). The dual
polyhedra of the Platonic solids
are not new polyhedra, but are themselves Platonic
solids.
A convex polyhedron is called semiregular if its faces have a similar arrangement of nonintersecting
regular planar convex polygons
of two or more different types about each polyhedron
vertex (Holden 1991, p. 41). These solids are more commonly called the Archimedean solids, and there
are 13 of them. The dual polyhedra
of the Archimedean solids
are 13 new (and beautiful) solids, sometimes called the Catalan solids.
A quasiregular polyhedron is the solid region interior to two dual regular polyhedra (Coxeter 1973,
pp. 17-20). There are only two convex quasiregular polyhedra:
the cuboctahedron and icosidodecahedron. There are also infinite families of prisms and antiprisms.
There exist exactly 92 convex polyhedra with regular polygonal
faces (and not necessarily equivalent vertices). They are known as the Johnson solids. Polyhedra with identical polyhedron vertices related by a symmetry operation are known
as uniform polyhedra. There
are 75 such polyhedra in which only two faces may meet at an polyhedron edge, and 76 in which any even number of faces may meet. Of these, 37 were discovered
by Badoureau in 1881 and 12 by Coxeter and Miller ca. 1930.
Polyhedra can be superposed on each other (with the sides allowed to pass through each other) to yield additional polyhedron
compounds. Those made from regular
polyhedra have symmetries which are especially aesthetically pleasing. The graphs
corresponding to polyhedra skeletons are called Schlegel
graphs.
Behnke et al. (1974) have determined the symmetry groups of all polyhedra symmetric with respect to their polyhedron
vertices.
Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13th ed. New York:
Dover, pp. 130-161, 1987.
Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2: Geometry. Cambridge,
MA: MIT Press, 1974.
Bulatov, V. "Polyhedra Collection." http://bulatov.org/polyhedra/.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Critchlow, K. Order in Space: A Design Source Book. New York: Viking
Press, 1970.
Cromwell, P. R. Polyhedra. New York: Cambridge University Press, 1997.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin
Pub., 1989.
Davie, T. "Books and Articles about Polyhedra and Polytopes." http://www.dcs.st-andrews.ac.uk/~ad/mathrecs/polyhedra/polyhedrabooks.html.
Davie, T. "The Regular (Platonic) and Semi-Regular (Archimedean) Solids." http://www.dcs.st-andrews.ac.uk/~ad/mathrecs/polyhedra/polyhedratopic.html.
Eppstein, D. "Geometric Models." http://www.ics.uci.edu/~eppstein/junkyard/model.html.
Eppstein, D. "Polyhedra and Polytopes." http://www.ics.uci.edu/~eppstein/junkyard/polytope.html.
Gabriel, J. F. (Ed.). Beyond the Cube: The Architecture of Space Frames and Polyhedra.
New York: Wiley, 1997.
Hart, G. "Annotated Bibliography." http://www.georgehart.com/virtual-polyhedra/references.html.
Hart, G. "Virtual Polyhedra." http://www.georgehart.com/virtual-polyhedra/vp.html.
Hilton, P. and Pedersen, J. Build Your Own Polyhedra. Reading, MA: Addison-Wesley,
1994.
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.
Kern, W. F. and Bland, J. R. "Polyhedrons." §41 in Solid
Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 115-119, 1948.
Kratochvil, P. "About Polyhedra." http://www.schulen.rosenheim.de/karolinengym/homepages/lehrer/kt/polyhed0.html.
Lyusternik, L. A. Convex Figures and Polyhedra. New York: Dover, 1963.
Malkevitch, J. "Milestones in the History of Polyhedra." In Shaping Space: A Polyhedral Approach (Ed. M. Senechal
and G. Fleck). Boston, MA: Birkhäuser, pp. 80-92, 1988.
Miyazaki, K. An Adventure in Multidimensional Space: The Art and Geometry of
Polygons, Polyhedra, and Polytopes. New York: Wiley, 1983.
Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books
Pub.,1993.
Paeth, A. W. "Exact Dihedral Metrics for Common Polyhedra." In Graphic
Gems II (Ed. J. Arvo). New York: Academic Press, 1991.
Pappas, T. "Crystals-Nature's Polyhedra." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 38-39, 1989.
Pearce, P. Structure in Nature Is a Strategy for Design. Cambridge,
MA: MIT Press, 1990.
Pedagoguery Software. Poly. http://www.peda.com/poly/.
Pegg, E. "Math Games: Supermagnetic Polyhedra." March 29, 2004. http://www.maa.org/editorial/mathgames/mathgames_03_29_04.html.
Pugh, A. Polyhedra: A Visual Approach. Berkeley: University of California
Press, 1976.
Schaaf, W. L. "Regular Polygons and Polyhedra." Ch. 3, §4 in A Bibliography of Recreational Mathematics. Washington,
DC: National Council of Teachers of Math., pp. 57-60, 1978.
Virtual Image. "Polytopia I" and "Polytopia II" CD-ROMs. http://ourworld.compuserve.com/homepages/vir_image/html/polytopiai.html
and http://ourworld.compuserve.com/homepages/vir_image/html/polytopiaii.html.
Weisstein, E. W. "Books about Solid Geometry." http://www.ericweisstein.com/encyclopedias/books/SolidGeometry.html.
Williams, R. The Geometrical Foundation of Natural Structure: A Source Book
of Design. New York: Dover, 1979.
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