TOPICS

# Polyhedron

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.

In the Wolfram Language, Polyhedron[] objects represent filled regions founded by closed surfaces with polygonal faces.

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 an 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 polyhedra. 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.

Acoptic Polyhedron, Apeirogon, Archimedean Solid, Canonical Polyhedron, Catalan Solid, Convex Polyhedron, Cube, Dice, Digon, Dodecahedron, Dual Polyhedron, Echidnahedron, Flexible Polyhedron, Haűy Construction, Hexahedron, Holyhedron, Hyperbolic Polyhedron, Icosahedron, Isohedron, Jessen's Orthogonal Icosahedron Johnson Solid, Kepler-Poinsot Polyhedron, Nolid, Octahedron, Petrie Polygon, Plaited Polyhedron, Platonic Solid, Polychoron, Polyhedron Coloring, Polyhedron Compound, Polytope, Prismatoid, Quadricorn, Quasiregular Polyhedron, Regular Polyhedron, Rigid Polyhedron, Rigidity Theorem, Schwarz's Polyhedron, Shaky Polyhedron, Semiregular Polyhedron, Skeleton, Stellation, Tetrahedron, Truncation, Uniform Polyhedron, Zonohedron Explore this topic in the MathWorld classroom

## Explore with Wolfram|Alpha

More things to try:

## References

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.

Polyhedron

## Cite this as:

Weisstein, Eric W. "Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Polyhedron.html