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Convex Polyhedron

A convex polyhedron can be defined algebraically 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 qualify as a convex polyhedron. A convex polyhedron may be obtained from an arbitrary set of points by computing the convex hull of the points. The surface defined by a set of inequalities may be visualized using the command RegionPlot3D[ineqs, x, xmin, xmax, y, ymin, ymax, z, zmin, zmax]. The method of vertex enumeration (Fukuda and Mizukoshi) can also be used to determine the faces of the resulting polyhedron directly.

An example of a convex polyhedron is illustrated above (Fukuda and Mizukoshi). A simpler example is the regular dodecahedron, which is given by a system with . Explicit examples are given in the following table.

 convex polyhedron regular tetrahedron 4 cube 6 regular octahedron 8

In general, given the matrices, the polyhedron vertices (and faces) can be found using an algorithmic procedure known as vertex enumeration.

Geometrically, a convex polyhedron can be defined as a polyhedron for which a line connecting any two (noncoplanar) points on the surface always lies in the interior of the polyhedron. The 92 convex polyhedra having only regular polygons as faces are called the Johnson solids, which include the Platonic solids and Archimedean solids. No method is known for computing the volume of a general convex polyhedron (Grünbaum and Klee 1967, p. 21; Ogilvy 1990, p. 173).

Every convex polyhedron can be represented in the plane or on the surface of a sphere by a 3-connected planar graph (called a polyhedral graph). Conversely, by a theorem of Steinitz as restated by Grünbaum, every 3-connected planar graph can be realized as a convex polyhedron (Duijvestijn and Federico 1981). The numbers of vertices , edges , and faces of a convex polyhedron are related by the polyhedral formula

Archimedean Solid, Convex Hull, Convex Polygon, Convex Polyomino, Convex Polytope, Deltahedron, Johnson Solid, Kepler-Poinsot Polyhedron, Platonic Solid, Polyhedral Formula, Polyhedral Graph, Polyhedron, Regular Polyhedron, Vertex Enumeration

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References

Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput. 37, 523-532, 1981.Grünbaum, B. and Klee, V. CUPM [Committee on the Undergraduate Program in Mathematics] Geometry Conference Proceedings, Part I: Convexity and Applications. Lectures by Branko Grünbaum and Victor Klee (Ed. L. K. Durst). Math. Assoc. Amer., No. 16, Aug. 1967. http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED024576. Fukuda, K. and Mizukoshi, I. "Vertex Enumeration Package for Convex Polytopes and Arrangements." http://library.wolfram.com/infocenter/MathSource/440/.Ogilvy, C. S. Excursions in Geometry. New York: Dover, 1990.Lyusternik, L. A. Convex Figures and Polyhedra. New York: Dover, 1963.Yaglom, I. M. and Boltianskii, V. G. Convex Figures. New York: Holt, Rinehart and Winston, 1961.

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Convex Polyhedron

Cite this as:

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