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A convex polyhedron is a polyhedron that forms a convex set as a solid. That being said, it is a three-dimensional solid whose every line segment connects two of its points lies its interior or on its boundary; none of its faces are coplanar (they do not share the same plane) and none of its edges are collinear (they are not segments of the same line).[1][2] A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces, or as the convex hull of finitely many points, in either case, restricted to intersections or hulls that have nonzero volume.[3][4]
Important classes of convex polyhedra include the family of prismatoid, the highly symmetrical Platonic solids, the Archimedean solids and their duals the Catalan solids, and the regular polygonal faces polyhedron. The prismatoids are the polyhedron whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles.[5] Examples of prismatoids are pyramids, wedges, parallelipipeds, prisms, antiprisms, cupolas, and frustums. The Platonic solids are the five ancientness polyhedrons—tetrahedron, octahedron, icosahedron, cube, and dodecahedron—classified by Plato in his Timaeus whose connecting four classical elements of nature.[6] The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other;[a] their dual polyhedrons are Catalan solids.[8] The class of regular polygonal faces polyhedron are the deltahedron (whose faces are all equilateral triangles, polycubes (whose faces are all squares), and Johnson solids (whose faces are arbitrary regular polygons).[9][10][11]
The convex polyhedron can be categorized into elementary polyhedron or composite polyhedron. An elementary polyhedron is a convex regular-faced polyhedron that cannot be produced into two or more polyhedrons by slicing it with a plane.[12] Quite opposite to a composite polyhedron, it can be alternatively defined as a polyhedron that can be constructed by attaching more elementary polyhedrons. For example, triaugmented triangular prism is a composite polyhedron since it can be constructed by attaching three equilateral square pyramids onto the square faces of a triangular prism; the square pyramids and the triangular prism are elementary.[13]
Some polyhedrons do not have the property of convexity, and they are called non-convex polyhedrons. Such polyhedrons are star polyhedrons, Kepler–Poinsot polyhedrons, and many more. They are constructed by either stellation (process of extending the faces—within their planes—so that they meet), or faceting (whose process of removing parts of a polyhedron to create new faces—or facets—without creating any new vertices).[14][15] A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face.[14] The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.
- ^ Boissonnat, J. D.; Yvinec, M. (June 1989), Probing a scene of non convex polyhedra, Proceedings of the fifth annual symposium on Computational geometry, pp. 237–246, doi:10.1145/73833.73860.
- ^ Litchenberg, D. R. (1988), "Pyramids, Prisms, Antiprisms, and Deltahedra", The Mathematics Teacher, 81 (4): 261–265, JSTOR 27965792.
- ^ Cite error: The named reference
polytope-bounded-1was invoked but never defined (see the help page). - ^ Cite error: The named reference
polytope-bounded-2was invoked but never defined (see the help page). - ^ Kern, William F.; Bland, James R. (1938), Solid Mensuration with proofs, p. 75.
- ^ Cromwell (1997), p. 51–52.
- ^ Grünbaum, Branko (2009), "An enduring error" (PDF), Elemente der Mathematik, 64 (3): 89–101, doi:10.4171/EM/120, MR 2520469. Reprinted in Pitici, Mircea, ed. (2011), The Best Writing on Mathematics 2010, Princeton University Press, pp. 18–31.
- ^ Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Springer, p. 39, doi:10.1007/978-3-319-64123-2, ISBN 978-3-319-64123-2.
- ^ Cundy, H. Martyn (1952), "Deltahedra", Mathematical Gazette, 36: 263–266, doi:10.2307/3608204, JSTOR 3608204
- ^ Lunnon, W. F. (1972), "Symmetry of Cubical and General Polyominoes", in Read, Ronald C. (ed.), Graph Theory and Computing, New York: Academic Press, pp. 101–108, ISBN 978-1-48325-512-5.
- ^ Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
- ^ Hartshorne (2000), p. 464.
- ^ Timofeenko, A. V. (2010), "Junction of Non-composite Polyhedra" (PDF), St. Petersburg Mathematical Journal, 21 (3): 483–512, doi:10.1090/S1061-0022-10-01105-2.
- ^ a b Bridge, N.J. (1974), "Facetting the dodecahedron", Acta Crystallographica, A30 (4): 548–552, Bibcode:1974AcCrA..30..548B, doi:10.1107/S0567739474001306.
- ^ Inchbald, G. (2006), "Facetting diagrams", The Mathematical Gazette, 90 (518): 253–261, doi:10.1017/S0025557200179653, S2CID 233358800.
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