Convex polyhedron
| Definition | bounded intersections of finitely many half-spaces |
|---|---|
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In geometry, a convex polyhedron (pl.: convex polyhedra) is a polyhedron whose interior is a convex set in the Euclidean space. It is a three-dimensional figure analogous to convex polygons in the Euclidean plane and convex polytopes in the Euclidean n-dimensional space. Many common families of polyhedra, such as cubes and pyramids, are convex.
Definition
A polyhedron has no universal agreement due to multiple definitions. Some authors defined it as a solid whose boundary can be covered by finitely many planes; by a manifold through the union of convex polygons, any two of which is a shared vertex or edge or the empty set; similar previous notion by the basis of topological definitions as subdivisions of a topological manifold into topological disks (the faces) whose pairwise intersections are required to be points (vertices), topological arcs (edges), or the empty set; or modernly by the theory of abstract polyhedra that can be defined as partially ordered sets containing the elements of vertices, edges, and faces. Respectively, nevertheless, these definitions do not admit the self-crossed star polyhedron, cannot be realized for a topological polyhedron, or accept the degenerate version. By all accounts, a polyhedron is understandable as a polytope in three-dimensional space.
Convex polyhedra are often defined as bounded intersections of finitely many half-spaces[1][2] or as the convex hull of finitely many points,[3] restricted in either case to intersections or hulls that have nonzero volume.
Characterizations
The Euler characteristic stated that for any convex polyhedron, the calculation of its vertices, edges, and faces is always equal to 2; that is, .[4] By Alexandrov's uniqueness theorem, every convex polyhedron is uniquely determined by the metric space of geodesic distances on its surface.
Some convex polyhedra possess a midsphere, a sphere tangent to each of their edges, which is intermediate in radius between the insphere and circumsphere for polyhedra in which all these spheres are present. Every convex polyhedron is combinatorially equivalent to a canonical polyhedron, a polyhedron that has a midsphere whose center coincides with the centroid of its tangent points with edges. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by the combinatorial structure of the given polyhedron.[5]
For every convex polyhedron, there exists a dual polyhedron having faces in place of the original's vertices and vice versa, and the same number of edges. The dual of a convex polyhedron can be obtained by the process of polar reciprocation.[6] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.[7]
By forgetting the face structure, any polyhedron gives rise to a graph, called the skeleton of the polyhedron, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C. Escher's print Stars.[10] One highlight of this approach is Steinitz's theorem, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a planar graph with three-connected. Every such graph is the skeleton of some convex polyhedron.[11]
Classificiations
Prismatoids are the polyhedra whose vertices lie on two parallel planes and whose faces are likely to be trapezoids and triangles.[12] Examples of prismatoids are pyramids, wedges, parallelipipeds, prisms, antiprisms, cupolas, and frustums. Platonic solids are the five ancient polyhedra—tetrahedron, octahedron, icosahedron, cube, and dodecahedron—described by Plato in the Timaeus.[13] Archimedean solids are the class of thirteen polyhedra whose faces are all regular polygons and whose vertices are symmetric to each other;[lower-alpha 1] their dual polyhedra are the Catalan solids.[15] Johnson solids are the class of convex polyhedra whose faces are all regular polygons.[16] These include the convex deltahedra, strictly convex polyhedra whose faces are all equilateral triangles.[17]
Convex polyhedra can be categorized into elementary polyhedra or composite polyhedra. Elementary polyhedra are convex, regular-faced polyhedra that cannot be produced into two or more polyhedra by slicing them with a plane.[18] Quite opposite to composite polyhedra, they can be alternatively defined as polyhedra constructed by attaching more elementary polyhedra. For example, triaugmented triangular prism is composite 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 elementaries.[19]
Convex polyhedra in which all vertices have integer coordinates are called lattice polyhedra (or integral polyhedra). A zonohedron is a convex polyhedron in which every face is a polygon that is symmetric under rotations through 180°.
Non-convex polyhedron
Prominent non-convex polyhedra include the star polyhedra. The regular star polyhedra, also known as the Kepler–Poinsot polyhedra, are constructible via stellation or faceting of regular convex polyhedra. Stellation is the process of extending the faces (within their planes) so that they meet. Faceting is the process of removing parts of a polyhedron to create new faces (or facets) without creating any new vertices.[20][21] A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face;[20] for example, a polygon involving diagonals (face diagonals or space diagonals). The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.[22] Other examples are the Chazelle polyhedron[23] and toroidal polyhedra.[24]
See also
- Cauchy's theorem (geometry)
- Composite polyhedron
- Parallelohedron
- Polyhedral graph
- Steinitz's theorem
Notes
- ↑ The Archimedean solids once had fourteenth solid known as the pseudorhombicuboctahedron, a mistaken construction of the rhombicuboctahedron. However, it was debarred for not having the vertex-transitive property, leading it to be instead classified as a Johnson solid.[14]
References
- ↑ Grünbaum, Branko (2003), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), New York: Springer-Verlag, p. 26, doi:10.1007/978-1-4613-0019-9, ISBN 978-0-387-00424-2, MR 1976856.
- ↑ Bruns, Winfried; Gubeladze, Joseph (2009), "Definition 1.1", Polytopes, Rings, and K-theory, Springer Monographs in Mathematics, Dordrecht: Springer, p. 5, CiteSeerX 10.1.1.693.2630, doi:10.1007/b105283, ISBN 978-0-387-76355-2, MR 2508056.
- ↑ Buldygin, V. V.; Kharazishvili, A. B. (2000), Geometric Aspects of Probability Theory and Mathematical Statistics, Springer, p. 2, doi:10.1007/978-94-017-1687-1, ISBN 978-94-017-1687-1
- ↑ Richeson, D. S. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, pp. 1–2, ISBN 9780691126777.
- ↑ Schramm, Oded (1992-12-01), "How to cage an egg", Inventiones Mathematicae, 107 (1): 543–560, Bibcode:1992InMat.107..543S, doi:10.1007/BF01231901, ISSN 1432-1297.
- ↑ Cundy, H. Martyn; Rollett, A.P. (1961), "3.2 Duality", Mathematical models (2nd ed.), Oxford: Clarendon Press, pp. 78–79, MR 0124167.
- ↑ Grünbaum, B.; Shephard, G.C. (1969), "Convex polytopes" (PDF), Bulletin of the London Mathematical Society, 1 (3): 257–300, doi:10.1112/blms/1.3.257, MR 0250188, archived from the original (PDF) on 2017-02-22, retrieved 2017-02-21. See in particular the bottom of page 260.
- ↑ Lawson-Perfect, Christian (13 October 2013), "An enneahedron for Herschel", The Aperiodical, retrieved 7 December 2016
- ↑ Coxeter, H. S. M. (1948), Regular Polytopes, London: Methuen, p. 8
- ↑ Coxeter, H.S.M. (1985), "A special book review: M.C. Escher: His life and complete graphic work", The Mathematical Intelligencer, 7 (1): 59–69, doi:10.1007/BF03023010 Unknown parameter
|s2cid=ignored (help) Coxeter's analysis of Stars is on pp. 61–62. - ↑ Grünbaum (2003), pp. 235–244.
- ↑ 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, Carbon Materials: Chemistry and Physics, 10, Springer, p. 39, doi:10.1007/978-3-319-64123-2, ISBN 978-3-319-64123-2.
- ↑ 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.
- ↑ Cundy, H. Martyn (1952), "Deltahedra", Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204.
- ↑ 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.
- ↑ 20.0 20.1 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 Unknown parameter
|s2cid=ignored (help). - ↑ Malloy, Kaoime (2014), The Art of Theatrical Design: Elements of Visual Composition, Methods, and Practice, Taylor & Francis, p. 203, ISBN 978-1-317-69426-7.
- ↑ Si, Hang; Goerigk, Nadja (2016), "On Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points", Procedia Engineering, 163: 33–45, doi:10.1016/j.proeng.2016.11.013
- ↑ Croft, Hallard; Falconer, Kenneth; Guy, Richard (1991), Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics, p. 76, doi:10.1007/978-1-4612-0963-8, ISBN 978-1-4612-0963-8
Further reading
- Shephard, G. C. (1968). "Twenty Problems on Convex Polyhedra: Part I". The Mathematical Gazette. Mathematical Association. 52 (380): 136–147. ISSN 0025-5572. JSTOR 3612678. Retrieved 2025-11-22.
- Shephard, G. C. (1968). "Twenty Problems on Convex Polyhedra Part II". The Mathematical Gazette. Mathematical Association. 52 (382): 359–367. ISSN 0025-5572. JSTOR 3611851. Retrieved 2025-11-22.
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