Editing Polyhedron (section)

==Definition==
[[convex polyhedron|Convex polyhedra]] are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.
Many definitions of "polyhedron" have been given within particular contexts,<ref name="lakatos">{{citation
 | last = Lakatos | first = Imre | editor2-first = Elie | editor2-last = Zahar | editor1-first = John | editor1-last = Worrall | author-link = Imre Lakatos
 | doi = 10.1017/CBO9781316286425
 | isbn = 978-1-107-53405-6
 | location = Cambridge
 | mr = 3469698
 | orig-date = 1976
 | page = 16
 | publisher = Cambridge University Press
 | quote = definitions are frequently proposed and argued about
 | series = Cambridge Philosophy Classics
 | title = Proofs and Refutations: The logic of mathematical discovery
 | year = 2015| title-link = Proofs and Refutations }}.</ref> some more rigorous than others, and there is no universal agreement over which of these to choose.
Some of these definitions exclude shapes that have often been counted as polyhedra (such as the [[star polyhedron|self-crossing polyhedra]]) or include
shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not [[manifold]]s). As [[Branko Grünbaum]] observed,
{{Blockquote|"The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the polyhedra".<ref name=sin>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | editor1-last = Bisztriczky | editor1-first = Tibor
 | editor2-last = McMullen | editor2-first = Peter 
 | editor3-last = Schneider|editor3-first = Rolf
 | editor4-last = Weiss | editor4-first = A.
 | contribution = Polyhedra with hollow faces
 | doi = 10.1007/978-94-011-0924-6_3
 | isbn = 978-94-010-4398-4
 | location = Dordrecht
 | mr = 1322057
 | pages = 43–70
 | publisher = Kluwer Acad. Publ.
 | title = Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational
 | year = 1994}}; for quote, see p.&nbsp;43.</ref>}}
Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its [[vertex (geometry)|vertices]] (corner points), [[edge (geometry)|edges]] (line segments connecting certain pairs of vertices),
[[face (geometry)|faces]] (two-dimensional [[polygon]]s), and that it sometimes can be said to have a particular three-dimensional interior [[volume]].
One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its [[incidence geometry]].<ref>{{citation|contribution=Polyhedra: Surfaces or solids?|first=Arthur L.|last=Loeb|author-link= Arthur Lee Loeb|pages=65–75|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination|edition=2nd|editor-first=Marjorie|editor-last=Senechal|editor-link=Marjorie Senechal|publisher=Springer|year=2013|doi=10.1007/978-0-387-92714-5_5|isbn=978-0-387-92713-8 }}</ref>
* A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes<ref>{{citation|title=Solid Geometry|first=Joseph P.|last=McCormack|publisher=D. Appleton-Century Company|year=1931|page=416}}.</ref><ref>{{citation|title=Computational Geometry: Algorithms and Applications|last1=de Berg|first1=M.|author1-link= Mark de Berg |last2=van Kreveld|first2=M.|author2-link= Marc van Kreveld |last3=Overmars|first3=M.|author3-link=Mark Overmars|last4=Schwarzkopf|first4=O.|author4-link=Otfried Cheong|edition=2nd|publisher=Springer|year=2000|page=64}}.</ref> or that it is a solid formed as the union of finitely many convex polyhedra.<ref>{{SpringerEOM|title=Polyhedron, abstract|id=Polyhedron,_abstract&oldid=25452|first=S.V.|last=Matveev}}</ref> Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. The faces of such a polyhedron can be defined as the [[Connected space|connected components]] of the parts of the boundary within each of the planes that cover it, and the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, whose faces may not form [[simple polygon]]s, and some of whose edges may belong to more than two faces.<ref>{{citation|title=Adventures Among the Toroids: A study of orientable polyhedra with regular faces|title-link= Adventures Among the Toroids |first=B. M.|last=Stewart|author-link= Bonnie Stewart |edition=2nd|year=1980|page=6}}.</ref>
* Definitions based on the idea of a bounding surface rather than a solid are also common.<ref name=cromwell>{{citation
 | last = Cromwell | first = Peter R.
 | isbn = 978-0-521-55432-9
 | location = Cambridge
 | mr = 1458063
 | publisher = Cambridge University Press
 | title = Polyhedra | title-link = Polyhedra (book)
 | year = 1997}}; for definitions of polyhedra, see pp. 206–209; for polyhedra with equal regular faces, see p. 86.</ref> For instance, {{harvtxt|O'Rourke|1993}} defines a polyhedron as a union of [[convex polygon]]s (its faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the [[empty set]] and so that their union is a [[manifold]].<ref>{{citation|title=Computational Geometry in C|journal=Computers in Physics|volume=9|issue=1|first=Joseph|last=O'Rourke|author-link=Joseph O'Rourke (professor)|year=1993|pages=113–116|bibcode=1995ComPh...9...55O|doi=10.1063/1.4823371|url=http://www.gbv.de/dms/goettingen/241632501.pdf }}.</ref> If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat [[dihedral angle]]s between them. Somewhat more generally, Grünbaum defines an ''acoptic polyhedron'' to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each.<ref name=acoptic>{{citation
 | last = Grünbaum
 | first = Branko
 | author-link = Branko Grünbaum
 | contribution = Acoptic polyhedra
 | doi = 10.1090/conm/223/03137
 | mr = 1661382
 | pages = 163–199
 | publisher = American Mathematical Society
 | location = Providence, Rhode Island
 | series = Contemporary Mathematics
 | title = Advances in discrete and computational geometry (South Hadley, MA, 1996)
 | contribution-url = https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | volume = 223
 | year = 1999
 | isbn = 978-0-8218-0674-6
 | access-date = 2022-07-01
 | archive-date = 2021-08-30
 | archive-url = https://web.archive.org/web/20210830211936/https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | url-status = dead
 }}.</ref> Cromwell's ''[[Polyhedra (book)|Polyhedra]]'' gives a similar definition but without the restriction of at least three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra.<ref name=cromwell/> Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into [[Disk (mathematics)|topological disks]] (the faces) whose pairwise intersections are required to be points (vertices), topological arcs (edges), or the empty set. However, there exist topological polyhedra (even with all faces triangles) that cannot be realized as acoptic polyhedra.<ref>{{citation
 | last1 = Bokowski | first1 = J.
 | last2 = Guedes de Oliveira | first2 = A.
 | doi = 10.1007/s004540010027
 | issue = 2–3
 | journal = [[Discrete and Computational Geometry]]
 | mr = 1756651
 | pages = 197–208
 | title = On the generation of oriented matroids
 | volume = 24
 | year = 2000| doi-access = free
 }}.</ref>
[[File:Pyramid abstract polytope.svg|thumb|340px|A square pyramid and the associated abstract polytope. Here, the elements of a square pyramid can be defined as the partially ordered set.]]
* One modern approach is based on the theory of [[abstract polyhedron|abstract polyhedra]]. These can be defined as [[partially ordered set]]s whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element (in this partial order) when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order (representing the empty set) and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart (that is, between each face and the bottom element, and between the top element and each vertex) have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron.{{cn|date=May 2023|reason=the [[11-cell]] and [[57-cell]] are valid abstract polytopes but not valid topological polytopes; the latter approach assumes simple balls but the former does not. How can these be said to carry the "same" information?}} However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment.<ref name="bursta">{{citation
 | last1 = Burgiel | first1 = H.
 | last2 = Stanton | first2 = D.
 | doi = 10.1007/s004540010030
 | issue = 2–3
 | journal = [[Discrete and Computational Geometry]]
 | mr = 1758047
 | pages = 241–255
 | title = Realizations of regular abstract polyhedra of types {3,6} and {6,3}
 | volume = 24
 | year = 2000| doi-access = free
 }}.</ref> (This means that each edge contains two vertices and belongs to two faces, and that each vertex on a face belongs to two edges of that face.) Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra. A ''realization'' of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron.<ref name=grunbaum-same>{{citation | last = Grünbaum | first = Branko | title = Discrete and Computational Geometry: The Goodman–Pollack Festschrift | author-link = Branko Grünbaum | editor1-last = Aronov | editor1-first = Boris | editor1-link = Boris Aronov | editor2-last = Basu | editor2-first = Saugata | editor3-last = Pach | editor3-first = János | editor3-link = János Pach | editor4-last = Sharir | editor4-first = Micha | editor4-link = Micha Sharir | contribution = Are your polyhedra the same as my polyhedra? | contribution-url = https://faculty.washington.edu/moishe/branko/BG249.Your%20polyh-my%20polyh.pdf | doi = 10.1007/978-3-642-55566-4_21 | mr = 2038487 | pages = 461–488 | publisher = Springer | location = Berlin | series = Algorithms and Combinatorics | volume = 25 | year = 2003 | isbn = 978-3-642-62442-1 | citeseerx = 10.1.1.102.755  }}</ref> Realizations that omit the requirement of face planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered.<ref name="bursta"/> Unlike the solid-based and surface-based definitions, this works perfectly well for star polyhedra. However, without additional restrictions, this definition allows [[Degeneracy (mathematics)|degenerate]] or unfaithful polyhedra (for instance, by mapping all vertices to a single point) and the question of how to constrain realizations to avoid these degeneracies has not been settled.

In all of these definitions, a polyhedron is typically understood as a three-dimensional example<!-- There are polyhedra in 4-dimensions like the [[regular skew polyhedron]] with flat faces and skew [[vertex figure]]s, and seen as an approximation of the surface of a 4D [[duocylinder]].--> of the more general [[polytope]] in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a [[4-polytope]] has a four-dimensional body and an additional set of three-dimensional "cells".
However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many [[Half-space (geometry)|half-spaces]], and a polytope to be a bounded polyhedron.<ref name="polytope-bounded-1">{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1007/978-1-4613-0019-9
 | edition = 2nd
 | isbn = 978-0-387-00424-2
 | location = New York
 | mr = 1976856
 | page = 26
 | publisher = Springer-Verlag
 | ref = Grünbaum-Convex-Polytopes<!-- Don't interfere with harv links to the other Grünbaum 2003 reference -->
 | series = Graduate Texts in Mathematics
 | title = Convex Polytopes
 | title-link = Convex Polytopes
 | volume = 221
 | year = 2003}}.</ref><ref name="polytope-bounded-2">{{citation
 | last1 = Bruns | first1 = Winfried
 | last2 = Gubeladze | first2 = Joseph
 | contribution = Definition 1.1
 | contribution-url = https://books.google.com/books?id=pbgg1pFxW8YC&pg=PA5
 | doi = 10.1007/b105283
 | isbn = 978-0-387-76355-2
 | mr = 2508056
 | page = 5
 | publisher = Springer | location = Dordrecht
 | series = Springer Monographs in Mathematics
 | title = Polytopes, Rings, and ''K''-theory
 | year = 2009| citeseerx = 10.1.1.693.2630
 }}.</ref> The remainder of this article considers only three-dimensional polyhedra.