Editing Polyhedron (section)

== Convex polyhedra ==
{{multiple image
 | total_width = 300
 | align = right
 | perrow = 2
 | image1 = Hexagonal pyramid.png
 | image2 = Afgeknotte driezijdige piramide.png
 | image3 = Triakisicosahedron.jpg
 | image4 = Triaugmented triangular prism (symmetric view).svg
 | footer = Top left to bottom right: [[hexagonal pyramid]] (a [[prismatoid]]), [[truncated tetrahedron]] (an [[Archimedean solid]]), [[triakis icosahedron]] (a [[Catalan solid]]), and [[triaugmented triangular prism]] (a [[Johnson solid]] and [[deltahedron#Strictly convex deltahedron|convex deltahedron]]). All of these classes are convex polyhedra.
}}
As [[#Definitions|mentioned above]], the convex polyhedra are well-defined, with several equivalent standard definitions. They are often defined as bounded intersections of finitely many [[Half-space (geometry)|half-spaces]],<ref name="polytope-bounded-1"/><ref name="polytope-bounded-2"/> or as the [[convex hull]] of finitely many points,<ref name=bk>{{citation
 | last1 = Buldygin | first1 = V. V.
 | last2 = Kharazishvili | first2 = A. B.
 | year = 2000
 | title = Geometric Aspects of Probability Theory and Mathematical Statistics
 | url = https://books.google.com/books?id=mGD9CAAAQBAJ&pg=PA2
 | page = 2
 | publisher = Springer
 | isbn = 978-94-017-1687-1
 | doi = 10.1007/978-94-017-1687-1
}}</ref> restricted in either case to intersections or hulls that have nonzero volume.

Important classes of convex polyhedra include the family of [[prismatoid]]s, the [[Platonic solid]]s, the [[Archimedean solid]]s and their duals the [[Catalan solid]]s, and the [[Johnson solid]]s. Prismatoids are the polyhedra whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles.<ref name="prismatoid">{{citation
 | last1 = Kern | first1 = William F.
 | last2 = Bland | first2 = James R.
 | title = Solid Mensuration with proofs
 | url = https://books.google.com/books?id=Y6cAAAAAMAAJ
 | year = 1938
 | page = 75
}}.</ref> Examples of prismatoids are [[Pyramid (geometry)|pyramid]]s, [[Wedge (geometry)|wedge]]s, [[parallelipiped]]s, [[Prism (geometry)|prism]]s, [[antiprism]]s, [[cupola]]s, and [[frustum]]s. Platonic solids are the five ancient polyhedra&mdash;[[regular tetrahedron|tetrahedron]], [[regular octahedron|octahedron]], [[regular icosahedron|icosahedron]], [[cube]], and [[regular dodecahedron|dodecahedron]]&mdash;described by [[Plato]] in the [[Timaeus (dialogue)|''Timaeus'']].{{sfnp|Cromwell|1997|p=51&ndash;52}} Archimedean solids are the class of thirteen polyhedra whose faces are all regular polygons and whose vertices are symmetric to each other;{{efn|The Archimedean solids once had fourteenth solid known as the [[pseudorhombicuboctahedron]], a mistaken construction of the [[rhombicuboctahedron]]. However, it was debarred for having no [[vertex-transitive]] property, leading it to instead be classified as a Johnson solid.<ref name="14th-archimedean">{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.4171/EM/120
 | issue = 3
 | journal = [[Elemente der Mathematik]]
 | mr = 2520469
 | pages = 89–101
 | title = An enduring error
 | url = https://digital.lib.washington.edu/dspace/bitstream/handle/1773/4592/An_enduring_error.pdf
 | volume = 64
 | year = 2009| doi-access = free
}}. Reprinted in {{citation|title=The Best Writing on Mathematics 2010|editor-first=Mircea|editor-last=Pitici|publisher=Princeton University Press|year=2011|pages=18–31}}.</ref>}} their dual polyhedra are the Catalan solids.<ref name="diudea">{{citation
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39
 | page = 39
}}.</ref> Johnson solids are the class of convex polyhedra whose faces are all regular polygons.<ref name="berman">{{citation
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}.</ref> These include the [[deltahedron#Strictly convex deltahedron|convex deltahedra]], strictly convex polyhedra whose faces are all equilateral triangles.<ref name="cundy">{{citation
 | last = Cundy | first = H. Martyn | author-link = Martyn Cundy
 | title = Deltahedra
 | journal = [[Mathematical Gazette]]
 | volume = 36 | pages = 263–266
 | year = 1952
 | issue = 318 | doi = 10.2307/3608204
 | jstor = 3608204
}}.</ref>

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 polyhedrons by slicing them with a plane.{{sfnp|Hartshorne|2000|p=[https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464 464]}} 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 pyramid]]s onto the square faces of a [[triangular prism]]; the square pyramids and the triangular prism are elementaries.<ref name="timofeenko-2010">{{citation
 | last = Timofeenko | first = A. V.
 | year = 2010
 | title = Junction of Non-composite Polyhedra
 | journal = St. Petersburg Mathematical Journal
 | volume = 21 | issue = 3 | pages = 483–512
 | doi = 10.1090/S1061-0022-10-01105-2
 | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
}}.</ref>

{{multiple image
 | image1 = Midsphere.png
 | caption1 = A canonical polyhedron
 | image2 = Fritsch map.svg
 | caption2 = The triangumented triangular prism's skeleton [[Fritsch graph|as a graph]]
 | total_width = 360
 | align = right
}}
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 for which all three of these spheres exist. 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.<ref>{{citation
 | last = Schramm | first = Oded
 | date = 1992-12-01
 | title = How to cage an egg
 | journal = Inventiones Mathematicae
 | language = en
 | volume = 107 | issue = 1 | pages = 543–560
 | doi = 10.1007/BF01231901
 | bibcode = 1992InMat.107..543S
 | issn = 1432-1297
}}.</ref>

By forgetting the face structure, any polyhedron gives rise to a [[Graph (discrete mathematics)|graph]], called its [[n-skeleton|skeleton]], 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 model|wire-frame]] polyhedra appear in [[M.C. Escher]]'s print [[Stars (M. C. Escher)|''Stars'']].<ref>{{citation | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | doi=10.1007/BF03023010 | issue=1 | journal=The Mathematical Intelligencer | pages=59–69 | title=A special book review: M.C. Escher: His life and complete graphic work | volume=7 | year=1985| s2cid=189887063 }} Coxeter's analysis of ''Stars'' is on pp. 61–62.</ref>  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 [[vertex connectivity|three-connected]], and every such graph is the skeleton of some convex polyhedron.{{sfnp|Grünbaum|2003|pp=235–244}}

Prominent ''non-convex polyhedra'' include the [[star polyhedra]]. The regular star polyhedra, also known as the [[Kepler&ndash;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).<ref name="bridge">{{citation
 | last = Bridge | first = N.J.
 | title = Facetting the dodecahedron
 | year = 1974
 | journal = Acta Crystallographica
 | volume = A30
 | issue = 4
 | doi = 10.1107/S0567739474001306
 | pages = 548–552| bibcode = 1974AcCrA..30..548B
}}.</ref><ref>{{citation
 | last = Inchbald | first = G.
 | title = Facetting diagrams
 | year = 2006
 | journal = The Mathematical Gazette
 | volume = 90
 | issue = 518
 | pages = 253–261
 | doi = 10.1017/S0025557200179653| s2cid = 233358800
}}.</ref> A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a ''[[Face (geometry)|face]]''.<ref name="bridge"/> The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.