Polyhedron#Abstract polyhedra

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,{{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 }}. 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 self-crossing polyhedra) or include

shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds). 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".{{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. 43.}}

Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices),

faces (two-dimensional polygons), 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.{{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 }}

• A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes{{citation|title=Solid Geometry|first=Joseph P.|last=McCormack|publisher=D. Appleton-Century Company|year=1931|page=416}}.{{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}}. or that it is a solid formed as the union of finitely many convex polyhedra.{{SpringerEOM|title=Polyhedron, abstract|id=Polyhedron,_abstract&oldid=25452|first=S.V.|last=Matveev}} 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 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 polygons, and some of whose edges may belong to more than two faces.{{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}}.
• Definitions based on the idea of a bounding surface rather than a solid are also common.{{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. For instance, {{harvtxt|O'Rourke|1993}} defines a polyhedron as a union of convex polygons (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.{{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 }}. 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 angles 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.{{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

}}. Cromwell's 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. Similar notions form the basis of topological definitions of polyhedra, 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. However, there exist topological polyhedra (even with all faces triangles) that cannot be realized as acoptic polyhedra.{{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

}}.

File:Pyramid abstract polytope.svg

• One modern approach is based on the theory of abstract polyhedra. These can be defined as partially ordered sets 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.{{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

}}. (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.{{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 }} 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. Unlike the solid-based and surface-based definitions, this works perfectly well for star polyhedra. However, without additional restrictions, this definition allows 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 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-spaces, and a polytope to be a bounded polyhedron.{{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

| series = Graduate Texts in Mathematics

| title = Convex Polytopes

| title-link = Convex Polytopes

| volume = 221

| year = 2003}}.{{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

}}. The remainder of this article considers only three-dimensional polyhedra.

Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc.{{citation|title=The New Elements of Mathematics, Volume II: Algebra and Geometry|first=Charles S.|last=Peirce|author-link=Charles Sanders Peirce|editor-first=Carolyn|editor-last=Eisele|editor-link=Carolyn Eisele|year=1976|publisher=Mouton Publishers & Humanities Press|page=297|isbn=9783110818840 |url=https://books.google.com/books?id=Z3ldDwAAQBAJ&pg=PA297}} For a complete list of the Greek numeral prefixes see {{slink|Numeral prefix|Table of number prefixes in English}}, in the column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (eight-sided polyhedra), dodecahedra (twelve-sided polyhedra), and icosahedra (twenty-sided polyhedra) are sometimes used without additional qualification to refer to the Platonic solids, and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry.{{citation|orig-date=1996|year=2020|title=Crystal Structures: Patterns and Symmetry|first1=Michael|last1=O'Keefe|first2=Bruce G.|last2=Hyde|publisher=Dover Publications|page=134|isbn=9780486836546 |url=https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA134}}

{{main|Toroidal polyhedron|Euler characteristic}}

File:Tetrahemihexahedron rotation.gif, a non-orientable self-intersecting polyhedron with four triangular faces (red) and three square faces (yellow). As with a Möbius strip or Klein bottle, a continuous path along the surface of this polyhedron can reach the point on the opposite side of the surface from its starting point, making it impossible to separate the surface into an inside and an outside. (Topologically, this polyhedron is a real projective plane.)]]

Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the tetrahemihexahedron, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours.

In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces.{{citation

| last = Ringel | first = Gerhard

| contribution = Classification of surfaces

| doi = 10.1007/978-3-642-65759-7_3

| pages = 34–53

| publisher = Springer

| title = Map Color Theorem

| year = 1974| isbn = 978-3-642-65761-0

}}

A more subtle distinction between polyhedron surfaces is given by their Euler characteristic, which combines the numbers of vertices $V$, edges $E$, and faces $F$ of a polyhedron into a single number $\backslash chi$ defined by the formula

:$\backslash chi=V-E+F.\backslash$

The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For a convex polyhedron, or more generally any simply connected polyhedron with the surface of a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2.{{citation

| last = Richeson | first = David S. | author-link = David Richeson

| isbn = 978-0-691-12677-7

| location = Princeton, NJ

| mr = 2440945

| publisher = Princeton University Press

| title = Euler's Gem: The polyhedron formula and the birth of topology

| title-link = Euler's Gem

| year = 2008}}, pp. 157, 180.

All polyhedra with odd-numbered Euler characteristics are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed toroid and the Klein bottle both have $\backslash chi\; =\; 0$, with the first being orientable and the other not.

For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a manifold. This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along a shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.

A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle.{{citation|first=B. M.|last=Stewart|title=Adventures Among the Toroids: A Study of Orientable Polyhedra with Regular Faces|title-link= Adventures Among the Toroids |edition=2nd|year=1980|isbn=978-0-686-11936-4|publisher=B. M. Stewart}}. One of the notable example is the Szilassi polyhedron, which has the geometrically ralizes the Heawood map.

{{main|Dual polyhedron}}

File:Dual Cube-Octahedron.svg

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.{{citation

| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy

| last2 = Rollett | first2 = A.P.

| edition = 2nd

| location = Oxford

| mr = 0124167

| publisher = Clarendon Press

| title = Mathematical models

| title-link = Mathematical Models (Cundy and Rollett)

| year = 1961

| contribution = 3.2 Duality

| pages = 78–79}}. 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.{{citation

| last1 = Grünbaum

| first1 = B.

| author1-link = Branko Grünbaum

| last2 = Shephard

| first2 = G.C.

| author2-link = Geoffrey Colin Shephard

| doi = 10.1112/blms/1.3.257

| journal = Bulletin of the London Mathematical Society

| mr = 0250188

| pages = 257–300

| title = Convex polytopes

| url = http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf

| volume = 1

| issue = 3

| year = 1969

| access-date = 2017-02-21

| archive-url = https://web.archive.org/web/20170222114014/http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf

| archive-date = 2017-02-22

}}. See in particular the bottom of page 260.

Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order. These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.

{{Main|Vertex figure}}

For every vertex one can define a vertex figure, which describes the local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex. For the Platonic solids and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex,{{citation| first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter|title=Regular Polytopes|title-link=Regular Polytopes (book)|publisher=Methuen|year=1947|page=[https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16 16]}} but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in convex position, this slice can be chosen as any plane separating the vertex from the other vertices.{{citation

| last = Barnette | first = David

| journal = Pacific Journal of Mathematics

| mr = 328773

| pages = 349–354

| title = A proof of the lower bound conjecture for convex polytopes

| url = https://projecteuclid.org/euclid.pjm/1102946311

| volume = 46

| year = 1973| issue = 2

| doi = 10.2140/pjm.1973.46.349

| doi-access = free

}} When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center;{{citation

| last = Luotoniemi | first = Taneli

| editor1-last = Swart | editor1-first = David

| editor2-last = Séquin | editor2-first = Carlo H.

| editor3-last = Fenyvesi | editor3-first = Kristóf

| contribution = Crooked houses: Visualizing the polychora with hyperbolic patchwork

| contribution-url = https://archive.bridgesmathart.org/2017/bridges2017-17.html

| isbn = 978-1-938664-22-9

| location = Phoenix, Arizona

| pages = 17–24

| publisher = Tessellations Publishing

| title = Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture

| year = 2017}} with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it is common instead to slice the polyhedron by a small sphere centered at the vertex.{{citation

| date = January 1930

| doi = 10.1098/rsta.1930.0009

| first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter

| issue = 670–680

| journal = Philosophical Transactions of the Royal Society of London, Series A

| pages = 329–425

| publisher = The Royal Society

| title = The polytopes with regular-prismatic vertex figures

| volume = 229| bibcode = 1930RSPTA.229..329C

}} Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes.

The surface area of a polyhedron is the sum of the areas of its faces, for definitions of polyhedra for which the area of a face is well-defined.

The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By Alexandrov's uniqueness theorem, every convex polyhedron is uniquely determined by the metric space of geodesic distances on its surface. However, non-convex polyhedra can have the same surface distances as each other, or the same as certain convex polyhedra.{{citation

| last = Hartshorne | first = Robin | author-link = Robin Hartshorne

| contribution = Example 44.2.3, the "punched-in icosahedron"

| doi = 10.1007/978-0-387-22676-7

| isbn = 0-387-98650-2

| mr = 1761093

| page = 442

| publisher = Springer-Verlag, New York

| series = Undergraduate Texts in Mathematics

| title = Geometry: Euclid and beyond

| year = 2000}}

When segment lines connect two vertices that are not in the same face, they form the diagonal lines.{{citation

| last1 = Posamentier | first1 = Alfred S.

| last2 = Bannister | first2 = Robert L.

| year = 2014

| edition = 2nd

| title = Geometry, Its Elements and Structure: Second Edition

| url = https://books.google.com/books?id=XktMBAAAQBAJ&pg=PA543

| page = 543

| publisher = Dover Publications

| isbn = 978-0-486-49267-4

}} However, not all polyhedra have diagonal lines, as in the family of pyramids,{{cn|date=February 2025}} Schönhardt polyhedron in which three diagonal lines lies entirely outside of it, and Császár polyhedron has no diagonal lines (rather, every pair of vertices is connected by an edge).{{citation

| last = Bagemihl | first = F. | authorlink = Frederick Bagemihl

| journal = American Mathematical Monthly

| pages = 411–413

| title = On indecomposable polyhedra

| volume = 55

| year = 1948

| doi = 10.2307/2306130

| issue = 7

| jstor = 2306130}}

Polyhedral solids have an associated quantity called volume that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and parallelepipeds can easily be expressed in terms of their edge lengths or other coordinates. (See Volume § Volume formulas for a list that includes many of these formulas.)

Volumes of more complicated polyhedra may not have simple formulas. The volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation). For example, the volume of a Platonic solid can be computed by dividing it into congruent pyramids, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex.

In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by

\frac{1}{3} \left| \sum_F (Q_F \cdot N_F) \operatorname{area}(F) \right|,

where the sum is over faces $F$ of the polyhedron, $Q\_F$ is an arbitrary point on face $F$, $N\_F$ is the unit vector perpendicular to $F$ pointing outside the solid, and the multiplication dot is the dot product.{{citation |last=Goldman |first=Ronald N.|author-link=Ron Goldman (mathematician) |editor-last=Arvo |editor-first=James |title=Graphic Gems Package: Graphics Gems II |publisher=Academic Press |year=1991 |pages=170–171 |chapter=Chapter IV.1: Area of planar polygons and volume of polyhedra}} In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized algorithms to determine the volume in these cases.{{citation | last1 = Büeler | first1 = B. | last2 = Enge | first2 = A. | last3 = Fukuda | first3 = K. | doi = 10.1007/978-3-0348-8438-9_6 | chapter = Exact Volume Computation for Polytopes: A Practical Study | title = Polytopes — Combinatorics and Computation | pages = 131–154 | year = 2000 | isbn = 978-3-7643-6351-2 | citeseerx = 10.1.1.39.7700 }}

{{main|Dehn invariant}}

In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them. The analogous question for polyhedra was the subject of Hilbert's third problem. Max Dehn solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other.{{citation |last=Sydler |first=J.-P. | author-link = Jean-Pierre Sydler |title=Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions |journal=Comment. Math. Helv.|language=fr |volume=40 |year=1965 |pages=43–80 |doi= 10.1007/bf02564364| mr = 0192407|s2cid=123317371 | url = https://eudml.org/doc/139296 }} The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges.{{SpringerEOM|first=M.|last=Hazewinkel|title=Dehn invariant|id=Dehn_invariant&oldid=35803}}

Another of Hilbert's problems, Hilbert's eighteenth problem, concerns (among other things) polyhedra that tile space. Every such polyhedron must have Dehn invariant zero.{{citation

| last = Debrunner | first = Hans E.

| doi = 10.1007/BF01235384

| issue = 6

| journal = Archiv der Mathematik

| language = de

| mr = 604258

| pages = 583–587

| title = Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln

| volume = 35

| year = 1980| s2cid = 121301319

}}. The Dehn invariant has also been connected to flexible polyhedra by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.{{citation

| last = Alexandrov | first = Victor

| arxiv = 0901.2989

| doi = 10.1007/s00022-011-0061-7

| issue = 1–2

| journal = Journal of Geometry

| mr = 2823098

| pages = 1–13

| title = The Dehn invariants of the Bricard octahedra

| volume = 99

| year = 2010| citeseerx = 10.1.1.243.7674

| s2cid = 17515249

}}.

File:Revolução de poliedros 03.webm)]]

Many of the most studied polyhedra are highly symmetrical. Their appearance is unchanged by some reflection by plane or rotation around the axes passing through two opposite vertices, edges, or faces in space. Each symmetry may change the location of a given element, but the set of all vertices (likewise faces and edges) is unchanged. The collection of symmetries of a polyhedron is called its symmetry group.{{citation

| last = Mal'cev | first = A. V.

| editor-last1 = Aleksandrov | editor-first1 = A. D.

| editor-last2 = Kolmogorov | editor-first2 = A. N.

| editor-last3 = Lavrent'ev | editor-first3 = M. A.

| year = 2012

| title = Mathematics: Its Content, Methods and Meaning

| url = https://books.google.com/books?id=2cPDAgAAQBAJ&pg=RA2-PA278

| page = Volume III, 278

| publisher = Dover Publications

| isbn = 978-0-486-15787-0

}}

All the elements (vertex, face, and edge) that can be superimposed on each other by symmetries are said to form a symmetry orbit. If these elements lie in the same orbit, the figure may be transitive on the orbit. Individually, they are isohedral (or face-transitive, meaning the symmetry transformations involve the polyhedra's faces in orbit),{{citation

| last = McLean | first = K. Robin

| year = 1990

| title = Dungeons, dragons, and dice

| journal = The Mathematical Gazette

| volume = 74 | issue = 469 | pages = 243–256

| doi = 10.2307/3619822

| jstor = 3619822

| s2cid = 195047512

}} See p. 247.{{efn|1=The topological property of an isohedral polyhedra can be represented by a face configuration. All five Platonic solids and thirteen Catalan solids are isohedra, as well as the infinite families of trapezohedra and bipyramids. Some definitions of isohedra allow geometric variations including concave and self-intersecting forms.}} isotoxal (or edge-transitive, which involves the edge's polyhedra),{{citation

| last = Grünbaum | first = Branko | authorlink = Branko Grünbaum

| year = 1997

| title = Isogonal Prismatoids

| journal = Discrete & Computational Geometry

| volume = 18 | issue = 1 | pages = 13–52

| doi = 10.1007/PL00009307

}} and isogonal (or vertex-transitive, which involves the polyhedra's vertices). For example, the cube in which all the faces are in one orbit and involving the rotation and reflections in the orbit remains unchanged in its appearance; hence, the cube is face-transitive. The cube also has the other two such symmetries.{{citation

| last = Senechal | first = Marjorie

| year = 1989

| contribution = A Brief Introduction to Tilings

| contribution-url = https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12

| editor-last = Jarić | editor-first = Marko

| title = Introduction to the Mathematics of Quasicrystals

| publisher = Academic Press

| page = 12

}}

File:Hexahedron.svg is regular polyhedron, because its faces, edges, and vertices are transitive to another, and the appearance is unchanged.]]

When three such symmetries belong to a polyhedron, it is known as regular polyhedron. There are nine regular polyhedra: five Platonic solids (cube, octahedron, icosahedron, tetrahedron, and dodecahedron—all of which have regular polygonal faces) and four Kepler–Poinsot polyhedrons. Nevertheless, some polyhedrons may not possess one or two of those symmetries:

• A polyhedron with vertex-transitive and edge-transitive is said to be quasiregular, although they have regular faces, and its dual is face-transitive and edge-transitive.
• A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be semiregular.{{efn|1=This is one of several definitions of the term, depending on the author. Some definitions overlap with the quasi-regular class.}} and such polyhedrons are the prisms and antiprisms. Its dual is face-transitive but not vertex-transitive, and every vertex is regular.
• A polyhedron with regular polygonal faces and vertex-transitive is said to be uniform. This class may be subdivided into a regular, quasi-regular, or semi-regular polyhedron, and may be convex or starry. The dual is face-transitive and has regular vertices but is not necessarily vertex-transitive. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not.
• A face- and vertex-transitive (but not necessarily edge-transitive) polyhedra is said to be noble. The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble.

Some polyhedra may have no reflection symmetry such that they have two enantiomorph forms, which are reflections of each other. Such symmetry is known for having chirality. Examples include the snub cuboctahedron and snub icosidodecahedron.

{{main|Point groups in three dimensions}}

The point group of polyhedra means a mathematical group endowed with its symmetry operations so that the appearance of polyhedra remains preserved while transforming in three-dimensional space. The indicated transformation here includes the rotation around the axes, reflection through the plane, inversion through a center point, and a combination of these three.{{citation

| last = Powell | first = R. C.

| year = 2010

| title = Symmetry, Group Theory, and the Physical Properties of Crystals

| series = Lecture Notes in Physics

| volume = 824

| publisher = Springer

| url = https://books.google.com/books?id=ojq5BQAAQBAJ&pg=PA27

| page = 27

| isbn = 978-1-441-97598-0

| doi = 10.1007/978-1-4419-7598-0

}}

Image:Symmetries of the tetrahedron.svg

The polyhedral group is the symmetry group originally derived from the three Platonic solids: tetrahedron, octahedron, and icosahedron. These three have point groups respectively known as tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry. Each of these focuses on the rotation group of polyhedra, known as the chiral polyhedral group, whereas the additional reflection symmetry is known as the full polyhedral group. One point group, pyritohedral symmetry, includes the rotation of tetrahedral symmetry and additionally has three planes of reflection symmetry and some rotoreflections. Overall, the mentioned polyhedral groups are summarized in the following bullets:{{citation

| last1 = Flusser | first1 = J.

| last2 = Suk | first2 = T.

| last3 = Zitofa | first3 = B.

| year = 2017

| title = 2D and 3D Image Analysis by Moments

| publisher = John Wiley & Sons

| isbn = 978-1-119-03935-8

| page = 127–128

| url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA128

}}

• chiral tetrahedral symmetry $\backslash mathrm\{T\}$, the rotation group for a regular tetrahedron and has the order of twelve.
• full tetrahedral symmetry $\backslash mathrm\{T\}\_\backslash mathrm\{d\}$, the symmetry group for a regular tetrahedron and has the order of twenty-four.
• pyritohedral symmetry $\backslash mathrm\{T\}\_\backslash mathrm\{h\}$, the symmetry of a pyritohedron and has the order of twenty-four.
• chiral octahedral symmetry $\backslash mathrm\{O\}$, the rotation group of both cube and regular octahedron and has the order twenty-four.
• full octahedral symmetry $\backslash mathrm\{O\}\_\backslash mathrm\{h\}$, the symmetry group of both cube and regular octahedron and has order forty-eight.
• chiral icosahedral symmetry $\backslash mathrm\{I\}$, the rotation group of both regular icosahedron and regular dodecahedron and has the order of sixty.
• full icosahedral symmetry $\backslash mathrm\{I\}\_\backslash mathrm\{h\}$, the symmetry group of both regular icosahedron and regular dodecahedron and has the order of a hundred-twenty.

File:Square pyramid.png has pyramidal symmetry $C\_\{4\backslash mathrm\{v\}\}$. It shows the appearance is invariant by rotating every quarter of a full turn around its axis and possesses mirror symmetric relative to any perpendicular plane passing through its base's bisector]]

Point groups in three dimensions may also allow the preservation of polyhedra's appearance by the circulation around an axis. There are three various of these point groups:

• pyramidal symmetry $C\_\{n\; \backslash mathrm\{v\}\}$, allowing rotate the axis passing through the apex and its base, as well as reflection relative to perpendicular planes passing through the bisector of a base. This point group symmetry can be found in pyramids,{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} cupolas, and rotundas.
• prismatic symmetry $D\_\{n\backslash mathrm\{h\}\}$, similar to the pyramidal symmetry, but with additional transformation by reflecting it across a horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} This may be achieved from the family of prisms and its dual bipyramids.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}}
• antiprismatic symmetry $D\_\{n\; \backslash mathrm\{v\}\}$, which preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} Examples can be found in antiprisms.

A point group $C\_\{n\; \backslash mathrm\{h\}\}$ consists of rotating around the axis of symmetry and reflection on the horizontal plane. In the case of $n\; =\; 1$, the symmetry group only preserves the symmetry by a full rotation solely, ordinarily denoting $C\_s$.{{citation

| last1 = Hergert | first1 = W.

| last2 = Geilhufe | first2 = M.

| year = 2018

| title = Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica

| url = https://books.google.com/books?id=6mvpCgAAQBAJ&pg=PA56

| publisher = John Wiley & Sons

| isbn = 978-3-527-41300-3

}} Polyhedra may have rotation only to preserve the symmetry, and the symmetry group may be considered as the cyclic group $C\_n$.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA125 125]}} Polyhedra with the rotoreflection and the rotation by the cyclic group is the point group $S\_n$.{{sfnp|Hergert|Geilhufe|2018|p=[https://books.google.com/books?id=6mvpCgAAQBAJ&pg=PA57 57]}}

{{-}}

{{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 convex deltahedron). All of these classes are convex polyhedra.

}}

As mentioned above, the convex polyhedra are well-defined, with several equivalent standard definitions. They are often defined as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points,{{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

}} restricted in either case to intersections or hulls that have nonzero volume.

Important classes of convex polyhedra include the family of prismatoids, the Platonic solids, the Archimedean solids and their duals the Catalan solids, and the Johnson solids. Prismatoids are the polyhedra whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles.{{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

}}. 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.{{sfnp|Cromwell|1997|p=51–52}} 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.{{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}}.}} their dual polyhedra are the Catalan solids.{{citation

| last = Diudea | first = M. V.

| year = 2018

| title = Multi-shell Polyhedral Clusters

| series = Carbon Materials: Chemistry and Physics

| volume = 10

| publisher = 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

}}. Johnson solids are the class of convex polyhedra whose faces are all regular polygons.{{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

}}. These include the convex deltahedra, strictly convex polyhedra whose faces are all equilateral triangles.{{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

}}.

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 pyramids onto the square faces of a triangular prism; the square pyramids and the triangular prism are elementaries.{{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

}}.

{{multiple image

| image1 = Midsphere.png

| caption1 = A canonical polyhedron

| image2 = Fritsch map.svg

| caption2 = The triangumented triangular prism's skeleton 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.{{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

}}.

By forgetting the face structure, any polyhedron gives rise to a graph, called its 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 polyhedra appear in M.C. Escher's print Stars.{{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. 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, 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–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).{{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

}}.{{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

}}. A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face. The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.

{{Main|Honeycomb (geometry)}}

A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a Dehn invariant equal to zero. Some honeycombs involve more than one kind of polyhedron.

{{main|Flexible polyhedron}}

It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem, flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem.{{citation

| last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine

| last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)

| contribution = 23.2 Flexible polyhedra

| doi = 10.1017/CBO9780511735172

| isbn = 978-0-521-85757-4

| mr = 2354878

| pages = 345–348

| publisher = Cambridge University Press, Cambridge

| title = Geometric Folding Algorithms: Linkages, origami, polyhedra

| title-link=Geometric Folding Algorithms

| year = 2007}}.

{{main article|Ideal polyhedron}}

Convex polyhedra can be defined in three-dimensional hyperbolic space in the same way as in Euclidean space, as the convex hulls of finite sets of points. However, in hyperbolic space, it is also possible to consider ideal points and the points within the space. An ideal polyhedron is the convex hull of a finite set of ideal points.{{citation

| last = Thurston | first = William P. | authorlink = William Thurston

| isbn = 0-691-08304-5

| mr = 1435975

| publisher = Princeton University Press, Princeton, NJ

| series = Princeton Mathematical Series

| title = Three-dimensional geometry and topology. Vol. 1

| volume = 35

| year = 1997

| page = 128

| url = https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA128

}} Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.

Convex polyhedra in which all vertices have integer coordinates are called lattice polyhedra or integral polyhedra. The Ehrhart polynomial of lattice a polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra.{{citation

| last = Stanley | first = Richard P. | author-link = Richard P. Stanley

| year = 1997

| title = Enumerative Combinatorics, Volume I

| edition = 1

| publisher = Cambridge University Press

| pages = 235–239

| isbn = 978-0-521-66351-9

}} An example is Reeve tetrahedron.{{citation

| last = Kołodziejczyk | first = Krzysztof

| doi = 10.1007/BF00150027

| issue = 3

| journal = Geometriae Dedicata

| mr = 1397808

| pages = 271–278

| title = An "odd" formula for the volume of three-dimensional lattice polyhedra

| volume = 61

| year = 1996| s2cid = 121162659

}}

There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties.{{citation |last=Cox |first=David A. |title=Toric varieties |date=2011 |publisher=American Mathematical Society |others=John B. Little, Henry K. Schenck |isbn=978-0-8218-4819-7 |location=Providence, R.I. |oclc=698027255}} This was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes.{{citation |last=Stanley |first=Richard P. |title=Combinatorics and commutative algebra |date=1996 |publisher=Birkhäuser |isbn=0-8176-3836-9 |edition=2nd |location=Boston |oclc=33080168}}

A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models.

A zonohedron is a convex polyhedron in which every face is a polygon that is symmetric under rotations through 180°. Zonohedra can also be characterized as the Minkowski sums of line segments, and include several important space-filling polyhedra.{{citation

| last = Taylor | first = Jean E. | author-link = Jean Taylor

| doi = 10.2307/2324178

| issue = 2

| journal = American Mathematical Monthly

| mr = 1144350

| pages = 108–111

| title = Zonohedra and generalized zonohedra

| volume = 99

| year = 1992| jstor = 2324178 }}.

{{main article|Orthogonal polyhedron}}

File:Soma_cube_figures.svg made of Soma cube pieces, themselves polycubes]]

{{anchor|Orthogonal polyhedra}}Polyhedra are said to be orthogonal because all of their edges are parallel to the axes of a Cartesian coordinate system. This implies that all faces meet at right angles, but this condition is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the rectangular cuboids, orthogonal polyhedra are nonconvex. They are the three-dimensional analogs of two-dimensional orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.{{citation

| last = O'Rourke | first = Joseph | author-link = Joseph O'Rourke (professor)

| contribution = Unfolding orthogonal polyhedra

| doi = 10.1090/conm/453/08805

| mr = 2405687

| pages = 307–317

| publisher = Amer. Math. Soc., Providence, RI

| series = Contemp. Math.

| title = Surveys on discrete and computational geometry

| volume = 453

| year = 2008| isbn =978-0-8218-4239-3 | doi-access = free

}}. Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar polyominoes.{{citation

| last = Gardner | first = Martin | author-link = Martin Gardner

| date = November 1966

| issue = 5

| journal = Scientific American

| jstor = 24931332

| pages = 138–143

| title = Mathematical Games: Is it possible to visualize a four-dimensional figure?

| volume = 215| doi = 10.1038/scientificamerican1166-138

}}

The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.

A classical polyhedral surface has a finite number of faces, joined in pairs along edges. The apeirohedra form a related class of objects with infinitely many faces. Examples of apeirohedra include:

• tilings or tessellations of the plane, and
• sponge-like structures called infinite skew polyhedra.

{{Main|Complex polytope}}

There are objects called complex polyhedra, for which the underlying space is a complex Hilbert space rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more closely related to configurations than to real polyhedra.{{citation

| last = Coxeter | first = H.S.M. | author-link = Harold Scott MacDonald Coxeter

| location = Cambridge

| mr = 0370328

| publisher = Cambridge University Press

| title = Regular Complex Polytopes

| year = 1974}}

Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow digonal faces to exist with a positive area.

• When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedra) have no flat-faced analogue.{{citation|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=978-1-4665-0430-1|page=463|url=https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA463|quote="A hosohedron is only possible on a sphere"}}.
• If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some of these curved polyhedra can pack together to fill space. Two important types are bubbles in froths and foams such as Weaire-Phelan bubbles,{{citation

| last1 = Kraynik | first1 = A.M.

| last2 = Reinelt | first2 = D.A.

| editor-last = Mortensen | editor-first = Andreas

| contribution = Foams, Microrheology of

| edition = 2nd

| pages = 402–407

| publisher = Elsevier

| title = Concise Encyclopedia of Composite Materials

| year = 2007}}. See in particular [https://books.google.com/books?id=zs_lGeGsuaAC&pg=PA403 p. 403]: "foams consist of polyhedral gas bubbles ... each face on a polyhedron is a minimal surface with uniform mean curvature ... no face can be a flat polygon with straight edges". and forms used in architecture.{{citation|last=Pearce|first=P.|title=Structure in nature is a strategy for design|publisher=MIT Press|year=1978|page=224|url=https://books.google.com/books?id=sfc2OEuE8oQC&pg=PA224|contribution=14 Saddle polyhedra and continuous surfaces as environmental structures|isbn=978-0-262-66045-7}}.

{{Main|n-dimensional polyhedron}}

From the latter half of the twentieth century, various mathematical constructs have been found to have properties also present in traditional polyhedra. Rather than confining the term "polyhedron" to describe a three-dimensional polytope, it has been adopted to describe various related but distinct kinds of structure.

A polyhedron has been defined as a set of points in real affine (or Euclidean) space of any dimension n that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a conventional polyhedron, it may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.

Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra in this way provides a geometric perspective for problems in linear programming.{{Cite Geometric Algorithms and Combinatorial Optimization}}{{Rp|page=9}}

File:Papyrus moscow 4676-problem 14 part 1.jpg, on calculating the volume of a frustum]]

Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided Egyptian pyramids dating from the 27th century BC.{{citation | last = Kitchen | first = K. A. | date = October 1991 | doi = 10.1080/00438243.1991.9980172 | issue = 2 | journal = World Archaeology | pages = 201–208 | title = The chronology of ancient Egypt | volume = 23}} The Moscow Mathematical Papyrus from approximately 1800–1650 BC includes an early written study of polyhedra and their volumes (specifically, the volume of a frustum).{{citation | last1 = Gunn | first1 = Battiscombe | last2 = Peet | first2 = T. Eric | date = May 1929 | doi = 10.1177/030751332901500130 | issue = 1 | journal = The Journal of Egyptian Archaeology | pages = 167–185 | title = Four Geometrical Problems from the Moscow Mathematical Papyrus | volume = 15| s2cid = 192278129 }} The mathematics of the Old Babylonian Empire, from roughly the same time period as the Moscow Papyrus, also included calculations of the volumes of cuboids (and of non-polyhedral cylinders), and calculations of the height of such a shape needed to attain a given volume.{{citation | last = Friberg | first = Jöran | issue = 2 | journal = Revue d'Assyriologie et d'archéologie orientale | jstor = 23281940 | pages = 97–188 | title = Mathematics at Ur in the Old Babylonian Period | volume = 94 | year = 2000}}

The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an Etruscan dodecahedron made of soapstone on Monte Loffa. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die.{{citation |title=An Etruscan dodecahedron|first=Amelia Carolina|last=Sparavigna|year=2012|arxiv=1205.0706}}

{{multiple image

| align = right |total_width=500

| image1 = Kepler Hexahedron Earth.jpg |width1=290|height1=304

| image2 = Kepler Icosahedron Water.jpg |width2=306|height2=328

| image3 = Kepler Octahedron Air.jpg |width3=328|height3=334

| image4 = Kepler Tetrahedron Fire.jpg |width4=367|height4=328

| image5 = Kepler Dodecahedron Universe.jpg |width5=330|height5=332

| footer = five elements in each Platonic solids, but the assignment drawing was by Kepler's Harmonices Mundi

}}

Ancient Greek mathematicians discovered and studied the convex regular polyhedra, which came to be known as the Platonic solids. Their first written description is in the Timaeus of Plato (circa 360 BC), which associates four of them with the four elements and the fifth to the overall shape of the universe. A more mathematical treatment of these five polyhedra was written soon after in the Elements of Euclid. An early commentator on Euclid (possibly Geminus) writes that the attribution of these shapes to Plato is incorrect: Pythagoras knew the tetrahedron, cube, and dodecahedron, and Theaetetus (circa 417 BC) discovered the other two, the octahedron and icosahedron.{{citation | last = Eves | first = Howard | date = January 1969 | department = Historically Speaking | doi = 10.5951/mt.62.1.0042 | issue = 1 | journal = The Mathematics Teacher | jstor = 27958041 | pages = 42–44 | title = A geometry capsule concerning the five platonic solids | volume = 62}} Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.{{citation | last = Field | first = J. V. | author-link = Judith V. Field | doi = 10.1007/BF00374595 | issue = 3–4 | journal = Archive for History of Exact Sciences | jstor = 41134110 | mr = 1457069 | pages = 241–289 | title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler | volume = 50 | year = 1997| s2cid = 118516740 }}

File:14-sided Chinese dice from warring states period.jpg]]

Both cubical dice and 14-sided dice in the shape of a truncated octahedron in China have been dated back as early as the Warring States period.{{citation

| last1 = Bréard | first1 = Andrea | author1-link = Andrea Bréard

| last2 = Cook | first2 = Constance A.

| date = December 2019

| doi = 10.1007/s00407-019-00245-9

| issue = 4

| journal = Archive for History of Exact Sciences

| pages = 313–343

| title = Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts

| volume = 74| s2cid = 253898304 }}

By 236 AD, Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.{{citation

| last = van der Waerden | first = B. L.

| contribution = Chapter 7: Liu Hui and Aryabhata

| doi = 10.1007/978-3-642-61779-9_7

| pages = 192–217

| publisher = Springer

| title = Geometry and Algebra in Ancient Civilizations

| year = 1983}}

After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam).{{citation | last = Knorr | first = Wilbur | author-link = Wilbur Knorr | doi = 10.1016/0315-0860(83)90034-4 | issue = 1 | journal = Historia Mathematica | mr = 698139 | pages = 71–78 | title = On the transmission of geometry from Greek into Arabic | volume = 10 | year = 1983| doi-access = free }} The 9th century scholar Thabit ibn Qurra included the calculation of volumes in his studies,{{citation | last = Rashed | first = Roshdi | editor-last = Rashed | editor-first = Roshdi | contribution = Thābit ibn Qurra et l'art de la mesure | contribution-url = https://books.google.com/books?id=V5PTZi77YxwC&pg=PA173 | language = fr | isbn = 9783110220780 | pages = 173–175 | publisher = Walter de Gruyter | series = Scientia Graeco-Arabica | title = Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad | volume = 4 | year = 2009}} and wrote a work on the cuboctahedron. Then in the 10th century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.{{citation | last1 = Hisarligil | first1 = Hakan | last2 = Hisarligil | first2 = Beyhan Bolak | date = December 2017 | doi = 10.1007/s00004-017-0363-7 | issue = 1 | journal = Nexus Network Journal | pages = 125–152 | title = The geometry of cuboctahedra in medieval art in Anatolia | volume = 20| doi-access = free }}

{{multiple image

| image1 = Pacioli.jpg

| caption1 = Doppio ritratto, attributed to Jacopo de' Barbari, depicting Luca Pacioli and a student studying a glass rhombicuboctahedron half-filled with water.{{citation | last = Gamba | first = Enrico | title = Imagine Math | editor-last = Emmer | editor-first = Michele | contribution = The mathematical ideas of Luca Pacioli depicted by Iacopo de' Barbari in the Doppio ritratto | doi = 10.1007/978-88-470-2427-4_25 | isbn = 978-88-470-2427-4 | pages = 267–271 | publisher = Springer | year = 2012}}

| image2 = Leonardo polyhedra.png

| caption2 = A skeletal polyhedron (specifically, a rhombicuboctahedron) drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli

| total_width = 400

}}

As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective.{{citation|title=The Polyhedrists: Art and Geometry in the Long Sixteenth Century|first=Noam|last=Andrews|publisher=MIT Press|year=2022|isbn=9780262046640}} Toroidal polyhedra, made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in marquetry panels of the period as a symbol of geometry.{{citation | last1 = Calvo-López | first1 = José | last2 = Alonso-Rodríguez | first2 = Miguel Ángel | date = February 2010 | doi = 10.1007/s00004-010-0018-4 | issue = 1 | journal = Nexus Network Journal | pages = 75–111 | title = Perspective versus stereotomy: From Quattrocento polyhedral rings to sixteenth-century Spanish torus vaults | volume = 12| doi-access = free }} Piero della Francesca wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids. Leonardo da Vinci illustrated skeletal models of several polyhedra for a book by Luca Pacioli,{{citation | last = Field | first = J. V. | authorlink = Judith V. Field | doi = 10.1007/BF00374595 | issue = 3–4 | journal = Archive for History of Exact Sciences | jstor = 41134110 | mr = 1457069 | pages = 241–289 | s2cid = 118516740 | title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler | volume = 50 | year = 1997}} with text largely plagiarized from della Francesca.{{citation | last = Montebelli | first = Vico | doi = 10.1007/s40329-015-0090-4 | issue = 3 | journal = Lettera Matematica | mr = 3402538 | pages = 135–141 | title = Luca Pacioli and perspective (part I) | volume = 3 | year = 2015 | s2cid = 193533200}} Polyhedral nets make an appearance in the work of Albrecht Dürer.{{citation | last = Ghomi | first = Mohammad | issue = 1 | journal = Notices of the American Mathematical Society | mr = 3726673 | pages = 25–27 | title = Dürer's unfolding problem for convex polyhedra | url = https://www.ams.org/publications/journals/notices/201801/rnoti-p25.pdf | volume = 65 | year = 2018| doi = 10.1090/noti1609 }}

Several works from this time investigate star polyhedra, and other elaborations of the basic Platonic forms. A marble tarsia in the floor of St. Mark's Basilica, Venice, designed by Paolo Uccello, depicts a stellated dodecahedron.{{citation | last = Saffaro | first = Lucio | editor1-last = Taliani | editor1-first = C. | editor2-last = Ruani | editor2-first = G. | editor3-last = Zamboni | editor3-first = R. | contribution = Cosmoids, fullerenes and continuous polygons | contribution-url = https://books.google.com/books?id=dOk7DwAAQBAJ&pg=PA55 | location = Singapore | pages = 55–64 | publisher = World Scientific | title = Fullerenes: Status and Perspectives, Proceedings of the 1st Italian Workshop, Bologna, Italy, 6–7 February | year = 1992}} As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings. Johannes Kepler (1571–1630) used star polygons, typically pentagrams, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polyhedra must be convex.{{citation | last = Field | first = J. V. | author-link = Judith V. Field | doi = 10.1016/0083-6656(79)90001-1 | issue = 2 | journal = Vistas in Astronomy | mr = 546797 | pages = 109–141 | title = Kepler's star polyhedra | volume = 23 | year = 1979| bibcode = 1979VA.....23..109F }}

In the same period, Euler's polyhedral formula, a linear equation relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico.{{citation|first= Michael|last=Friedman|publisher=Birkhäuser|year=2018|title=A History of Folding in Mathematics: Mathematizing the Margins|title-link=A History of Folding in Mathematics|series=Science Networks. Historical Studies|volume=59|isbn=978-3-319-72486-7|doi=10.1007/978-3-319-72487-4|page=71}}

René Descartes, in around 1630, wrote his book De solidorum elementis studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was Descartes' theorem on total angular defect, which is closely related to Euler's polyhedral formula.{{citation | last = Federico | first = Pasquale Joseph | isbn = 0-387-90760-2 | mr = 680214 | publisher = Springer-Verlag | series = Sources in the History of Mathematics and Physical Sciences | title = Descartes on Polyhedra: A Study of the "De solidorum elementis" | volume = 4 | year = 1982}} Leonhard Euler, for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof.{{citation

| last1 = Francese | first1 = Christopher

| last2 = Richeson | first2 = David

| doi = 10.1080/00029890.2007.11920417

| issue = 4

| journal = The American Mathematical Monthly

| mr = 2281926

| pages = 286–296

| title = The flaw in Euler's proof of his polyhedral formula

| volume = 114

| year = 2007| s2cid = 10023787

}} Euler's work (together with his earlier solution to the puzzle of the Seven Bridges of Königsberg) became the foundation of the new field of topology.{{citation

| last = Alexanderson | first = Gerald L.

| doi = 10.1090/S0273-0979-06-01130-X

| issue = 4

| journal = American Mathematical Society

| mr = 2247921

| pages = 567–573

| series = New Series

| title = About the cover: Euler and Königsberg's bridges: a historical view

| volume = 43

| year = 2006| doi-access = free

}} The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by Henri Poincaré, Enrico Betti, Bernhard Riemann, and others.{{citation | last = Eckmann | first = Beno | contribution = The Euler characteristic – a few highlights in its long history | doi = 10.1007/978-3-540-33791-1_15 | isbn = 978-3-540-33791-1 | mr = 2269092 | pages = 177–188 | publisher = Springer | title = Mathematical Survey Lectures 1943–2004 | year = 2006}}

In the early 19th century, Louis Poinsot extended Kepler's work, and discovered the remaining two regular star polyhedra. Soon after, Augustin-Louis Cauchy proved Poinsot's list complete, subject to an unstated assumption that the sequence of vertices and edges of each polygonal side cannot admit repetitions (an assumption that had been considered but rejected in the earlier work of A. F. L. Meister).{{citation | last = Grünbaum | first = Branko | editor-last = Grattan-Guinness | editor-first = I. | contribution = Regular polyhedra | contribution-url = https://books.google.com/books?id=ZptYDwAAQBAJpg | isbn = 0-415-03785-9 | mr = 1469978 | pages = 866–876 | publisher = Routledge | title = Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences | volume = 2 | year = 1994}} They became known as the Kepler–Poinsot polyhedra, and their usual names were given by Arthur Cayley.{{citation|title=Regular-faced polyhedra: remembering Norman Johnson|work=AMS Feature column|first=Joseph|last=Malkevitch|year=2018|url=https://www.ams.org/publicoutreach/feature-column/fc-2018-01|publisher=American Mathematical Society|access-date=2023-05-27}}

Meanwhile, the discovery of higher dimensions in the early 19th century led Ludwig Schläfli by 1853 to the idea of higher-dimensional polytopes.{{sfnp|Coxeter|1947|pages=141–143}} Additionally, in the late 19th century, Russian crystallographer Evgraf Fedorov completed the classification of parallelohedra, convex polyhedra that tile space by translations.{{citation|last=Austin|first=David|title=Fedorov's five parallelohedra|work=AMS Feature Column|publisher=American Mathematical Society|url=https://www.ams.org/samplings/feature-column/fc-2013-11|date=November 2013}}

Mathematics in the 20th century dawned with Hilbert's problems, one of which, Hilbert's third problem, concerned polyhedra and their dissections. It was quickly solved by Hilbert's student Max Dehn, introducing the Dehn invariant of polyhedra.{{citation

| last = Zeeman | first = E. C. | author-link = Christopher Zeeman

| date = July 2002

| doi = 10.2307/3621846

| issue = 506

| journal = The Mathematical Gazette

| jstor = 3621846

| pages = 241–247

| title = On Hilbert's third problem

| volume = 86| s2cid = 125593771 }} Steinitz's theorem, published by Ernst Steinitz in 1992, characterized the graphs of convex polyhedra, bringing modern ideas from graph theory and combinatorics into the study of polyhedra.{{citation

| last = Grünbaum | first = Branko | author-link = Branko Grünbaum

| doi = 10.1016/j.disc.2005.09.037

| hdl = 1773/2276 | hdl-access = free

| issue = 3–5

| journal = Discrete Mathematics

| mr = 2287486

| pages = 445–463

| title = Graphs of polyhedra; polyhedra as graphs

| volume = 307

| year = 2007}}

The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper The Fifty-Nine Icosahedra.{{citation

| last1 = Coxeter | first1 = H.S.M. | author1-link = Harold Scott MacDonald Coxeter

| last2 = Du Val | first2 = P.

| last3 = Flather | first3 = H.T.

| last4 = Petrie | first4 = J. F.

| isbn = 978-1-899618-32-3

| mr = 676126

| orig-date = 1938

| publisher = Tarquin Publications

| title = The Fifty-Nine Icosahedra

| title-link = The Fifty-Nine Icosahedra

| year = 1999}}. Coxeter's analysis signaled a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the regular skew polyhedra and to develop the theory of complex polyhedra first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry.{{citation|title=King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry|first=Siobhan|last=Roberts|author-link=Siobhan Roberts|publisher=Bloomsbury Publishing|year=2009|isbn=9780802718327}}

In the second part of the twentieth century, both Branko Grünbaum and Imre Lakatos pointed out the tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment. In a series of papers, Grünbaum broadened the accepted definition of a polyhedron, discovering many new regular polyhedra. At the close of the twentieth century, these latter ideas merged with other work on incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte.{{citation|last1=McMullen|first1=Peter|author1-link=Peter McMullen|last2=Schulte|first2=Egon|author2-link=Egon Schulte|title=Abstract Regular Polytopes|series=Encyclopedia of Mathematics and its Applications|volume=92|publisher=Cambridge University Press|year=2002}}

{{multiple image

| total_width = 350

| align = right

| image1 = Circogonia icosahedra.jpg

| caption1 = The radiolarian Circogonia icosahedra

| image2 = Dymaxion projection.png

| caption2 = Dymaxion map, created by the net of a regular icosahedron

}}

Polyhedra have been discovered in many fields of science. The Platonic solids appeared in biological creatures, as in The Braarudosphaera bigelowii has a regular dodecahedral structure.{{citation

| last1 = Hagino | first1 = K.

| last2 = Onuma | first2 = R.

| last3 = Kawachi | first3 = M.

| last4 = Horiguchi | first4 = T.

| year = 2013

| title = Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in Braarudosphaera bigelowii (Prymnesiophyceae)

| journal = PLOS ONE

| volume = 8

| issue = 12

| article-number = e81749

| doi = 10.1371/journal.pone.0081749

| doi-access = free

| pmid = 24324722

| pmc = 3852252

| bibcode = 2013PLoSO...881749H

}} Ernst Haeckel described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra.{{citation

| last = Haeckel | first = E

| year = 1904

| title = Kunstformen der Natur

}}. Available as Haeckel, E. Art forms in nature, Prestel USA (1998), {{isbn|3-7913-1990-6}}. Online version at [http://www.biolib.de/haeckel/kunstformen/index.html Kurt Stüber's Biolib] (in German) The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus.{{citation

| title = Virus Taxonomy

| chapter = Myoviridae

| publisher = Elsevier

| year = 2012

| pages = 46–62

| doi = 10.1016/b978-0-12-384684-6.00002-1

| isbn = 9780123846846

| ref={{sfnref|Elsevier|2012}}

}}{{citation

| last1 = Strauss | first1 = James H.

| last2 = Strauss | first2 = Ellen G.

| title = Viruses and Human Disease

| chapter = The Structure of Viruses

| publisher = Elsevier

| year = 2008

| pages = 35–62

| doi = 10.1016/b978-0-12-373741-0.50005-2

| pmc = 7173534

| isbn = 9780123737410

| s2cid = 80803624

}} The regular icosahedron may also appeared in the applications of cartography when R. Buckminster Fuller used its net to his project known as Dymaxion map, frustatedly realized that the Greenland size is smaller than the South America.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/7/mode/1up?view=theater 7]}}

Polyhedra make a frequent appearance in modern computational geometry, computer graphics, and geometric design with topics including the reconstruction of polyhedral surfaces or surface meshes from scattered data points,{{citation

| last1 = Lim | first1 = Seng Poh

| last2 = Haron | first2 = Habibollah

| date = March 2012

| doi = 10.1007/s10462-012-9329-z

| issue = 1

| journal = Artificial Intelligence Review

| pages = 59–78

| title = Surface reconstruction techniques: a review

| volume = 42| s2cid = 254232891

}} geodesics on polyhedral surfaces,{{citation

| last1 = Mitchell | first1 = Joseph S. B. | author1-link = Joseph S. B. Mitchell

| last2 = Mount | first2 = David M. | author2-link = David Mount

| last3 = Papadimitriou | first3 = Christos H. | author3-link = Christos Papadimitriou

| doi = 10.1137/0216045

| issue = 4

| journal = SIAM Journal on Computing

| mr = 899694

| pages = 647–668

| title = The discrete geodesic problem

| volume = 16

| year = 1987}} visibility and illumination in polyhedral scenes,{{citation

| last1 = Teller | first1 = Seth J. | author1-link = Seth J. Teller

| last2 = Hanrahan | first2 = Pat | author2-link = Pat Hanrahan

| editor-last = Whitton | editor-first = Mary C. | editor-link = Mary Whitton

| contribution = Global visibility algorithms for illumination computations

| doi = 10.1145/166117.166148

| pages = 239–246

| publisher = Association for Computing Machinery

| title = Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1993, Anaheim, CA, USA, August 2–6, 1993

| year = 1993| isbn = 0-89791-601-8 | s2cid = 7957200 }} polycubes and other non-convex polyhedra with axis-parallel sides,{{citation|title=Polycube Optimization and Applications: From the Digital World to Manufacturing|hdl=11584/261570|first=Gianmarco|last=Cherchi|date=February 2019|type=Doctoral dissertation|publisher=University of Cagliari}} algorithmic forms of Steinitz's theorem,{{citation

| last = Rote | first = Günter

| editor1-last = van Kreveld | editor1-first = Marc J.

| editor2-last = Speckmann | editor2-first = Bettina

| contribution = Realizing planar graphs as convex polytopes

| doi = 10.1007/978-3-642-25878-7_23

| pages = 238–241

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Graph Drawing – 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21–23, 2011, Revised Selected Papers

| volume = 7034

| year = 2011| doi-access = free

| isbn = 978-3-642-25877-0

}} and the still-unsolved problem of the existence of polyhedral nets for convex polyhedra.{{citation|last1=Demaine|first1=Erik|author1-link=Erik Demaine|last2=O'Rourke|first2=Joseph|author2-link=Joseph O'Rourke (professor)|title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra|title-link=Geometric Folding Algorithms|publisher=Cambridge University Press|year=2007}}

• Chazelle polyhedron
• Extension of a polyhedron
• Goldberg polyhedron
• List of books about polyhedra
• Near-miss Johnson solid
• Polyhedron model
• Polyhedral number
• Polyhedral skeletal electron pair theory
• Polyhedral space
• Polyhedral symbol
• Polyhedral terrain
• Polytope model
• Stella (software)

{{notelist|group="lower-alpha"}}

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{{Wiktionary|polyhedron}}

{{Commons|Polyhedra}}

{{Wikisource|1911 Encyclopædia Britannica/Polyhedron|Polyhedron}}

• {{Mathworld | urlname=Polyhedron | title=Polyhedron | mode=cs2 }}
• [http://www.steelpillow.com/polyhedra/ Polyhedra Pages]
• [https://www.math.technion.ac.il/S/rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra by Dr. Zvi Har'El] {{Webarchive|url=https://web.archive.org/web/20151127053535/http://www.math.technion.ac.il/S/rl/docs/uniform.pdf |date=2015-11-27 }}
• [https://web.archive.org/web/20170224151555/http://www.uwgb.edu/dutchs/symmetry/symmetry.htm Symmetry, Crystals and Polyhedra]

• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra.
• [http://www.eg-models.de/index.html Electronic Geometry Models] – Contains a peer reviewed selection of polyhedra with unusual properties.
• [http://www.orchidpalms.com/polyhedra/index.html Polyhedron Models] – Virtual polyhedra.
• [http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]

• [https://web.archive.org/web/20170616023727/http://www.uff.br/cdme/pdp/pdp-html/pdp-en.html A Plethora of Polyhedra] – An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.
• [http://dogfeathers.com/java/hyperstar.html Hyperspace Star Polytope Slicer] – Explorer java applet, includes a variety of 3d viewer options.
• [http://www.openscad.org/ openSCAD] – Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The openSCAD User Manual is also available.
• [http://www.openvolumemesh.org/ OpenVolumeMesh] – An open source cross-platform C++ library for handling polyhedral meshes. Developed by the Aachen Computer Graphics Group, RWTH Aachen University.
• [https://levskaya.github.com/polyhedronisme/ Polyhedronisme] {{Webarchive|url=https://web.archive.org/web/20120425234840/http://levskaya.github.com/polyhedronisme/ |date=2012-04-25 }} – Web-based tool for generating polyhedra models using Conway Polyhedron Notation. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files.

• [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Free nets of polyhedra.
• [http://ldlewis.com/How-to-Build-Polyhedra/ Simple instructions for building over 30 paper polyhedra]
• [http://hbmeyer.de/flechten/indexeng.htm Polyhedra plaited with paper strips] – Polyhedra models constructed without use of glue.

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