|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{redirect|Polyhedra|the relational database system|Polyhedra DBMS}}
| | Hi, everybody! My name is Sandy. <br>It is a little about myself: I live in Switzerland, my city of Gasenried. <br>It's called often Eastern or cultural capital of . I've married 4 years ago.<br>I have two children - a son (Clinton) and the daughter (Royal). We all like Paintball.<br><br>Feel free to surf to my weblog; renovation ideas [[http://www.homeimprovementdaily.com homeimprovementdaily.com]] |
| {{about||the game magazine|Polyhedron (magazine)|the scientific journal|Polyhedron (journal)}}
| |
| {| style="border:8px solid white; float:right;"
| |
| |-
| |
| |
| |
| {| class="wikitable"
| |
| |+Some Polyhedra
| |
| |align=center|[[Image:POV-Ray-Dodecahedron.svg|120px]]<br>[[Dodecahedron]]<br>([[Platonic solid|Regular polyhedron]])
| |
| |align=center|[[Image:Small stellated dodecahedron.png|120px]]<br>[[Small stellated dodecahedron]]<br>([[Kepler-Poinsot polyhedron|Regular star]])
| |
| |-
| |
| |align=center|[[Image:Icosidodecahedron.png|120px]]<br>[[Icosidodecahedron]]<br>([[Archimedean solid|Uniform/Quasiregular]])
| |
| |align=center|[[Image:Great cubicuboctahedron.png|120px]]<br>[[Great cubicuboctahedron]]<br>([[Nonconvex uniform polyhedron|Uniform star]])
| |
| |-
| |
| |align=center|[[Image:Rhombictriacontahedron.svg|120px]]<br>[[Rhombic triacontahedron]]<br>([[Catalan solid|Uniform/Quasiregular dual]])
| |
| |align=center|[[Image:Elongated pentagonal cupola.png|120px]]<br>[[Elongated pentagonal cupola]]<br>([[Johnson solid|Convex regular-faced]])
| |
| |-
| |
| |align=center|[[Image:Octagonal prism.png|120px]]<br>[[Octagonal prism]]<br>([[Uniform prism]])
| |
| |align=center|[[Image:Square antiprism.png|120px]]<br>[[Square antiprism]]<br>([[Uniform antiprism]])
| |
| |}
| |
| |}
| |
| In [[geometry]], a '''polyhedron''' (plural '''polyhedra''' or '''polyhedrons''') is a solid in [[Three-dimensional space|three dimensions]] with flat [[Face (geometry)|faces]] and straight [[Edge (geometry)|edges]]. The word '''polyhedron''' comes from the [[Greek language|Classical Greek]] πολύεδρον, as ''poly-'' (stem of πολύς, "many") + ''-hedron'' (form of έδρα, "base", "seat", or "face").
| |
| | |
| A polyhedron is a 3-dimensional example of the more general [[polytope]] in any number of dimensions.
| |
| | |
| ==Basis for definition==
| |
| [[File:UniversumUNAM19.JPG|thumb|Polyhedron blocks on display at the [[Universum (UNAM)|Universum museum]] in Mexico City]]
| |
| Defining a polyhedron as a solid bounded by flat faces and straight edges is not very precise and, to a modern [[mathematician]], quite unsatisfactory, for example it is difficult to reconcile with [[star polyhedra]]. [[Branko Grünbaum|Grünbaum]] (1994, p. 43) observed, "The [[Original Sin]] in the theory of polyhedra goes back to [[Euclid]], and through [[Kepler]], [[Poinsot]], [[Cauchy]] and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others.<ref>Lakatos, I.; ''Proofs and refutations: The logic of mathematical discovery'' (2nd Ed.), CUP, 1977.</ref> For example definitions based on the idea of a boundary surface rather than a solid are common.<ref name="cromwell">Cromwell, P.; ''Polyhedra'', CUP, 1997.</ref> However such definitions are not always compatible in other mathematical contexts.
| |
| | |
| One modern approach treats a geometric polyhedron as a ''realisation'' of some [[#Abstract polyhedra|abstract polyhedron]]. Any such polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:
| |
| *3 dimensions: The '''body''' is bounded by the faces, and is usually the [[volume]] enclosed by them.
| |
| *2 dimensions: A '''[[face (geometry)|face]]''' is a ''[[polygon]]'' bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral '''surface'''.
| |
| *1 dimension: An '''[[edge (geometry)|edge]]''' joins one vertex to another and one face to another, and is usually a [[line (mathematics)|line]] segment. The edges together make up the polyhedral '''skeleton'''.
| |
| *0 dimensions: A '''[[vertex (geometry)|vertex]]''' (plural '''vertices''') is a corner [[point (geometry)|point]].
| |
| *−1 dimension: The '''null polytope''' is a kind of mathematical convention required by abstract set-based definitions, and may be represented geometrically as the [[empty set]] of points. From an [[inductive dimension]] approach, the boundary of a point is the empty set and it has dimension one less than a point, leading to a dimension of −1.
| |
| | |
| More generally in [[mathematics]] and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely [[algebra]]ic or abstract.
| |
| | |
| == Characteristics ==
| |
| | |
| ===Polyhedral surface===
| |
| A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.
| |
| | |
| ===Edges===
| |
| Edges have two important characteristics (unless the polyhedron is [[#Complex polyhedra|complex]]):
| |
| *An edge joins just two vertices.
| |
| *An edge joins just two faces.
| |
| These two characteristics are [[#Duality|dual]] to each other.
| |
| | |
| ===Euler characteristic===
| |
| The '''[[Euler characteristic]]''' χ relates the number of vertices ''V'', edges ''E'', and faces ''F'' of a polyhedron:
| |
| :<math>\chi=V-E+F.\ </math>
| |
| | |
| For a [[convex polyhedron]] or more generally for any [[simply connected space|simply connected]] polyhedron whose faces are also simply connected and whose boundary is a [[manifold]], χ = 2. For a detailed discussion, see [[Proofs and Refutations]] by [[Imre Lakatos]].
| |
| | |
| ===Orientability===
| |
| Some polyhedra, such as all [[convex polytope|convex polyhedra]], have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is [[Orientability|orientable]].
| |
| | |
| But for some polyhedra this is not possible, and the figure is said to be non-orientable. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even χ < 2 may or may not be orientable.
| |
| | |
| ===Vertex figure===
| |
| For every vertex one can define a [[vertex figure]], which describes the local structure of the figure around the vertex. If the vertex figure is a [[regular polygon]], then the vertex itself is said to be ''regular''.
| |
| | |
| ===Duality===
| |
| [[Image:Dual Cube-Octahedron.svg|75px]]
| |
| | |
| For every polyhedron there exists a [[dual polyhedron]] having:
| |
| *faces in place of the original's vertices and vice versa,
| |
| *the same number of edges
| |
| *the same Euler characteristic and orientability
| |
| | |
| The dual of a convex polyhedron can be obtained by the process of [[polar reciprocation]].
| |
| | |
| ===Volume===
| |
| | |
| ;Elementary calculation
| |
| Any regular polyhedron can be divided up into congruent [[Pyramid (geometry)|pyramids]], with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex. The height of a pyramid is equal to the [[inradius]] of the polyhedron. If the area of a face is <math>A</math> and the in-radius is <math>r</math> then the volume of the pyramid is one-third of the base times the height, or <math>Ar/3</math>. For a regular polyhedron with <math>n</math> faces, its volume is then simply
| |
| :<math>
| |
| \text{volume} = nAr/3
| |
| </math>.
| |
| | |
| For instance, a cube with edges of length <math>L</math> has six faces, each face being a square with area <math>A = L^2</math>. The inradius from the center of the face to the center of the cube is <math> r = L/2</math>. Then the volume is given by
| |
| :<math>
| |
| \text{volume} = \frac{6\cdot L^2 \cdot \frac{L}{2}}{3} = L^3,
| |
| </math>
| |
| the usual formula for the volume of a cube.
| |
| | |
| ;Advanced calculation
| |
| The volume of any [[orientability|orientable]] polyhedron can be calculated using the [[divergence theorem]]. Consider the [[vector field]] <math>\vec F(\vec x) = \frac{1}{3} \vec x = (\frac{x_1}{3}, \frac{x_2}{3}, \frac{x_3}{3})</math>, whose [[divergence]] is identically 1. The divergence theorem implies that the volume is equal to a surface integral of <math>F(x)</math>:
| |
| :<math>
| |
| \text{volume}(\Omega) = \int_\Omega \nabla\cdot\vec F d\Omega = \oint_S \vec F \cdot \hat n dS.
| |
| </math>
| |
| | |
| When Ω is the region enclosed by a polyhedron, since the faces of a polyhedron are planar and have piecewise constant [[Normal (geometry)|normal vectors]], this simplifies to
| |
| | |
| :<math>
| |
| \text{volume} = \frac{1}{3}\sum_{\text{face } i} \vec x_i \cdot \hat n_i A_i
| |
| </math>
| |
| | |
| where for the ''i'''th face, <math>\vec x_i</math> is the face [[barycenter]], <math>\hat n_i</math> is its normal vector, and <math>A_i</math> is its area.<ref>{{cite book |last=Arvo |first=James |year=1991 |title=Graphic Gems Package: Graphics Gems II | publisher=Academic Press}}</ref>
| |
| Once the faces are decomposed in a set of non-overlapping triangles with surface normals pointing away from the volume, the volume is a sixths of the sum over the [[Vector triple product#Scalar triple product|triple products]] of the nine Cartesian vertex coordinates of the triangles.
| |
| | |
| Since it may be difficult to [[enumerate]] the faces, volume computation may be challenging, and hence there exist specialized [[algorithm]]s to determine the volume (many of these generalize to [[convex polytope]]s in higher dimensions).<ref>{{cite doi|10.1007/978-3-0348-8438-9_6}}</ref>
| |
| | |
| ==Names of polyhedra==
| |
| Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example [[tetrahedron]] (4), [[pentahedron]] (5), [[hexahedron]] (6), [[heptahedron]] (7), [[triacontahedron]] (30), and so on. Sometimes this is qualified by a description of the kinds of faces present, for example the [[Rhombic dodecahedron]] vs. the [[Dodecahedron|Pentagonal dodecahedron]].
| |
| | |
| Some polyhedra have gained common names, for example the regular hexahedron is commonly known as the [[cube]]. Others are named after their discoverer, such as [[Miller's monster]] or the [[Szilassi polyhedron]].
| |
| | |
| Other common names indicate that some operation has been performed on a simpler polyhedron, for example the [[truncated cube]] looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron or tetradecahedron).
| |
| | |
| == Traditional polyhedra ==
| |
| [[Image:POV-Ray-Dodecahedron.svg|right|thumb|169px|A dodecahedron]]
| |
| In [[geometry]], a '''polyhedron''' is traditionally a three-dimensional shape that is made up of a finite number of [[polygon]]al ''[[Face (geometry)|faces]]'' which are parts of [[plane (mathematics)|planes]]; the faces meet in pairs along ''[[Edge (geometry)|edges]]'' which are [[straight line|straight-line]] segments, and the edges meet in points called ''[[Vertex (geometry)|vertices]]''. [[Cube (geometry)|Cubes]], [[Prism (geometry)|prisms]] and [[Pyramid (geometry)|pyramids]] are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.
| |
| | |
| A polyhedron is said to be '''[[Convex polyhedron|convex]]''' if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.
| |
| | |
| ===Symmetrical polyhedra===
| |
| Many of the most studied polyhedra are highly [[Symmetry|symmetrical]].
| |
| | |
| Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as [[icosidodecahedron]], the most symmetrical geometry is almost always implied, unless otherwise stated.
| |
| | |
| Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid or [[tetrahedron]], [[cube]] or hexahedron, [[octahedron]], [[dodecahedron]] and [[icosahedron]]:
| |
| | |
| :{| class="wikitable"
| |
| |[[Image:Tetrahedron.svg|50px]]
| |
| |[[Image:Hexahedron.svg|50px]]
| |
| |[[Image:Octahedron.svg|50px]]
| |
| |[[Image:POV-Ray-Dodecahedron.svg|50px]]
| |
| |[[Image:Icosahedron.svg|50px]]
| |
| |}
| |
| | |
| There are several types of highly-symmetric polyhedron, classified by which kind of element - faces, edges and/or vertices - belong to a single [[Symmetry orbit#Orbits and stabilizers|symmetry orbit]]:
| |
| * '''[[Isogonal figure|Isogonal]]''' or '''[[Vertex-transitive]]''' if all vertices are the same, in the sense that for any two vertices there exists a [[symmetry group|symmetry]] of the polyhedron mapping the first [[isometry|isometrically]] onto the second.
| |
| * '''[[Isotoxal]]''' or '''[[Edge-transitive]]''' if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
| |
| * '''[[Isohedral]]''' or '''[[Face-transitive]]''' if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
| |
| * '''[[Regular polyhedron|Regular]]''' if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same [[regular polygon]]; it also implies that every vertex is regular).
| |
| * '''[[Quasiregular polyhedron|Quasi-regular]]''' if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A '''quasi-regular dual''' is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.
| |
| * '''[[Semiregular polyhedron|Semi-regular]]''' if it is vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class). A '''semi-regular dual''' is face-transitive but not vertex-transitive, and every vertex is regular.
| |
| * '''[[Uniform polyhedron|Uniform]]''' if it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular. A '''uniform dual''' is face-transitive and has regular vertices, but is not necessarily vertex-transitive).
| |
| * '''[[Noble polyhedron|Noble]]''' if it is face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra.
| |
| | |
| A polyhedron can belong to the same overall [[symmetry group]] as one of higher symmetry, but will have several groups of elements (for example faces) in different symmetry orbits.
| |
| | |
| ====Uniform polyhedra and their duals====
| |
| {{Main|Uniform polyhedron}}
| |
| | |
| '''Uniform polyhedra''' are ''[[vertex-transitive]]'' and every face is a [[regular polygon]].
| |
| They may be [[Regular polyhedron|regular]], [[Quasiregular polyhedron|quasi-regular]], or [[Semiregular polyhedron|semi-regular]], and may be convex or starry.
| |
| | |
| The '''uniform [[Dual polyhedron|duals]]''' are '''[[face-transitive]]''' and every [[vertex figure]] is a regular polygon.
| |
| | |
| Face-transitivity of a polyhedron corresponds to vertex-transitivity of the dual and conversely, and edge-transitivity of a polyhedron corresponds to edge-transitivity of the dual. The dual of a regular polyhedron is also regular. The dual of a non-regular uniform polyhedron (called a [[Catalan solid]] if convex) has irregular faces.
| |
| | |
| Each uniform polyhedron shares the same symmetry as its dual, with the symmetries of faces and vertices simply swapped over. Because of this some authorities regard the duals as uniform too. But this idea is not held widely: a polyhedron and its symmetries are not the same thing.
| |
| | |
| The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are [[Convex polyhedron|convex]] or not.
| |
| | |
| {| class="wikitable"
| |
| !
| |
| !Convex uniform
| |
| !Convex uniform dual
| |
| !Star uniform
| |
| !Star uniform dual
| |
| |-
| |
| ![[Regular polyhedron|Regular]]
| |
| | style="text-align:center;" colspan="2"|[[Platonic solid]]s
| |
| | style="text-align:center;" colspan="2"|[[Kepler-Poinsot polyhedra]]
| |
| |-
| |
| ![[Quasiregular polyhedron|Quasiregular]]
| |
| |rowspan="2"|[[Archimedean solid]]s
| |
| |rowspan="2"|[[Catalan solids]]
| |
| | (no special name)
| |
| | (no special name)
| |
| |-
| |
| !rowspan="3"|[[Semiregular polyhedron|Semiregular]]
| |
| | (no special name)
| |
| | (no special name)
| |
| |-
| |
| |[[Prism (geometry)|Prisms]]
| |
| |[[Bipyramids|Dipyramids]]
| |
| |Star [[Prism (geometry)|Prisms]]
| |
| |Star [[Bipyramids|Dipyramids]]
| |
| |-
| |
| |[[Antiprism]]s
| |
| |[[Trapezohedra]]
| |
| |Star [[Antiprism]]s
| |
| |Star [[Trapezohedra]]
| |
| |}
| |
| | |
| ====Noble polyhedra====
| |
| {{main|Noble polyhedron}}
| |
| | |
| A '''[[Noble polyhedron|noble]]''' polyhedron is both [[isohedral]] (equal-faced) and [[isogonal figure|isogonal]] (equal-cornered). Besides the regular polyhedra, there are many other examples.
| |
| | |
| The [[Dual polyhedron|dual]] of a noble polyhedron is also noble.
| |
| | |
| ====Symmetry groups====
| |
| The polyhedral symmetry groups (using [[Schoenflies notation]]) are all [[Point groups in three dimensions#The seven remaining point groups|point groups]] and include:
| |
| *'''T''' - '''chiral [[tetrahedral symmetry]]'''; the rotation group for a regular [[tetrahedron]]; order 12.
| |
| *'''T<sub>d</sub>''' - '''full [[tetrahedral symmetry]]'''; the symmetry group for a regular [[tetrahedron]]; order 24.
| |
| *'''T<sub>h</sub>''' - '''[[tetrahedral symmetry|pyritohedral symmetry]]'''; order 24. The symmetry of a [[pyritohedron]].
| |
| *'''O''' - '''chiral [[octahedral symmetry]]''';the rotation group of the [[Cube (geometry)|cube]] and [[octahedron]]; order 24.
| |
| *'''O<sub>h</sub>''' - '''full [[octahedral symmetry]]'''; the symmetry group of the [[Cube (geometry)|cube]] and [[octahedron]]; order 48.
| |
| *'''I''' - '''chiral [[icosahedral symmetry]]'''; the rotation group of the [[icosahedron]] and the [[dodecahedron]]; order 60.
| |
| *'''I<sub>h</sub>''' - '''full [[icosahedral symmetry]]'''; the symmetry group of the [[icosahedron]] and the [[dodecahedron]]; order 120.
| |
| *'''C<sub>nv</sub>''' - [[Cyclic symmetries|''n''-fold pyramidal symmetry]]
| |
| *'''D<sub>nh</sub>''' - [[Dihedral symmetry|''n''-fold prismatic symmetry]]
| |
| *'''D<sub>nv</sub>''' - [[Dihedral symmetry|''n''-fold antiprismatic symmetry]].
| |
| | |
| Those with [[Chirality (mathematics)|chiral]] symmetry do not have [[reflection symmetry]] and hence have two [[Chirality (mathematics)|enantiomorphous]] forms which are reflections of each other. The '''snub''' Archimedean polyhedra have this property.
| |
| | |
| === Other polyhedra with regular faces ===
| |
| | |
| ====Equal regular faces====
| |
| A few families of polyhedra, where every face is the same kind of polygon:
| |
| | |
| *'''[[#Deltahedra|Deltahedra]]''' have equilateral triangles for faces.
| |
| | |
| *With regard to polyhedra whose faces are all squares: if [[coplanar]] faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (''a'',''b'',''c''), with integers ''a'',''b'',''c'' of which at most one is even.
| |
| | |
| *There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.
| |
| | |
| There exists no polyhedron whose faces are all identical and are regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (See [[infinite skew polyhedron]] for exceptions with zig-zagging [[vertex figure]]s.)
| |
| | |
| ===== Deltahedra =====
| |
| A '''[[deltahedron]]''' (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
| |
| | |
| * 3 regular convex polyhedra (3 of the Platonic solids)
| |
| ** [[Tetrahedron]]
| |
| ** [[Octahedron]]
| |
| ** [[Icosahedron]]
| |
| * 5 non-uniform convex polyhedra (5 of the Johnson solids)
| |
| ** [[Triangular dipyramid]]
| |
| ** [[Pentagonal dipyramid]]
| |
| ** [[Snub disphenoid]]
| |
| ** [[Triaugmented triangular prism]]
| |
| ** [[Gyroelongated square dipyramid]].
| |
| | |
| ==== Johnson solids ====
| |
| {{main|Johnson solid}}
| |
| [[Norman Johnson (mathematician)|Norman Johnson]] sought which convex non-uniform polyhedra had regular faces. In 1966, he published a list of 92 such solids, gave them names and numbers, and conjectured that there were no others. [[Victor Zalgaller]] proved in 1969 that the list of these '''[[Johnson solid]]s''' was complete.
| |
| | |
| === Other important families of polyhedra ===
| |
| | |
| ==== Pyramids ====
| |
| {{main|Pyramid (geometry)}}
| |
| Pyramids include some of the most time-honoured and famous of all polyhedra.
| |
| | |
| ==== Stellations and facettings ====
| |
| {{Main|Stellation}}
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| |[[Image:First stellation of octahedron.png|50px]]
| |
| |[[Image:First stellation of dodecahedron.png|50px]]
| |
| |[[Image:Second stellation of dodecahedron.png|50px]]
| |
| |[[Image:Third stellation of dodecahedron.png|50px]]
| |
| |[[Image:Sixteenth stellation of icosahedron.png|50px]]
| |
| |[[Image:First stellation of icosahedron.png|50px]]
| |
| |[[Image:Seventeenth stellation of icosahedron.png|50px]]
| |
| |}
| |
| | |
| '''Stellation''' of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.
| |
| | |
| It is the exact reciprocal to the process of '''[[facetting]]''' which is the process of removing parts of a polyhedron without creating any new vertices.
| |
| | |
| ==== Zonohedra ====
| |
| {{main|Zonohedron}}
| |
| A '''zonohedron''' is a convex polyhedron where every face is a [[polygon]] with inversion [[symmetry]] or, equivalently, symmetry under [[rotation]]s through 180°.
| |
| | |
| ==== Toroidal polyhedra ====
| |
| {{main|Toroidal polyhedron}}
| |
| A '''toroidal polyhedron''' is a polyhedron with an [[Euler characteristic]] of 0 or smaller, equivalent to a [[Genus (mathematics)|Genus]] of 1 or greater, representing a [[torus]] surface having one or more holes through the middle.
| |
| | |
| ==== Compounds ====
| |
| {{main|Polyhedral compound}}
| |
| Polyhedral compounds are formed as compounds of two or more polyhedra.
| |
| | |
| These compounds often share the same vertices as other polyhedra and are often formed by stellation. Some are listed in the [[list of Wenninger polyhedron models]].
| |
| | |
| ==== Orthogonal polyhedra ====
| |
| An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D [[orthogonal polygon]]s, also known as [[rectilinear polygon]]s. Orthogonal polyhedra are used in [[computational geometry]], where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a [[polygonal net]].
| |
| | |
| ==Generalisations of polyhedra==
| |
| The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.
| |
| | |
| ===Apeirohedra===
| |
| A classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called an '''[[apeirohedron]]'''. Examples include:
| |
| *'''Tilings''' or '''[[tessellation]]s''' of the plane.
| |
| *Sponge-like structures called '''[[infinite skew polyhedron|infinite skew polyhedra]]'''.
| |
| | |
| See also: [[Apeirogon]] - infinite regular polygon: {∞}
| |
| | |
| ===Complex polyhedra===
| |
| A [[complex polytope|complex polyhedron]] is one which is constructed in complex [[Hilbert space|Hilbert]] 3-space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).
| |
| | |
| ===Curved polyhedra===
| |
| Some fields of study allow polyhedra to have curved faces and edges.
| |
| | |
| ====Spherical polyhedra====
| |
| {{Main|Spherical polyhedron}}
| |
| | |
| The surface of a sphere may be divided by line segments into bounded regions, to form a '''spherical polyhedron'''. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
| |
| | |
| Spherical polyhedra have a long and respectable history:
| |
| * The first known man-made polyhedra are spherical polyhedra carved in stone.
| |
| * Poinsot used spherical polyhedra to discover the four regular star polyhedra.
| |
| * Coxeter used them to enumerate all but one of the uniform polyhedra.
| |
| | |
| Some polyhedra, such as [[hosohedron|hosohedra]] and [[dihedron|dihedra]], exist only as spherical polyhedra and have no flat-faced analogue.
| |
| | |
| ====Curved spacefilling polyhedra====
| |
| Two important types are:
| |
| *Bubbles in froths and foams, such as [[Weaire–Phelan structure|Weaire-Phelan bubbles]].
| |
| *Spacefilling forms used in architecture. See for example Pearce (1978).
| |
| <!--More needs to be said about these, too.--> | |
| | |
| === General polyhedra ===
| |
| More recently [[mathematics]] has defined a '''polyhedron''' as a set in [[real number|real]] [[affine space|affine]] (or [[Euclidean space|Euclidean]]) space of any dimensional ''n'' that has flat sides. It may alternatively be defined as the union of a finite number of convex polyhedra, where a ''convex polyhedron'' is any set that is the intersection of a finite number of [[half-space (geometry)|half-space]]s. 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]].
| |
| | |
| Many traditional polyhedral forms are general polyhedra. Other examples include:
| |
| | |
| * A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { ( ''x'', ''y'' ) : x ≥ 0, y ≥ 0 }. Its sides are the two positive axes.
| |
| * An octant in Euclidean 3-space, { ( ''x'', ''y'', ''z'' ) : x ≥ 0, y ≥ 0, z ≥ 0 }.
| |
| * A prism of infinite extent. For instance a doubly infinite square prism in 3-space, consisting of a square in the ''xy''-plane swept along the ''z''-axis: { ( ''x'', ''y'', ''z'' ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }.
| |
| * Each [[cell (geometry)|cell]] in a [[Voronoi diagram|Voronoi tessellation]] is a convex polyhedron. In the Voronoi tessellation of a set ''S'', the cell ''A'' corresponding to a point ''c''∈''S'' is bounded (hence a traditional polyhedron) when ''c'' lies in the [[interior (topology)|interior]] of the [[convex hull]] of ''S'', and otherwise (when ''c'' lies on the [[Boundary (topology)|boundary]] of the convex hull of ''S'') ''A'' is unbounded.
| |
| | |
| ===Hollow-faced or skeletal polyhedra===
| |
| It is not necessary to fill in the face of a figure before we can call it a polyhedron. For example [[Leonardo da Vinci]] devised frame models of the regular solids, which he drew for [[Pacioli]]'s book ''Divina Proportione''. In modern times, [[Branko Grünbaum]] (1994) made a special study of this class of polyhedra, in which he developed an early idea of [[#Abstract polyhedra|abstract polyhedra]]. He defined a '''face''' as a cyclically ordered set of vertices, and allowed faces to be [[Skew polygon|skew]] as well as planar.
| |
| | |
| ==Non-geometric polyhedra==
| |
| Various mathematical constructs have been found to have properties also present in traditional polyhedra.
| |
| | |
| === Topological polyhedra ===
| |
| A '''topological polytope''' is a topological space given along with a specific decomposition into shapes that are topologically equivalent to [[convex polytope]]s and that are attached to each other in a regular way.
| |
| | |
| Such a figure is called ''simplicial'' if each of its regions is a [[simplex]], i.e. in an ''n''-dimensional space each region has ''n''+1 vertices. The dual of a simplicial polytope is called ''simple''. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an ''n''-dimensional cube.
| |
| | |
| ===Abstract polyhedra===
| |
| An '''abstract polyhedron''' is a [[partially ordered set]] (poset) of elements whose partial ordering obeys certain rules. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, or ''nullitope'', which has a dimensionality of −1. These posets belong to the larger family of [[abstract polytope]]s in any number of dimensions.
| |
| | |
| ===Polyhedra as graphs===
| |
| Any polyhedron gives rise to a '''[[Graph (mathematics)|graph]]''', or '''skeleton''', with corresponding vertices and edges. Thus [[Glossary of graph theory|graph terminology]] and properties can be applied to polyhedra. For example:
| |
| * Due to [[Steinitz theorem]] convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.
| |
| * The [[tetrahedron]] gives rise to a [[complete graph]] (K<sub>4</sub>). It is the only polyhedron to do so.
| |
| * The [[octahedron]] gives rise to a [[strongly regular graph]], because adjacent vertices always have two common neighbors, and non-adjacent vertices have four.
| |
| * The [[Archimedean solid]]s give rise to [[regular graph]]s: 7 of the Archimedean solids are of [[Degree (graph theory)|degree]] 3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.
| |
| | |
| ==History==
| |
| | |
| ===Prehistory===
| |
| Stones carved in the shape of a cluster of spheres or similar objects have been found in [[Scotland]] and may be as much as 4,000 years old. These stones show the symmetries of various polyhedra, but have curved surfaces. Examples of these stones are on display in the [http://www.ashmol.ox.ac.uk/ash/guide/t-text/room29.html John Evans room] of the [[Ashmolean Museum]] at [[Oxford University]]. It is not known why these objects were made, or how the sculptor gained the inspiration for them.
| |
| | |
| Polyhedra appeared in early [[architecture|architectural forms]] such as cubes and cuboids, with the earliest four-sided pyramids of ancient [[Egypt]] also dating from the Stone Age.
| |
| | |
| The [[Etruscan civilization|Etruscans]] preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near [[Padua]] (in Northern [[Italy]]) in the late 19th century of a [[dodecahedron]] made of [[soapstone]], and dating back more than 2,500 years (Lindemann, 1987).
| |
| | |
| ===Greeks===
| |
| The earliest known ''written'' records of these shapes come from Classical [[Ancient Greece|Greek]] authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the [[Regular polyhedron#History|convex regular polyhedra]], which came to be known as the [[Platonic solid]]s. [[Pythagoras]] knew at least three of them, and [[Theaetetus (mathematician)|Theaetetus]] (circa 417 B. C.) described all five. Eventually, [[Euclid]] described their construction in his ''[[Euclid's Elements|Elements]]''. Later, [[Archimedes]] expanded his study to the [[Uniform polyhedron|convex uniform polyhedra]] which now bear his name. His original work is lost and his solids come down to us through [[Pappus of Alexandria|Pappus]].
| |
| | |
| ===Chinese ===
| |
| By 236 AD, in China 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.
| |
| | |
| ===Islamic===
| |
| After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see [[Mathematics in medieval Islam]]).
| |
| | |
| The 9th century scholar [[Thabit ibn Qurra]] gave formulae for calculating the volumes of polyhedra such as truncated pyramids.
| |
| | |
| Then in the 10th century [[Abūl Wafā' Būzjānī|Abu'l Wafa]] described the convex regular and quasiregular spherical polyhedra.
| |
| | |
| ===Renaissance===
| |
| 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 (graphical)|perspective]]. Several appear in marquetry panels of the period. [[Piero della Francesca]] gave the first written description of direct geometrical construction of such perspective views of polyhedra. [[Leonardo da Vinci]] made skeletal models of several polyhedra and drew illustrations of them for a book by Pacioli. A painting by an anonymous artist of Pacioli and a pupil depicts a glass [[rhombicuboctahedron]] half-filled with water.
| |
| | |
| As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of various kinds, many of them novel, in imaginative etchings.
| |
| | |
| === Star polyhedra ===
| |
| For almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians.
| |
| | |
| During the [[Renaissance]] star forms were discovered. A marble tarsia in the floor of [[St. Mark's Basilica]], Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel star-like forms of increasing complexity.
| |
| | |
| [[Johannes Kepler]] realized that [[star polygon]]s, typically [[pentagram]]s, could be used to build star polyhedra. Some of these star polyhedra 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 polytopes be convex. Later, [[Louis Poinsot]] realized that star [[vertex figure]]s (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the [[small stellated dodecahedron]] and [[great stellated dodecahedron]], and (Poinsot's) the [[great icosahedron]] and [[great dodecahedron]]. Collectively they are called the [[Kepler-Poinsot polyhedra]].
| |
| | |
| 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 59 icosahedra''. This work has recently been re-published (Coxeter, 1999).
| |
| | |
| The reciprocal process to stellation is called [[facetting]] (or faceting). Every stellation of one polytope is [[Dual polyhedron|dual]], or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. {{Harvnb|Bridge|1974|p=}} listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.
| |
| <!-- [[star polyhedron|star polyhedra]]. -->
| |
| | |
| '''See also:'''
| |
| *[[Regular polyhedron#History|Regular polyhedron: History]]
| |
| *[[Regular polytope#History of discovery|Regular polytope: History of discovery]].
| |
| | |
| ==Polyhedra in nature==
| |
| <!--This needs a bit of writing, esp. re '''''crystals'''''.-->
| |
| | |
| For natural occurrences of regular polyhedra, ''see [[Regular polyhedron#Regular polyhedra in nature|Regular polyhedron: Regular polyhedra in nature]].''
| |
| | |
| Irregular polyhedra appear in nature as [[crystal]]s.
| |
| | |
| == See also ==
| |
| {{commons|1=Polyhedron}}
| |
| <div style="-moz-column-count:3; column-count:3;">
| |
| *[[Antiprism]]
| |
| *[[Archimedean solid]]
| |
| *[[Bipyramid]]
| |
| *[[Conway polyhedron notation]] (a notation for describing construction of polyhedra)
| |
| *[[defect (geometry)|Defect]]
| |
| *[[Deltahedron]]
| |
| *[[Deltohedron]]
| |
| *[[M. C. Escher|Escher]]
| |
| *[[Flexible polyhedra]]
| |
| *[[Extension of a polyhedron]]
| |
| *[[:Category:Images of polyhedra|Images of polyhedra]]
| |
| *[[Johnson solid]]
| |
| *[[Kepler–Poinsot polyhedra]]
| |
| *[[Near-miss Johnson solid]]
| |
| *[[Net (polyhedron)]]
| |
| *[[Platonic solid]]
| |
| *[[Polychoron]] (4 dimensional analogues to polyhedra)
| |
| *[[Polyhedral compound]]
| |
| *[[Polyhedron model]]s
| |
| *[[Prism (geometry)|Prism]]
| |
| *[[Schlegel diagram]]
| |
| *[[Semiregular polyhedron]]
| |
| *[[Spidron]]
| |
| *[[Tessellation]]
| |
| *[[Trapezohedron]]
| |
| *[[Uniform polyhedron]]
| |
| *[[Waterman polyhedron]]
| |
| *[[Zonohedron]]
| |
| </div>
| |
| | |
| ==References==
| |
| {{More footnotes|date=December 2010}}
| |
| {{reflist}}
| |
| *[[Coxeter|Coxeter, H.S.M.]]; ''Regular complex Polytopes'', CUP (1974).
| |
| *Cromwell, P.;''Polyhedra'', CUP hbk (1997), pbk. (1999).
| |
| *[[Branko Grünbaum|Grünbaum, B.]]; Polyhedra with Hollow Faces, ''Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993)'', ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
| |
| *Grünbaum, B.; Are your polyhedra the same as my polyhedra? ''Discrete and comput. geom: the Goodman-Pollack festschrift'', ed. Aronov et al. Springer (2003) pp. 461–488.'' ([http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf pdf])
| |
| *Pearce, P.; ''Structure in nature is a strategy for design'', MIT (1978)
| |
| | |
| ==Books on polyhedra==
| |
| {{Main|List of books about polyhedra}}
| |
| | |
| == External links ==
| |
| {{Wiktionary|polyhedron}}
| |
| | |
| === General theory ===
| |
| *{{Mathworld | urlname=Polyhedron | title=Polyhedron }}
| |
| * [http://www.steelpillow.com/polyhedra/ Polyhedra Pages]
| |
| * [http://www.math.technion.ac.il/~rl/kaleido/ uniform solution for uniform polyhedra by Dr. Zvi Har'El]
| |
| * [http://www.uwgb.edu/dutchs/symmetry/symmetry.htm Symmetry, Crystals and Polyhedra]
| |
| *[http://www.cgl.uwaterloo.ca/~csk/washington/tile/papers/kaplan_hart_bridges2001.pdf Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons]
| |
| | |
| === Lists and databases of polyhedra ===
| |
| * [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
| |
| * [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] - The Encyclopedia of Polyhedra
| |
| ** [http://www.georgehart.com/virtual-polyhedra/pyramids-info.html Polyhedra and Pyramids]
| |
| * [http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra in Java]
| |
| * [http://www.eg-models.de/index.html Electronic Geometry Models] - Contains a peer reviewed selection of polyhedra with unusual properties.
| |
| * [http://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] - Models made with Modular Origami
| |
| * [http://bulatov.org/polyhedra/ Polyhedra Collection] - Various virtual and physical polyhedra models.
| |
| * [http://hbmeyer.de/flechten/indexeng.htm Polyhedra plaited with paper strips] - Polyhedra models constructed without use of glue.
| |
| * [http://polyhedra.org/poly/ Rotatable polyhedron models] (uses Javascript)
| |
| * [http://www.robertinventor.com/software/virtualflower/polyhedra/thumbnails_1.htm Rotatable self-intersecting polyhedra] (uses Java)
| |
| * [http://www.orchidpalms.com/polyhedra/index.html Polyhedron Models] - Virtual polyhedra
| |
| * [http://homepage.mac.com/dmccooey/polyhedra/ Java Applets for Visualizing Polyhedra] - Systematically explicit formulae given!
| |
| | |
| === Software ===
| |
| * [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://www.software3d.com/Stella.php Stella: Polyhedron Navigator] - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
| |
| * [http://www.viviasoft.com/polyhedra/index.htm World of Polyhedra] - Comprehensive polyhedra in Flash applet, showing vertices and edges (but not shaded faces)
| |
| * [http://dogfeathers.com/java/hyperstar.html Hyperspace Star Polytope Slicer] - Explorer java applet, includes a variety of 3d viewer options.
| |
| * [http://www.orchidpalms.com/polyheda/hedron.html HEDRON] - Polyhedron modelling software
| |
| * [http://public.beuth-hochschule.de/~meiko/pentatope.html Uniform Polyhedra Java Applets with sources]
| |
| * [http://www.openscad.org/ openSCAD] - Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The [http://en.wikibooks.org/wiki/OpenSCAD_User_Manual 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.
| |
| *[http://levskaya.github.com/polyhedronisme/ Polyhedronisme] - Web based tool for generating polyhedra models using [[Conway polyhedron notation|Conway Polyhedron Notation]]. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files. The 3D files can be opened in CAD software, or uploaded for 3D printing at services such as [http://www.shapeways.com Shapeways].
| |
| | |
| === Resources for making physical models, and models for sale ===
| |
| * [http://mathforum.org/sum95/math_and/poly/polyhedra.html Making Polyhedra]
| |
| * [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Free nets of polyhedra
| |
| * [http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]
| |
| * [http://www.bamboocorps.com/ Kits and instructions for making models of polyhedra out of bamboo]
| |
| * [http://www.peda.com/poly Polyhedra software, die-cast models, & posters]
| |
| * [http://ldlewis.com/How-to-Build-Polyhedra/ Simple instructions for building over 30 paper polyhedra]
| |
| * [http://peak.org/~jeremy/hexagons/ Kits to print out, cut, and fold into various polyhedra]
| |
| | |
| === Miscellaneous ===
| |
| * [http://www.picturespice.com PictureSpice] - A site that lets you make polyhedra with your own uploaded pictures.
| |
| * [http://www.youtube.com/watch?v=KWB6s5FBBdo Lattice Polyhedra]
| |
| | |
| {{Polyhedra}}
| |
| {{Polyhedron navigator}}
| |
| {{Polytopes}}
| |
| | |
| [[Category:Polyhedra|*]]
| |
| | |
| {{Link FA|it}}
| |
| {{Link FA|ca}}
| |