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Encyclopedia > Polyhedron
Some Polyhedra

dodecahedron
(Regular polyhedron)

Small stellated dodecahedron
(Regular star)

Icosidodecahedron
(Uniform)

Great cubicuboctahedron
(Uniform star)

Rhombic triacontahedron
(Uniform dual)

Elongated pentagonal cupola
(Convex regular-faced)

Octagonal prism
(Uniform prism)

Square antiprism
(Uniform antiprism)

A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. The Polyhedron magazine is the official publication of the RPGA, or Roleplaying Gamers Association. ... Image File history File links This is a lossless scalable vector image. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... In geometry, a Platonic solid is a convex regular polyhedron. ... Image File history File links Download high resolution version (639x641, 18 KB) Summary Small stellated dodecahedron, U34 Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... A single face is colored yellow and outlined in red to help identify the faces. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 258 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... Image File history File links Download high resolution version (640x640, 20 KB) Summary Stellated truncated hexahedron, U19 Licensing I, the creator of this work, hereby release it into the public domain. ... In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... Image File history File links Rhombictriacontahedron. ... The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. ... A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ... Image File history File links Elongated_pentagonal_cupola. ... In geometry, the elongated pentagonal cupola is one of the Johnson solids (J20). ... The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... Image File history File links Square_antiprism. ... In geometry, the Square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...


The word polyhedron comes from the Classical Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face". Greek ( IPA: or simply IPA: — Hellenic) has a documented history of 3,500 years, the longest of any single language in the Indo-European language family. ...


Although that might seem clear enough for most of us, mathematicians do not agree as to exactly what makes something a polyhedron. In an oft-quoted but seldom respected remark, Grünbaum (1994) observed that: Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...


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' ... For other uses, see Euclid (disambiguation). ... Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ... Louis Poinsot (1777 - 1859) was a French mathematician and physicist. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

Contents

What is a polyhedron?

We can at least say that a polyhedron is 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 inside them.
  • 2 dimensions: A face is bounded by a circuit of edges, and is usually a flat (plane) region called a polygon. The faces together make up the polyhedral surface.
  • 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line of some kind. The edges together make up the polyhedral skeleton.
  • 0 dimensions: A vertex (plural vertices) is a corner point.
  • -1 dimension: The nullity is a kind of non-entity required by abstract theories.

More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract. For other uses, see Body (disambiguation). ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... Look up polygon in Wiktionary, the free dictionary. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...


A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...


Characteristics

Naming 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. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A hexahedron is a polyhedron with six faces. ... A heptahedron is a polyhedron having seven sides, or faces. ... The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. ...


Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron. The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...


Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron. In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75. ... The Szilassi polyhedron. ...


Edges


Edges have two important characteristics (unless the polyhedron is complex):

  • An edge joins just two vertices.
  • An edge joins just two faces.

These two characteristics are dual to each other.


Euler characteristic


The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron: It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

χ = V - E + F.

For a simply connected polyhedron χ = 2.


Duality


Image File history File links Dual_Cube-Octahedron. ...


For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa. In most cases the dual can be obtained by the process of spherical reciprocation. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...


Vertex figure


For every vertex one can define a vertex figure consisting of the vertices joined to it. The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron. In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...


Traditional polyhedra

A dodecahedron

In geometry, a polyhedron is traditionally a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Look up polygon in Wiktionary, the free dictionary. ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... This article is about the mathematical construct. ... An edge between two vertices For edge in graph theory, see Edge (graph theory) In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertices in a polytope. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ...


A polyhedron is said to be 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 and surface. Look up Convex set in Wiktionary, the free dictionary. ...


Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetrical. Sphere symmetry group o. ...


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. An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...


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 tetrahedron, cube, octahedron, dodecahedron and icosahedron: A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

Polyhedra of the highest symmetries have all of some kind of element - faces, edges and/or vertices, within a single symmetry orbit. There are various classes of such polyhedra: Image File history File links Tetrahedron. ... Image File history File links Hexahedron. ... Image File history File links No higher resolution available. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links Icosahedron. ...

  • Isogonal or Vertex-transitive if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first 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 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).
  • 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.
  • 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 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 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. In geometry, a polyhedron (or tiling) is isogonal or vertex-transitive if all its vertices are the same. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... The symmetry group of an object (e. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. ... In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... // A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular. ... A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). ...


Uniform polyhedra and their duals

Main article: Uniform polyhedron

Uniform polyhedra are vertex-transitive and every face is a regular polygon. They may be regular, quasi-regular, or semi-regular, and may be convex or starry. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... // A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular. ... A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. ...


The uniform duals are face-transitive and every vertex figure is a regular polygon. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...


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. In most duals of uniform polyhedra, faces are irregular polygons. The regular polyhedra are an exception, because they are dual to each other.


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 or not. Look up convex in Wiktionary, the free dictionary. ...

Convex uniform Convex uniform dual Star uniform Star uniform dual
Regular Platonic solids Kepler-Poinsot polyhedra
Quasiregular Archimedean solids Catalan solids (no special name) (no special name)
Semiregular (no special name) (no special name)
Prisms Dipyramids Star Prisms Star Dipyramids
Antiprisms Trapezohedra Star Antiprisms Star Trapezohedra

In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... In geometry, a Platonic solid is a convex regular polyhedron. ... A single face is colored yellow and outlined in red to help identify the faces. ... // A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... Wikipedia does not yet have an article with this exact name. ... A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... Regular octahedron triangular dipyramid J12 Pentagonal dipyramid J13 An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image base-to-base. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... Regular octahedron triangular dipyramid J12 Pentagonal dipyramid J13 An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image base-to-base. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of a regular n-gonal antiprism. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of a regular n-gonal antiprism. ...

Noble polyhedra

Main article: Noble polyhedron

A noble polyhedron is both isohedral (equal-faced) and isogonal (equal-cornered). Besides the regular polyhedra, there are many other examples. A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). ... A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ... In geometry, a polyhedron (or tiling) is isogonal or vertex-transitive if all its vertices are the same. ...


The dual of a noble polyhedron is also noble. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...


Symmetry groups

The polyhedral symmetry groups are all point groups and include: The symmetry group of an object (e. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub Archimedean polyhedra have this property. The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... A pyritohedron is an irregular dodecahedron. ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... This article deals with the four infinite series of point groups in three dimensions (n≥1) with n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and no other rotational symmetry (n=1 covers the cases of no rotational symmetry... This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... Figures with the axes of symmetry drawn in. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...


Other polyhedra with regular faces

Equal regular faces

A few families of polyhedra, where every face is the same kind of polygon:

  • 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 figures.) A set of points is said to be coplanar if and only if they lie on the same geometric plane. ... A simple (prismatic) semiregular skew polyhedron with vertex configuration 4. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...


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: The triaugmented triangular prism, a convex deltahedron A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. ...

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... In geometry, the triangular dipyramid is a polyhedron made entirely out of 6 faces, which are all equilateral triangles, 9 edges, and 5 vertexes. ... In geometry, the pentagonal dipyramid is one of the Johnson solids (J13). ... In geometry, the Snub disphenoid is one of the Johnson solids (J84). ... In geometry, the triaugmented triangular prism is one of the Johnson solids (J51). ... In geometry, the gyroelongated square dipyramid is one of the Johnson solids (J19). ...

Johnson solids

Main article: Johnson solid

Norman Johnson sought which non-uniform polyhedra had regular faces. In 1966, he published a list of 92 convex solids, now known as the Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ... Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ... Year 1966 (MCMLXVI) was a common year starting on Saturday (link will display full calendar) of the 1966 Gregorian calendar. ... The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ... Victor (Viktor) Abramovich Zalgaller (Russian: ) (born on December 25, 1920 in Parfino, Novgorod oblast) is a mathematician in the fields of geometry and optimization. ...


Other important families of polyhedra

Pyramids

Main article: Pyramid (geometry)

Pyramids include some of the most time-honoured and famous of all polyhedra. This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ...


Stellations and facettings

Main article: Stellation

Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron. Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... Image File history File links Download high resolution version (900x900, 24 KB) Summary First stellation of octahedron External links Polyhedra Stellations Applet Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (899x901, 33 KB) Summary First stellation of dodecahedron External links Polyhedra Stellations Applet Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (899x900, 34 KB) Summary Second stellation of dodecahedron External links Polyhedra Stellations Applet Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (900x900, 33 KB) Summary Third stellation of dodecahedron External links Polyhedra Stellations Applet Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (940x900, 44 KB) Summary Sixteenth stellation of icosahedron Generated by Polyhedra Stellations Applet Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (920x900, 34 KB) Summary First stellation of icosahedron External links Polyhedra Stellations Applet Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high resolution version (909x910, 53 KB) Summary Image of Wenningers 17th (and last) stellation of the icosahedron, Model 42 Generated by: Polyhedra Stellations Applet Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or...


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. In geometry, facetting (also spelled faceting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. ...


Zonohedra

Main article: Zonohedron

A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°. A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic... A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic... Look up polygon in Wiktionary, the free dictionary. ... Sphere symmetry group o. ... A sphere rotating around its axis. ...


Compounds

Main article: Polyhedral compound

Polyhedral compounds are formed as compounds of two or more polyhedra. A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...


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. This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ...


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 polygons (also known as rectilinear polygons). 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 net (polyhedron). An rectilinear polygon is a polygon all of whose edges meet at right angles. ... An rectilinear polygon is a polygon all of whose edges meet at right angles. ... In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. ... Categories: Polyhedra | Stub ...


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: An apeirohedron is a polyhedron having infinitely many faces. ...

See also: Apeirogon - infinite regular polygon: {∞} A tessellated plane seen in street pavement. ... A simple (prismatic) semiregular skew polyhedron with vertex configuration 4. ... An apeirogon is a degenerate polygon with an infinite number of sides. ...


Complex polyhedra

A complex polyhedron is one which is constructed in unitary 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). A complex polytope is one which exists in a unitary space, where each real dimension is accompanied by an imaginary one. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. ...


Curved polyhedra

Some fields of study allow polyhedra to have curved faces and edges.


Spherical polyhedra

Main article: 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. The surface of a sphere may be divided by line segments into bounded regions, to form a spherical tiling or spherical polyhedron. ...


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 hosohedra, exist only as spherical polyhedra and have no flat-faced analogue.


Curved spacefilling polyhedra

Two important types are:

  • Bubbles in froths and foams.
  • 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 affine (or Euclidean) space of any dimensional n that has flat sides. It could 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-spaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...


All traditional polyhedra are general polyhedra, and in addition there are examples like:

  • 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 in a Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point cS is bounded (hence a traditional polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded.

This is the Voronoi diagram of a random set of points in the plane (all points lie within the image). ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

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. He defined a face as a cyclically ordeded set of vertices, and allowed faces to be skew as well as planar. “Da Vinci” redirects here. ... Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... In geometry, a skew polygon is a polygon whose vertices do no lie in a plane. ...


Tessellations or tilings

Tessellations or tilings of the plane are sometimes treated as polyhedra, because they have quite a lot in common. For example the regular ones can be given Schläfli symbols. A tessellated plane seen in street pavement. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...


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 polytopes and that are attached to each other in a regular way. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...


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. A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...


Abstract polyhedra

An abstract polyhedron is a partially ordered set (poset) of elements. 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 polytopes in any number of dimensions. In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. ...


Polyhedra as graphs

Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra. For example: Graph theory is a growth area in mathematical research, and has a large specialized vocabulary. ...

  • 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 (K4). 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 solids give rise to regular graphs: 7 of the Archimedean solids are of degree 3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In graph theory, a regular graph is a graph where each vertex has the same number of neighbors. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. ... In graph theory, the degree (or valency) of a vertex is the number of edges incident to the vertex. ...

History

Prehistory

Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyehdra, but also the relations of duality amongst some of them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them. This article is about the country. ... Ashmolean Museum main entrance. ... The University of Oxford, located in the city of Oxford in England, is the oldest university in the English-speaking world. ...


Other polyhedra have of course made their mark in architecture - cubes and cuboids being obvious examples, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age.


The 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 1800s of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy[citation needed]. Extent of Etruscan civilization and the twelve Etruscan League cities. ... Padua, Italy, (Italian: IPA: , Latin: Patavium, Venetian: ) is a city in the Veneto, northern Italy, the economic and communications hub of the region. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... The lid of a pyrophyllite box. ...


Greeks

The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the convex regular polyhedra, while Archimedes later expanded his study to the convex uniform polyhedra. In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... For other uses, see Archimedes (disambiguation). ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...


Muslims and Chinese

After the end of the Classical era, Islamic scholars continued to make advances, for example in the tenth century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra. Meanwhile in China, dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids was used as the basis for calculating volumes of earth to be moved during engineering excavations.


Renaissance

Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.


Star polyhedra

For almost 2000 years, the concept of a polyhedron had remained as developed by the ancient Greek mathematicians.


Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons, typically pentagrams as faces. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Louis Poinsot realised that star vertex figures (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. Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Louis Poinsot (1777 - 1859) was a French mathematician and physicist. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ... A single face is colored yellow and outlined in red to help identify the faces. ...


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). Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... Harold Scott MacDonald Donald Coxeter, CC , Ph. ...


The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. Bridge 1974 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. In geometry, facetting (also spelled faceting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...


See also:

In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A dodecahedron, one of the five Platonic solids. ...

Polyhedra in nature

For natural occurrences of regular polyhedra, see Regular polyhedron: History. In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...


Irregular polyhedra appear in nature as crystals. For other uses, see Crystal (disambiguation). ...


References

  • Coxeter, H.S.M.; Regular complex Polytopes, CUP (1974).
  • Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
  • 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. (pdf)
  • Pearce, P.; Structure in nature is a strategy for design, MIT (1978)

H.S.M. Coxeter. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...

Books on polyhedra

Introductory books, also suitable for school use

  • Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
  • Cundy, H.M. & Rollett, A.P.; Mathematical models, 1st Edn. hbk OUP (1951), 2nd Edn. hbk OUP (1961), 3rd Ed. pbk.
  • Holden; Shapes, space and symmetry, (1971), Dover pbk (1991).
  • Tarquin publications: books of cut-out and make card models.
  • Wenninger, M.; Polyhedron models, CUP hbk (1971), pbk (1974).
  • Wenninger, M.; Spherical models, CUP.
  • Wenninger, M.; Dual models, CUP.

Undergraduate level

  • Coxeter, H.S.M. DuVal, Flather & Petrie; The fifty-nine icosahedra, 3rd Edn. Tarquin.
  • Coxeter, H.S.M. Twelve geometric essays. Republished as The beauty of geometry, Dover.
  • Thompson, Sir D'A. W. On growth and form, (1943). (not sure if this is the right category for this one, I haven't read it).

H.S.M. Coxeter. ... H.S.M. Coxeter. ...

Design and architecture bias

  • Critchlow, K.; Order in space.
  • Pearce, P.; Structure in nature is a strategy for design, MIT (1978)
  • Williams, R.; The geometrical foundation of natural structure, Dover (1979).

Advanced mathematical texts

H.S.M. Coxeter. ... H.S.M. Coxeter. ...

Historical books

  • Brückner, Vielecke und Vielflache (Polygons and polyhedra), (1900).
  • Fejes Toth, L.;

See also

Wikimedia Commons has media related to:
Look up polyhedron in Wiktionary, the free dictionary.

Image File history File links Commons-logo. ... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operators. ... In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ... The triaugmented triangular prism, a convex deltahedron A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. ... The trapezohedra are the Dual polyhedrons of the antiprisms. ... Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), 1935. ... Flexible polyhedra are polyhedral surfaces which allow continuous non-rigid deformations such that all faces remain rigid. ... The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ... A single face is colored yellow and outlined in red to help identify the faces. ... In geometry, a near-miss Johnson solid is a convex polyhedron, where every face is a regular or nearly regular polygon, and excluding the 5 Platonic solids, the 13 Archimedean solids, the infinite set of prisms, the infinite set of antiprisms, and the 92 Johnson solids. ... Categories: Polyhedra | Stub ... In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ... A sculpture of the small stellated dodecahedron in M. C. Eschers Gravitation, near the Mesa+ Institute of Universiteit Twente A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. ... Examples colored by the number of sides on each face. ... A simple spidron In geometry, a spidron is a continuous flat geometric figure composed entirely of triangles, where, for every pair of intersecting triangles, each has a leg of the other as one of its legs, and neither has any point inside the interior of the other. ... A tessellated plane seen in street pavement. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... Waterman polyhedra were invented, around 1990, by Steve Waterman. ... A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic...

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