A polyhedron is a geometric shape which in mathematics is defined by three related meanings. 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. Further generalizing the latter, there are topological polyhedra. ==Classical polyhedron== In older (and still current) mathematics, a polyhedron (from Greek language πολυεδρον, from ''poly-'', stem of πολυς, "many," + ''-edron'', form of εδρον, "base", "seat", or "face") is a three-dimensional shape that is made up of a finite number of polygonal ''Face (mathematics)'' which are parts of plane (mathematics), the faces meet in ''edges'' which are straight line segments, and the edges meet in points called ''vertex''. Cube (geometry)s, Prism (geometry) 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. A polyhedron is a three-dimensional analog of a polygon. The general term for polygons, polyhedra and even higher dimensional analogs is polytope. A polyhedron is * convex if the line segment joining any two points of the polyhedron is contained in the polyhedron's interior * vertex-uniform if all vertices are the same, in the sense that for any two vertices there exists a symmetry group of the polyhedron mapping the first onto the second * edge-uniform if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first onto the second * face-uniform if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first onto the second * regular if it is vertex-uniform, edge-uniform and face-uniform * uniform if it is vertex-uniform and every face is a regular polygon. These are semiregular in the same way that the Archimedean solids are, but the faces and vertex figures need not be convex. In addition to the prisms, antiprisms and crossed antiprisms, there are 75 uniform polyhedra, as conjectured by H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling. [http://mathworld.wolfram.com/UniformPolyhedron.html] The Euler characteristic relates the number of edges ''E'', vertices ''V'', and faces ''F'' of a simply connected polyhedron: ''F'' - ''E'' + ''V'' = 2. There are only five regular convex polyhedra. These have been known since ancient times, and are called the Platonic solids (see pictures there):
Interestingly, there are also more convex figures made entirely out of equilateral triangle known as deltahedron. The reason only three are mentioned above is that in the others, the number of faces that meet at each vertex varies.
The regular polyhedra come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself. These are called duals, and can be obtained by connecting the midpoints of each other's faces, among other interesting things. There are also five regular polyhedral compounds.
If you allow the polyhedra to be non-convex, there are four more, called the Kepler-Poinsot solids.
Polyhedra which are vertex- and edge-uniform, but not necessarily face-uniform, are called quasi-regular and include two more convex forms (the cuboctahedron and icosidodecahedron), as well as a few non-convex forms. The duals of these are the edge- and face-uniform polyhedra: the rhombic dodecahedron, rhombic triacontahedron, plus whatever the non-convex ones are. No other convex edge-uniform polyhedra exist.
Any polyhedron which is vertex-uniform can be deformed slightly to form a vertex-uniform polyhedron with regular polygons as faces. These are called semi-regular polyhedra. Convex forms include two infinite series, one of prism (geometry)s and one of antiprisms, as well as the thirteen Archimedean solids. The duals of these are of course the face-uniform polyhedra, with the two infinite convex series becoming the bipyramids and trapezohedron. These don't have regular faces, but do have regular vertices.
Another thing to consider is what kind of polyhedra, of any symmetry, can be made of regular polygons. There are an infinite number of non-convex forms, but surprisingly only a finite number of convex shapes other than the prisms and antiprisms. These include the Platonic solids, Archimedean solids, and 92 extra shapes called Johnson solids.
Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron. This is a version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1900.
==General polyhedron==
More recently mathematics has defined a polyhedron as a set in real number affine geometry (or Euclidean geometry) 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-space. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.
All classical 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 diagram 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 classical polyhedron) when ''c'' lies in the interior (topology) of the convex hull of ''S'', and otherwise (when ''c'' lies on the boundary of the convex hull of ''S'') ''A'' is unbounded.
==Topological polyhedron==
A topological polyhedron 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 that needs better description.
==See also==
*defect (geometry)
*M. C. Escher
*polyhedral compound
*prism
*antiprism
*Platonic solid
*Archimedean solid
*Johnson solid
*Kepler-Poinsot solid
*trapezohedron
*bipyramid
*deltohedron
*deltahedron
*zonohedron
*spidron
==External links==
*[http://www.queenhill.demon.co.uk/polyhedra/ Polyhedra Index Page]
*[http://www.software3d.com/Stella.html Stella: Polyhedron Navigator]
*[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.korthalsaltes.com/ Paper Models of Polyhedra] Many links
*[http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]
*[http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra in Java]
Polyhedra
The terminology problems mentioned in the first paragraph should be moved to the second paragraph. The first paragraph should contain the concise definition used in this encyclopedia. Right now, it is not explained at all what a polyhedron is. --AxelBoldt ---- I came on this with it saying only "convex lenses are cool", so I grabbed the earlier form from the diff and restored it as best I could. The table didn't seem to work very well, so I redid it using html table markup and it seems to me to work better this way. Feel free to tinker with it. -- User:Blain ---- I need help about the English term ''shape''. In this article it is written: A Polyhedron is a shape... What ''a shape'' really means? Since English is not my native and since I've started an article geometric shape I understand ''a shape'' to mean something planar, but here and also in many places ''a shape'' is used for bodies or solids. Slovene language -- my native -- distinguishes between ''a shape'' (''lik'') and ''a body'' or ''a solid'' (''telo'' or ''trdno telo''). What term should we use here? I know from CADs there are solids. Is a non planar surface also a shape of some sort? Basically as I've learned geometry from my early days I know for geometric shapes (e.g. triangle, square, rectangle, ...) and I couln't find this term here, so I've made an article on it. Someone might think it is too trivial, but anyway. How can we talk about mathematical realities, if we still do not know how many Fermat primes are there -- and such. So I guess we still have to look back on simple roots of pure mathematical objects. And as professor Dragan Marusic recently stole words from my lips that we just discover anew things which are there already from the eternity. I think ''a solid'' might be just fine, since it is used commonly in geometry (and CADs) and ''a body'' is used for example much more in astronomy, physiology and in related fields. Best regards. --User:XJamRastafire 01:44 Jan 24, 2003 (UTC) Shape can be either planar or three dimemsional. -- User:Karl Palmen 3 Oct 2003 (UTC) ---- I think we are in deep trouble here with the definitions. The Kepler solids are supposed to be regular polyhedra, but that only works if the terms "face" and "vertex" are properly understood. Not all intersections of edges apparently are vertices. How should one define polyhedra so that both the "normal" polyhedra and the Kepler solids fit the bill? User:AxelBoldt 18:23, 2 Oct 2003 (UTC) :Agreed. The trouble is that of having two viewpoints with regards to classical polyhedra which generalise in different directions: :*You can say that a polyhedron is a bounded solid body such that its boundary is made up of planar facets (e.g. the boundary is contained within a finite union of affine subspaces of codimension 1.) Then the "general polyhedron" definition is fine - something in the algebra of sets generated by half-spaces. (Actually there is still a slight problem, in that "has flat sides" stated in the article does not imply the finiteness condition.) :*You can say that a polyhedron is an arrangement of vertices, edges and faces in space with some combinatorial relationships, such as a a 2-to-1 surjection (ends of edges) -> (vertices of faces) and a 2-to-1 surjection (edges of faces) -> (edges of polyhedron). (Sorry, this suboptimal --- off the top of my head, but I was trying to remember how to set up simplicial geometry and then generalise it.) This case can be generalised to include polygrams (stars) as the sides and thus fit in the other regular and uniform polyhedra as generalisations of the platonic and archimedian polyhedra. :So yes, some modification of the definition to include two viewpoints would be useful. Something for my todo or some other brave soul. :Also I think the section on "Topological polyhedra" sucks somewhat. WTF is being defined here? I can't tell sufficiently well even to fix it up. Is it referring to a construction like a simplicial complex?? If so, I think "Topological polyhedron" is not standard nomenclature, but it is not my field. : -- User:AndrewKepert 08:52, 6 Aug 2004 (UTC) ---- We have a big problem in cartography. A flat piece of paper cannot be curved to cover a sphere exactly without some stretching or wrinkling. Is there a general term for the sorts of shapes that paper *can* cover without stretching or wrinkling ? In other words, I'm looking for terms to fill in these blanks: * A polyhedron (such as the pentagonal pyramid) is made of flat plates (facets) (in this case, triangles and a pentagon) stitched together. * A __________ (such as the quonset hut) is made of constant-curvature cylinders or planes (in this case, 2 half circles and 2 rectangles) stiched together. * A __________ (such as the cone) is make of ____ surfaces ( developable surfaces ?) (in this case, a circle with a sector cut out, and another circle) stitched together. * A __________ (such as the sphere) is made of ___ surfaces (such as Nonuniform rational B-splines) stitched together. --User:DavidCary 20:29, 25 Jan 2005 (UTC) ---- This article seems to imply that polyhedra are only used in some obscure, archaic branch of mathematics. It also focuses on "regular polyhedra" and "convex polyhedra", which are idealized shapes that have few practical applications. Most polyhedra are irregular and concave. I want it to say more about how polyhedra are used all the time in Computer-aided design (in particular, Solid modelling), video games, etc. I supposed all I want to say can be summarized as ''The visual appearance of any physical object can be duplicated by a sufficiently detailed polyhedron.'' How can I emphasize how important this is? --User:DavidCary 20:29, 25 Jan 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Polyhedron:
Polyhedron
Polyhedron
Polyhedron_(magazine)
Polyhedron_magazine
Polyhedron
A polyhedron is a geometric shape which in mathematics is defined by three related meanings. 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. Further generalizing the latter, there are topological polyhedra. ==Classical polyhedron== In older (and still current) mathematics, a polyhedron (from Greek language πολυεδρον, from ''poly-'', stem of πολυς, "many," + ''-edron'', form of εδρον, "base", "seat", or "face") is a three-dimensional shape that is made up of a finite number of polygonal ''Face (mathematics)'' which are parts of plane (mathematics), the faces meet in ''edges'' which are straight line segments, and the edges meet in points called ''vertex''. Cube (geometry)s, Prism (geometry) 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. A polyhedron is a three-dimensional analog of a polygon. The general term for polygons, polyhedra and even higher dimensional analogs is polytope. A polyhedron is * convex if the line segment joining any two points of the polyhedron is contained in the polyhedron's interior * vertex-uniform if all vertices are the same, in the sense that for any two vertices there exists a symmetry group of the polyhedron mapping the first onto the second * edge-uniform if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first onto the second * face-uniform if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first onto the second * regular if it is vertex-uniform, edge-uniform and face-uniform * uniform if it is vertex-uniform and every face is a regular polygon. These are semiregular in the same way that the Archimedean solids are, but the faces and vertex figures need not be convex. In addition to the prisms, antiprisms and crossed antiprisms, there are 75 uniform polyhedra, as conjectured by H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling. [http://mathworld.wolfram.com/UniformPolyhedron.html] The Euler characteristic relates the number of edges ''E'', vertices ''V'', and faces ''F'' of a simply connected polyhedron: ''F'' - ''E'' + ''V'' = 2. There are only five regular convex polyhedra. These have been known since ancient times, and are called the Platonic solids (see pictures there):
| Name | Faces | Edges | Vertices | Symmetry group | ||
|---|---|---|---|---|---|---|
| Tetrahedron | 4 | 6 | 4 | 3 | 3 | Td |
| Hexahedron or Cube (geometry) | 6 | 12 | 8 | 4 | 3 | Oh |
| Octahedron | 8 | 12 | 6 | 3 | 4 | Oh |
| Dodecahedron | 12 | 30 | 20 | 5 | 3 | Ih |
| Icosahedron | 20 | 30 | 12 | 3 | 5 | Ih |
Polyhedron
The terminology problems mentioned in the first paragraph should be moved to the second paragraph. The first paragraph should contain the concise definition used in this encyclopedia. Right now, it is not explained at all what a polyhedron is. --AxelBoldt ---- I came on this with it saying only "convex lenses are cool", so I grabbed the earlier form from the diff and restored it as best I could. The table didn't seem to work very well, so I redid it using html table markup and it seems to me to work better this way. Feel free to tinker with it. -- User:Blain ---- I need help about the English term ''shape''. In this article it is written: A Polyhedron is a shape... What ''a shape'' really means? Since English is not my native and since I've started an article geometric shape I understand ''a shape'' to mean something planar, but here and also in many places ''a shape'' is used for bodies or solids. Slovene language -- my native -- distinguishes between ''a shape'' (''lik'') and ''a body'' or ''a solid'' (''telo'' or ''trdno telo''). What term should we use here? I know from CADs there are solids. Is a non planar surface also a shape of some sort? Basically as I've learned geometry from my early days I know for geometric shapes (e.g. triangle, square, rectangle, ...) and I couln't find this term here, so I've made an article on it. Someone might think it is too trivial, but anyway. How can we talk about mathematical realities, if we still do not know how many Fermat primes are there -- and such. So I guess we still have to look back on simple roots of pure mathematical objects. And as professor Dragan Marusic recently stole words from my lips that we just discover anew things which are there already from the eternity. I think ''a solid'' might be just fine, since it is used commonly in geometry (and CADs) and ''a body'' is used for example much more in astronomy, physiology and in related fields. Best regards. --User:XJamRastafire 01:44 Jan 24, 2003 (UTC) Shape can be either planar or three dimemsional. -- User:Karl Palmen 3 Oct 2003 (UTC) ---- I think we are in deep trouble here with the definitions. The Kepler solids are supposed to be regular polyhedra, but that only works if the terms "face" and "vertex" are properly understood. Not all intersections of edges apparently are vertices. How should one define polyhedra so that both the "normal" polyhedra and the Kepler solids fit the bill? User:AxelBoldt 18:23, 2 Oct 2003 (UTC) :Agreed. The trouble is that of having two viewpoints with regards to classical polyhedra which generalise in different directions: :*You can say that a polyhedron is a bounded solid body such that its boundary is made up of planar facets (e.g. the boundary is contained within a finite union of affine subspaces of codimension 1.) Then the "general polyhedron" definition is fine - something in the algebra of sets generated by half-spaces. (Actually there is still a slight problem, in that "has flat sides" stated in the article does not imply the finiteness condition.) :*You can say that a polyhedron is an arrangement of vertices, edges and faces in space with some combinatorial relationships, such as a a 2-to-1 surjection (ends of edges) -> (vertices of faces) and a 2-to-1 surjection (edges of faces) -> (edges of polyhedron). (Sorry, this suboptimal --- off the top of my head, but I was trying to remember how to set up simplicial geometry and then generalise it.) This case can be generalised to include polygrams (stars) as the sides and thus fit in the other regular and uniform polyhedra as generalisations of the platonic and archimedian polyhedra. :So yes, some modification of the definition to include two viewpoints would be useful. Something for my todo or some other brave soul. :Also I think the section on "Topological polyhedra" sucks somewhat. WTF is being defined here? I can't tell sufficiently well even to fix it up. Is it referring to a construction like a simplicial complex?? If so, I think "Topological polyhedron" is not standard nomenclature, but it is not my field. : -- User:AndrewKepert 08:52, 6 Aug 2004 (UTC) ---- We have a big problem in cartography. A flat piece of paper cannot be curved to cover a sphere exactly without some stretching or wrinkling. Is there a general term for the sorts of shapes that paper *can* cover without stretching or wrinkling ? In other words, I'm looking for terms to fill in these blanks: * A polyhedron (such as the pentagonal pyramid) is made of flat plates (facets) (in this case, triangles and a pentagon) stitched together. * A __________ (such as the quonset hut) is made of constant-curvature cylinders or planes (in this case, 2 half circles and 2 rectangles) stiched together. * A __________ (such as the cone) is make of ____ surfaces ( developable surfaces ?) (in this case, a circle with a sector cut out, and another circle) stitched together. * A __________ (such as the sphere) is made of ___ surfaces (such as Nonuniform rational B-splines) stitched together. --User:DavidCary 20:29, 25 Jan 2005 (UTC) ---- This article seems to imply that polyhedra are only used in some obscure, archaic branch of mathematics. It also focuses on "regular polyhedra" and "convex polyhedra", which are idealized shapes that have few practical applications. Most polyhedra are irregular and concave. I want it to say more about how polyhedra are used all the time in Computer-aided design (in particular, Solid modelling), video games, etc. I supposed all I want to say can be summarized as ''The visual appearance of any physical object can be duplicated by a sufficiently detailed polyhedron.'' How can I emphasize how important this is? --User:DavidCary 20:29, 25 Jan 2005 (UTC)
See other meanings of words starting from letter:
P
PA | PB | PC | PD | PE | PF | PG | PH | PI | PJ | PK | PL | PM | PN | PO | PR | PS | PT | PU | PW | PX | PY | PZ |Words begining with Polyhedron:
Polyhedron
Polyhedron
Polyhedron_(magazine)
Polyhedron_magazine
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