The polyhedron occurs as geometrical shape which within mathematics is defined by three related meanings. In the traditional meaning these are the Three-three-dimensional polytope, and around the recently meaning that lives alongside the older the single these are a delimited or even limitless generalization of a polytope of any dimension. Farther generalizing a latter, there are topologic polyhedra.

Classical polyhedron

A dodecahedron Within authoritative maths, the polyhedron (from either Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or even "face") occurs as 3-cubic shape that is manufactured higher of the finite total of polygonal faces which are parts of planes, the faces meet around edges which are straight-line segments, and a edges meet within points known as vertices. Cubes, prisms and pyramids are examples of polyhedra. the polyhedron surrounds a delimited volume inside 3-cubic space; occasionally this interior volume is considered to exist as a share of the polyhedron. The polyhedron occurs as 3-cubic parallel of the polygon. A general term for polygonal shape, polyhedra & potentially higher miscreate parallel is polytope.

List of polyhedra by total of faces come tetrahedron, pentahedron, hexahedron, etc. Such terms come particularly utilized by using "regular" ahead or even implied (inside a 5 legal actions where this is applicable) because for both there are different types which use non good deal in green except with the equivalent total of faces. For the tetrahedron this applies to the good deal lesser extent, these are universally the triangular pyramid.

Characteristics

The polyhedron is convex if the line section joining any ii points of the polyhedron is contained in the polyhedron or even its interior vertex-uniform if tons vertices come the equivalent, in the feel that for any deuce vertices there is a symmetry of the polyhedron mapping the 1st isometrically onto the second edge-uniform if completely edges come a equivalent, in a feel that for any 2 edges there is a symmetry of the polyhedron mapping the foremost isometrically onto the second face-uniform if tons faces come a equivalent, in a feel that for any 2 faces there is a symmetry of the polyhedron mapping the foremost isometrically onto the second regular whenever these are vertex-uniform, edge-uniform & face-uniform; this implies that each face occurs as regular polygon quasi-regular whenever these are vertex-uniform & edge-uniform but not face-uniform, & each face occurs as regular polygon semi-regular in case these are vertex-uniform however neither edge-uniform nor face-uniform, & each face occurs as regular polygon uniform if these are vertex-uniform & each face occurs as regular polygon, we.e. these are regular, quasi-regular, or even semi-regular.

A Euler characteristic relates the total of edges E, vertices V, & faces F of the just attached polyhedron: F - E + V = Ii.

Uniform polyhedra

When conjectured by H. S. M. Coxeter et al. around 1954 & late confirmed by J. Skilling, there are exactly 75 uniform polyhedra, + an infinite total of prisms & antiprisms. Uniform polyhedra may be organized when follows: Niner regular polyhedra: Fivesome bulging solids: Platonic solids Four non-convex solids: Kepler-Poinsot solids Fifteen quasi-regular polyhedra: Ii bulging solids Thirteen non-convex solids semi-regular polyhedra: semi-regular bulging polyhedra: an infinite total of prism and antiprisms Xi more bulging solids Forty semi-regular non-convex polyhedra: Seventeen stellated Archimedean solids Xxiii more semi-regular non-convex solids

According to this organisation, except a prisms & antiprisms, there are exactly Thirteen bulging semi-regular & quasi-regular polyhedra; it is known as a Archimedean solids. When shown on this text, there are exactly 53 non-convex semi-regular & quasi-regular polyhedra.

Regular polyhedra

Regular polyhedra come vertex-uniform, edge-uniform & face-uniform -- this implies that each face occurs as regular polygon.

Platonic solids

There are exactly 5 regular convex polyhedra. These use at days been known since ancient times, & come known as a Platonic solids: a tetrahedron a hexahedron or cube a octahedron a dodecahedron a icosahedron

Kepler-Poinsot solids

There are exactly iv regular non-convex polyhedra: a Kepler-Poinsot solids: a small stellated dodecahedron a great stellated dodecahedron a great icosahedron a great dodecahedron

Quasi-regular polyhedra

Quasi-regular means vertex- & edge-uniform but not face-uniform; this implies that there are deuce "kinds" of faces,& that at both edge one of both meet; too that at each vertex quaternity faces meet: alternatingly deuce of each variety.

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There are exactly 2 quasi-regular bulging polyhedra; each come likewise Archimedean solids: a cuboctahedron (with triangles & squares) a icosidodecahedron (with triangles & pentagons)

There are no more bulging edge-uniform polyhedra survive than a 5 regular & ii quasi-regular bulging polyhedra, thus edge uniformity by having convexity implies vertex-uniformity.

There are exactly Thirteen quasi-regular non-convex polyhedra (Hart): a dodecadodecahedron a great icosidodecahedron Tercet come triambic: a small triambic icosidodecahedron a triambic dodecadodecahedron a great triambic icosidodecahedron Nina from carolina come hemihedra: a tetrahemihexahedron a octahemioctahedron a cubohemioctahedron a small icosihemidodecahedron a small dodecahemidodecahedron a great dodecahemiicosahedron a small dodecahemiicosahedron a great dodecahemidodecahedron a great icosihemidodecahedron

Semi-regular polyhedra

Semi-regular means vertex-uniform but not edge-uniform.

There are infinitely numerous semi-regular bulging polyhedra due to 2 infinite series: a prisms (with 2 north-gons & n squares) and

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a antiprisms (with Two north-gons & Iin triangles)

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A others come Eleven of the Archimedean solids:

a truncated cube a truncated dodecahedron a truncated tetrahedron a truncated octahedron a truncated icosahedron a truncated cuboctahedron a truncated icosidodecahedron a snub cube or snub cuboctahedron a snub dodecahedron or a rhombicuboctahedron a rhombicosidodecahedron

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A semi-regular non-convex polyhedra include a Xvii stellated Archimedean solids: Threesome develop three-dimensional symmetry: a great cubicuboctahedron a cuboctatruncated cuboctahedron or cubitruncated cuboctahedron a quasitruncated hexahedron or stellated truncated cube Eleven stand icosahedral symmetry: a quasitruncated small stellated dodecahedron or small stellated truncated dodecahedron a quasitruncated great stellated dodecahedron or great stellated truncated dodecahedron a great truncated dodecahedron a great truncated icosahedron a rhombidodecadodecahedron a icosidodecatruncated icosidodecahedron ** a small ditrigonal icosidodecahedron ** a great ditrigonal icosidodecahedron ** a great quasitruncated icosidodecahedron or great truncated icosidodecahedron a great dodecicosidodecahedron a great ditrigonal dodecicosidodecahedron ** Trio use a symmetry of the forget about dodecahedron: a snub dodecadodecahedron a great inverted snub icosidodecahedron or great vertisnub icosidodecahedron a great inverted retrosnub icosidodecahedron or great retrosnub icosidodecahedron

Additionally to the Xvii stellated Archimedean solids, there are Twenty-three additional semi-regular non-convex polyhedra: ? stand octahedral symmetry: a small cubicuboctahedron a small rhombicube a great rhombicube a great truncated cuboctahedron ? develop icosahedral symmetry: a rhombicosahedron a small rhombidodecahedron a great rhombidodecahedron a small dodecicosahedron a great dodecicosahedron a small dodecicosidodecahedron a icositruncated dodecadodecahedron a great truncated dodecadodecahedron a vertisnub dodecadodecahedron a icosidodecadodecahedron a snub icosidodecadodecahedron a small dodekic icosidodecahedron ?? a great dodekic icosidodecahedron ?? a small icosic icosidodecahedron ?? a great icosic icosidodecahedron ?? a great snub icosidodecahedron a snub disicosidodecahedron a retrosnub disicosidodecahedron a great snub icosidisdodecahedron a great snub disicosidisdodecahedron

TODO: Prevent this listing for duplicates/alternate names

Given deuce polyhedra of equal volume, of these might ask whether it is so universally conceivable to cut a foremost into polyhedral pieces which may be reassembled to yield a 2nd polyhedron. This occurs as version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1900.

Polyhedron duals

For each polyhedron there is a dual polyhedron which can be found by connecting a center of the faces. Face-uniformity of the polyhedron corresponds to vertex-uniformity of the dual & on the other h&, and edge-uniformity of the polyhedron corresponds to edge-uniformity of the dual. So a regular polyhedra are around natural pairs: a dodecahedron by having a icosahedron, a cube by using a octahedron, & a tetrahedron by having itself.

Inside virtually all duals of uniform polyhedra, faces come irregular polygonal shape. A exceptions come: a cube, which is a favorite trapezohedron a triangular dipyramid, the octahedron and the pentagonal dipyramid, which are favorite bipyramids.

Quasi-regular duals

A duals of the quasi-regular polyhedra come edge- & face-uniform. Which are actually, correspondingly: a rhombic dodecahedron a rhombic triacontahedron & Xiii others.

Semi-regular duals

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A duals of the semi-regular polyhedra come face-uniform. Which are actually, correspondingly: a bipyramids a trapezohedra Eleven of the Catalan solids

Other polyhedra with regular faces

Norman Johnson sought which non-uniform polyhedra experienced regular faces. Inside 1966, he published a names of 92 bulging solids, a Johnson solids, & gave the two their list & cost. He did non prove there were exclusively 92, however he did conjecture that there were there are no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

The deltahedron (plural deltahedra) occurs as polyhedron whose faces come a lot equilateral triangles. There come infinitely numerous deltahedra, but only eight one are bulging:

Triad regular convex polyhedra (Triad of the Platonic solids) a tetrahedron a octahedron a icosahedron Five non-inhomogeneous bulging polyhedra (Cinque of the Johnson solids) a triangular dipyramid a pentagonal dipyramid a snub disphenoid a triaugmented triangular prism a gyroelongated square dipyramid

By having regard to polyhedra whose faces come a lot squares: whenever coplanar faces are not allowed, potentially whenever it is disconnected, there exists merely a cube. Otherwise there exists likewise a symptom of pasting sixer cubes to the sides of a single, 100% 7 of the equivalent size; it hwhen Thirty square faces (counting staccato faces in the equivalent plane as separate). This may be extended around of these, 2, or even 3 directions: i personally may assume a union of indiscriminately numerous copies of these structures, found by translations of (verbalised inside cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence using from each one adjacent pair getting one most common cube. the symptom may be any attached placed of cubes by owning positions (a,b,c), sustaining whole number a,b,hundred of which at the most of these is potentially.

No favorite title for even polyhedra whose faces come a lot equilateral pentagons or pentacle. There are infinitely numerous of these, but only one is bulging: a dodecahedron. A rest come assembled by (pasting) combinations of a regular polyhedra described earliest: a dodecahedron, a little stellated dodecahedron, a nifty stellated dodecahedron & the groovy icosahedron.

There is there come no polyhedron whose faces are a lot regular polygons sustaining sixer or even other sides.

There survive an infinite total of non-inhomogeneous non-convex polyhedra.

General polyhedron

Other recently mathematics has defined a polyhedron as a placed inside real affine (or Euclidean) space of any dimensional n that has flat sides. It can be defined when the union of a finite total of convex polyhedrthe, where a convex polyhedron is any placed that is the intersection of the finite total of half-spaces. It can be bounded or even limitless. In that meaning, the polytope is a bounded polyhedron.

Totally definitive polyhedra come general polyhedra, & additionally there are examples like A quadrant in the plane. E.g., a region of a cartesian plane consisting of everthing points above the horizontal axis & to the perfect of the vertical axis: . Its sides come them caring axe. An octant within Euclidian Three-space, *A prism of infinite extent. E.g. the doubly-infinite square prism within Three-space, consisting of the square in the xy-plane swept along a z-axis: *Each cell within the Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of the placed S, a cell The corresponding to the point c∈S is bounded (hence the authoritative polyhedron) once c lies in the interior of the convex hull of S, and otherwise (after c lies on the boundary of the convex hull of S) The is limitless.

Topological polyhedron
The topologic polyhedron occurs as topological space given along using the specific decomposition into shapes that come topologically same to convex polytopes and that come bound to every more around the regular way that needs better description.

Relation with graphs
Any polyhedron produce to the graph, called skeleton, using corresponding vertices & edges. So graph terminology and properties can be applied to polyhedra:

A Archimedean solids bring about to regular graphs: 7 Archimedean solids come degree 3, Tetrad solids come degree 4, & a unexpended Deuce come chiral pairs of degree Little phoebe.

the octahedron produce to a strongly regular graph, because adjacent vertices have universally deuce most common neighbors, & non-adjacent vertices universally quaternary.

Lone the tetrahedron bring about to a complete graph (K_Quatern).

Due to Steinitz theorem convex polyhedra are inside 1-to-of these correspondence using Three-attached two-dimensional graphical record.