# 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.

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## Classical polyhedron

In older (and still current) mathematics, a polyhedron (from Greek 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 faces which are parts of planes, the faces meet in 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. 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 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 FacesEdgesVertices Edges/FaceEdges/VertexSymmetry group
Tetrahedron46433Td
Hexahedron or Cube61284 3Oh
Octahedron812634Oh
Dodecahedron12302053Ih
Icosahedron20301235Ih

Interestingly, there are also more convex figures made entirely out of equilateral triangles known as deltahedra. 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 prisms 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 trapezohedra. 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 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.

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 tesselation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point cS is bounded (hence a classical 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.

## 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.

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