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Polyhedron

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For the game magazine, see Polyhedron (magazine).

A polyhedron (plural: polyhedra) 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

A dodecahedron

In classical 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 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. A polyhedron is a three-dimensional analog of a polygon. The general term for polygons, polyhedra and even higher dimensional analogs is polytope.

Names of polyhedra by number of faces are tetrahedron, pentahedron, hexahedron, octahedron, decahedron, etc. Such terms are in particular used with "regular" in front or implied (in the five cases in which this is applicable) because for each there are different types which have not much in common except having the same number of faces. For a tetrahedron this applies to a much lesser extent, it is always a triangular pyramid.

Classical polyhedra include the five regular convex polyhedra: tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces) and icosahedron (20 faces). Other classical polyhedra are the four regular non-convex polyhedra (the Kepler-Poinsot solids), the thirteen convex Archimedean solids and the 53 remaining uniform polyhedra. Dual polyhedra of the classical polyhedra can also be considered classical.

Characteristics

A polyhedron is:

  • Convex if its boundary (including its faces and edges) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the polyhedron or its 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 isometrically 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 isometrically 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 isometrically onto the second.
  • Regular if it is vertex-uniform, edge-uniform and face-uniform. (Vertex-uniformity and edge-uniformity combined imply that the faces are regular.)
  • Quasi-regular if it is edge-uniform but either not face-uniform or not vertex-uniform.
  • Semi-regular if it is vertex-uniform but not face-uniform, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Overlaps with quasi-regular category)
  • Uniform if it is vertex-uniform and every face is a regular polygon, i.e. it is regular or semi-regular.

The Euler characteristic χ relates the number of edges E, vertices V, and faces F of a polyhedron: χ = V - E + F. For a simply connected polyhedron χ = 2.

Symmetry

Many polyhedra are highly symmetric, their symmetry groups are all point groups and include:

Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub polyhedra have this property.

Uniform polyhedra

Main article: Uniform polyhedron

Uniform polyhedra are vertex uniform and every face is a regular polygon. They may be regular, quasi-regular, or semi-regular but not necessarily convex. The Uniform polyhedra include all the polyhedra mentioned above.

As conjectured by H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling, there are exactly 75 uniform polyhedra, plus an infinite number of prisms and antiprisms. Some of the antiprisms are non-convex.

The full list of uniform polyhedra contains details of all uniform polyhedra and List of uniform polyhedra by vertex figure exhibits some relations between the polyhedra.

Of the 39 non-convex, non-prismatic uniform polyhedra, 17 are stellations of Archimedean solids.

Two examples of non-convex uniform polyhedra are the

Regular polyhedra

Platonic solids

Main article: platonic solid

There are exactly five regular convex polyhedra. These have been known since ancient times, and are called the Platonic solids:

Kepler-Poinsot solids

Main article: Kepler-Poinsot solid

There are exactly four regular non-convex polyhedra, known as the Kepler-Poinsot solids:

Semi-regular convex polyhedron

The term Semi-regular is variously defined. One definition is "convex vertex-uniform polyhedra with two or more types of regular polygon faces."

This includes the set of uniform prisms and antiprisms and the Archimedean solids.

Prisms and antiprisms

There are infinitely many uniform polyhedra in two infinite series:

  • Prisms (with 2 n-gons and n squares) and
  • Antiprisms (with 2 n-gons and 2n triangles)

4.4.3
4.4.4
4.4.5
4.4.6
4.4.8
4.4.10
4.4.12

3.3.3.3
3.3.3.4
3.3.3.5
3.3.3.6
3.3.3.8
3.3.3.10
3.3.3.12
3.3.3.17

Archimedean solid

There are 13 Archimedean solids.

Two are quasi-regular convex polyhedra, which have the additional property of being edge-uniform:


cuboctahedron
icosidodecahedron

The 11 others are also convex polyhedra:

No other convex edge-uniform polyhedra composed of regular polygons exist than the five regular and two quasi-regular convex polyhedra, so edge uniformity and face regularity with convexity implies vertex-uniformity. (There are two other edge-uniform convex polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, but they are not face-regular and not vertex-uniform. These are the duals of the quasi-regular convex polyhedra, and are both members of the Catalan solids.)

Polyhedron duals

For every polyhedron there is a dual polyhedron which can be obtained, for regular polyhedra, by connecting the midpoints of the faces. For an arbitrary polyhedron, the more complicated process of spherical reciprocation is required (see dual polyhedron). Face-uniformity of a polyhedron corresponds to vertex-uniformity of the dual and conversely, and edge-uniformity of a polyhedron corresponds to edge-uniformity of the dual.

Thus the regular polyhedra come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself.

In most duals of uniform polyhedra, faces are irregular polygons. The exceptions are:

Quasi-regular duals

The duals of the quasi-regular polyhedra are edge- and face-uniform. These are, correspondingly:

There are 13 other nonconvex duals. QUESTION: Is this the intended meaning? If not, to what did "ones" refer? Finell (Talk)

Semi-regular duals

The set of Catalan solids, bipyramids, and trapezohedra.

Bipyramids

Trapezohedra

Catalan solids

Other Families of polyhedra

Pyramids

Stellations

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.

It the exact reciprocal to the process of facetting which is the removal of parts of a polyhedron without creating any new vertices.

Compounds

Polyhedral compounds are formed as compounds of two or more polyhedra. These include:

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.

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.

Deltahedron

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:

Other polyhedra with regular 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.)

Zonohedron

A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°.

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

Relation with graphs

Any polyhedron gives rise to a graph, called skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra:

  • The Archimedean solids give rise to regular graphs: 7 Archimedean solids are degree 3, 4 solids are degree 4, and the remaining 2 are chiral pairs of degree 5.
  • The octahedron gives rise to a strongly regular graph, because adjacent vertices have always two common neighbors, and non-adjacent vertices always four.
  • Only the tetrahedron gives rise to a complete graph (K4).
  • Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.

History and Polytopes in nature

Much of the history of polyhedra is covered in Regular polytope: History of discovery. For natural occurrences of polyhedra, see Regular polytope: Polytopes in nature: Polyhedra.

See also

Wikimedia Commons has media related to Polyhedron.
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