Polyhedron

A polyhedron (plural: polyhedra or polyhedrons) is a 3-dimensional geometric shape having flat faces that meet along straight edges. The word polyhedron comes from the Classical Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face".

Polyhedra have fascinated mankind since prehistory, were first studied formally by the ancient Greeks, and continute to fascinate students, mathematicians and artists today.

The above definition of a polyhedron might seem clear enough for most of us, but not for a mathematician. In an oft-quoted but seldom respected remark, Grünbaum (1994) observed that:

"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' ..."

And to this day mathematicians cannot agree as to exactly what makes something a polyhedron.

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. Most of us can forget about it.

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.

A polyhedron is a 3-dimensional example of the more general polytope in any number of 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.

Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.

Some special polyhedra have grown their own names over the years, such as Miller's monster.

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:

χ = V - E + F.

For a simply connected polyhedron χ = 2.

Duality

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.

Traditional polyhedra

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.

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.

Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetric.

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.

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:

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:

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. (Vertex-transitivity and edge-transitivity combined imply that the faces are regular.)
Quasi-regular if it is edge-transitive but either not face-transitive or not vertex-transitive.
Semi-regular if it is vertex-transitive but not face-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).
Uniform if it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular.

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.

Uniform polyhedra and their duals

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.

The Uniform duals are face-transitive and every vertex figure is a regular polygon.

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, including Cromwell, 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.

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

Noble polyhedra

A noble polyhedron is both isohedral (equal-faced) and isogonal (equal-cornered). Besides the regular polyhedra, there are many other examples.

The dual of a noble polyhedron is also noble.

Symmetry groups

The polyhedral symmetry groups are all point groups and include:

T - chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12.
T_d - full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
T_h - pyritohedral symmetry; order 24. The symmetry of a pyritohedron.
O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
O_h - full octahedral symmetry; the symmetry group of the cube and octahedron; order 48.
I - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.
I_h - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
C_nv - n-fold pyramidal symmetry
D_nh - n-fold prismatic symmetry
D_nv - n-fold antiprismatic symmetry

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.

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.)

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:

3 regular convex polyhedra (3 of the Platonic solids)
5 non-uniform convex polyhedra (5 of the Johnson solids)

Johnson solids

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.

Other Families of polyhedra

Pyramids

Pyramids are self dual.

Stellations and facettings

Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.

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.

Compounds

Polyhedral compounds are formed as compounds of two or more polyhedra.

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.

Zonohedra

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

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:

Unbounded tilings and tessellations of the plane.
Honeycomb-like structures.

More needs to be said about these - probably on a new page.

See: Apeirogon - infinite regular polygon: {∞}

Complex polyhedra

A complex polyhedron is one which is constructed in complex 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).

Curved polyhedra

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

Spherical polyhedra

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.

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 the uniform polyhedra. This topic could do with its own page.

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.

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 c∈S 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.

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.

Tesellations 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.

Non-geometric polyhedra

Various mathematical constructs have been found to have properties also present in traditional polyhedra.

Topological polyhedra

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.

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.

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:

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.

More about polyhedra

History

Much of the history of polyhedra is covered in Regular polytope: History of discovery.

Polyhedra in nature

For natural occurrences of polyhedra, see Regular polytope: Polytopes in nature: Polyhedra.

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.
Pearce, P.; Structure in nature is a strategy for design, MIT (1978)

External links

Weisstein, Eric W. "Polyhedron". MathWorld.
Making Polyhedra
Polyhedra Pages
Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
The Uniform Polyhedra
Virtual Reality Polyhedra - The Encyclopedia of Polyhedra
- Polyhedra and Pyramids
Paper Models of Polyhedra Many links
Paper Models of Uniform (and other) Polyhedra
Interactive 3D polyhedra in Java
World of Polyhedra - Comprehensive polyhedra in flash applet, showing vertices and edges (but not shaded faces)
Polyhedra software, die-cast models, & posters
Electronic Geometry Models - Contains a peer reviewed selection of polyhedra with unusual properties.
Symmetry, Crystals and Polyhedra
uniform solution for uniform polyhedra by Dr. Zvi Har'El
Java applet with the use of kaleido
Origami Polyhedra - Models made with Modular Origami