Polyhedron
2008/9 Schools Wikipedia Selection. Related subjects: Mathematics
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A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric object with flat faces and 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").
This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory. Grünbaum (1994, p.43) 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' ...
Modern mathematicians do not even 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.
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 defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.
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.
Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).
Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron.
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. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.
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.
Vertex figure
For every vertex one can define a vertex figure consisting of the vertices joined to it. The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron.
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 symmetrical.
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 (this implies that every face is the same regular polygon; it also implies that every vertex is regular).
- Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.
- Semi-regular if it is vertex-transitive but not edge-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). A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular.
- Uniform if it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive).
- Noble if it is face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra.
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 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 uniform dual | Star uniform | Star uniform dual | |
|---|---|---|---|---|
| Regular | Platonic solids | Kepler-Poinsot polyhedra | ||
| Quasiregular | Archimedean solids | Catalan solids | (no special name) | (no special name) |
| Semiregular | (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.
- Td - full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
- Th - pyritohedral symmetry; order 24. The symmetry of a pyritohedron.
- O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
- Oh - 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.
- Ih - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
- Cnv - n-fold pyramidal symmetry
- Dnh - n-fold prismatic symmetry
- Dnv - 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 Archimedean 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)
- Tetrahedron
- Octahedron
- Icosahedron
- 5 non-uniform convex polyhedra (5 of the Johnson solids)
- Triangular dipyramid
- Pentagonal dipyramid
- Snub disphenoid
- Triaugmented triangular prism
- Gyroelongated square dipyramid
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 important families of polyhedra
Pyramids
Pyramids include some of the most time-honoured and famous of all polyhedra.
Stellations and facettings
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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.
Zonohedra
A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°.
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.
Orthogonal Polyhedra
An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons (also known as rectilinear polygons). Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a net (polyhedron).
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:
- Tilings or tessellations of the plane.
- Sponge-like structures called infinite skew polyhedra.
See also: Apeirogon - infinite regular polygon: {∞}
Complex polyhedra
A complex polyhedron is one which is constructed in unitary 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 all but one of the uniform polyhedra.
Some polyhedra, such as hosohedra, exist only as spherical polyhedra and have no flat-faced analogue.
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 ordered set of vertices, and allowed faces to be skew as well as planar.
Tessellations 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 polytope 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.
Such a figure is called simplicial if each of its regions is a simplex, i.e. in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an n-dimensional cube.
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.
History
Prehistory
Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyehdra, but also the relations of duality amongst some of them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them.
Other polyhedra have of course made their mark in architecture - cubes and cuboids being obvious examples, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age.
The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 1800s of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy.
Greeks
The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the convex regular polyhedra, while Archimedes later expanded his study to the convex uniform polyhedra.
Muslims and Chinese
After the end of the Classical era, Islamic scholars continued to make advances, for example in the tenth century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra. Meanwhile in China, dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids was used as the basis for calculating volumes of earth to be moved during engineering excavations.
Renaissance
Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.
Star polyhedra
For almost 2000 years, the concept of a polyhedron had remained as developed by the ancient Greek mathematicians.
Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons, typically pentagrams as faces. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Louis Poinsot realised that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they are called the Kepler-Poinsot polyhedra.
The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been re-published (Coxeter, 1999).
The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. Bridge 1974 listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.
See also:
- Regular polyhedron: History
- Regular polytope: History of discovery.
Polyhedra in nature
For natural occurrences of regular polyhedra, see Regular polyhedron: History.
Irregular polyhedra appear in nature as crystals.
Books on polyhedra
Introductory books, also suitable for school use
- Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
- Cundy, H.M. & Rollett, A.P.; Mathematical models, 1st Edn. hbk OUP (1951), 2nd Edn. hbk OUP (1961), 3rd Edn. pbk Tarquin (1981).
- Holden; Shapes, space and symmetry, (1971), Dover pbk (1991).
- Pearce, P and Pearce, S: Polyhedra primer, Van Nost. Reinhold (May 1979), ISBN-10: 0442264968, ISBN-13: 978-0442264963.
- Tarquin publications: books of cut-out and make card models.
- Wenninger, M.; Polyhedron models for the classroom, pbk (1974)
- Wenninger, M.; Polyhedron models, CUP hbk (1971), pbk (1974).
- Wenninger, M.; Spherical models, CUP.
- Wenninger, M.; Dual models, CUP.
Undergraduate level
- Coxeter, H.S.M. DuVal, Flather & Petrie; The fifty-nine icosahedra, 3rd Edn. Tarquin.
- Coxeter, H.S.M. Twelve geometric essays. Republished as The beauty of geometry, Dover.
- Thompson, Sir D'A. W. On growth and form, (1943). (not sure if this is the right category for this one, I haven't read it).
Design and architecture bias
- Critchlow, K.; Order in space.
- Pearce, P.; Structure in nature is a strategy for design, MIT (1978)
- Williams, R.; The geometrical foundation of natural structure, Dover (1979).
Advanced mathematical texts
- Coxeter, H.S.M.; Regular Polytopes 3rd Edn. Dover (1973).
- Coxeter, H.S.M.; Regular complex polytopes, CUP (1974).
- Lakatos, Imre; Proofs and Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic
- Several more to add here.
Historical books
- Brückner, Vielecke und Vielflache (Polygons and polyhedra), (1900).
- Fejes Toth, L.;