For the game magazine, see Polyhedron (magazine)."Polyhedra" redirects here. For the relational database system, see Polyhedra DBMS.

A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric solid with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολύεδρον, from poly-, stem of πολύς, "many," + -edron, form of έδρα, "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, "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' ...." Mathematicians still do not agree as to exactly what makes something a polyhedron.

Contents

[hide]

1 Basis for definition 2 Characteristics

o 2.1 Names of polyhedra

o 2.2 Edgeso 2.3 Euler

characteristico 2.4 Orientabilityo 2.5 Vertex figureo 2.6 Dualityo 2.7 Volume

3 Traditional polyhedra

Some Polyhedra

Dodecahedron(Regular polyhedron)

Small stellated dodecahedron(Regular star)

Icosidodecahedron(Uniform)

Great cubicuboctahedron(Uniform star)

Rhombic triacontahedron(Uniform dual)

Elongated pentagonal cupola(Convex regular-faced)

Octagonal prism(Uniform prism)

Square antiprism(Uniform antiprism)

o 3.1 Symmetrical polyhedra

3.1.1 Uniform polyhedra and their duals

3.1.2 Noble polyhedra

3.1.3 Symmetry groups

o 3.2 Other polyhedra with regular faces

3.2.1 Equal regular faces

3.2.1.1 Deltahedra

3.2.2 Johnson solids

o 3.3 Other important families of polyhedra

3.3.1 Pyramids 3.3.2 Stellations

and facettings 3.3.3

Zonohedra 3.3.4 Toroidal

polyhedra 3.3.5

Compounds 3.3.6

Orthogonal polyhedra

4 Generalisations of polyhedra o 4.1 Apeirohedrao 4.2 Complex polyhedrao 4.3 Curved polyhedra

4.3.1 Spherical polyhedra

4.3.2 Curved spacefilling polyhedra

o 4.4 General polyhedrao 4.5 Hollow faced or

skeletal polyhedra 5 Non-geometric polyhedra

o 5.1 Topological

polyhedrao 5.2 Abstract polyhedrao 5.3 Polyhedra as

graphs 6 History

o 6.1 Prehistoryo 6.2 Greekso 6.3 Muslims and

Chineseo 6.4 Renaissanceo 6.5 Star polyhedra

7 Polyhedra in nature 8 See also 9 References 10 Books on polyhedra

o 10.1 Introductory books, also suitable for school use

o 10.2 Undergraduate level

o 10.3 Design and architecture bias

o 10.4 Advanced mathematical texts

o 10.5 Historic books 11 External links

o 11.1 General theoryo 11.2 Lists and

databases of polyhedrao 11.3 Softwareo 11.4 Resources for

making models, and models for sale

o 11.5 Miscellaneous

[edit] Basis for definition

Any polyhedron can be 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 enclosed by them.

2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal 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 segment. 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.

[edit] Characteristics

[edit] Names of 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.

[edit] 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.

[edit] Euler characteristic

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.

[edit] Orientability

Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable.

But for some polyhedra this is not possible, and the figure is said to be non-orientable. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even χ < 2 may or may not be orientable.

[edit] Vertex figure

For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.

[edit] Duality

For every polyhedron we can construct a dual polyhedron having:

faces in place of the original's vertices and vice versa, the same number of edges the same Euler characteristic and orientability

For a convex polyhedron the dual can be obtained by the process of polar reciprocation.

[edit] Volume

The volume of an orientable polyhedron having an identifiable centroid can be calculated using Green's theorem:

by choosing the function

where (x,y,z) is the centroid of the surface enclosing the volume under consideration. As we have,

Hence the volume can be calculated as:

where the normal of the surface pointing outwards is given by:

The final expression can be written as

where S is the surface area of the polyhedron.

[edit] Traditional polyhedra

A dodecahedron

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 or surface.

[edit] 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 triangular pyramid or tetrahedron, cube or hexahedron, 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.

[edit] Uniform polyhedra and their duals

Main article: Uniform polyhedron

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 dualRegular Platonic solids Kepler-Poinsot polyhedra

QuasiregularArchimedean solids Catalan solids

(no special name) (no special name)

Semiregular(no special name) (no special name)

Prisms Dipyramids Star Prisms Star DipyramidsAntiprisms Trapezohedra Star Antiprisms Star Trapezohedra

[edit] Noble polyhedra

Main article: Noble polyhedron

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.

[edit] 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.

[edit] Other polyhedra with regular faces

[edit] 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.)

[edit] 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) o Tetrahedron o Octahedron o Icosahedron

5 non-uniform convex polyhedra (5 of the Johnson solids) o Triangular dipyramid o Pentagonal dipyramid o Snub disphenoid o Triaugmented triangular prism o Gyroelongated square dipyramid

[edit] 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.

[edit] Other important families of polyhedra

[edit] Pyramids

Main article: Pyramid (geometry)

Pyramids include some of the most time-honoured and famous of all polyhedra.

[edit] Stellations and facettings

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

[edit] Zonohedra

Main article: Zonohedron

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

[edit] Toroidal polyhedra

Main article: Toroidal polyhedron

A toroidal polyhedron is a polyhedra with an Euler characteristic of 0 or smaller, representing a torus surface.

[edit] Compounds

Main article: Polyhedral compound

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.

[edit] 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 polygonal net.

[edit] Generalisations of polyhedra

The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.

[edit] 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: {∞}

[edit] Complex polyhedra

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

[edit] Curved polyhedra

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

[edit] Spherical polyhedra

Main article: Spherical polyhedron

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 and dihedra, exist only as spherical polyhedra and have no flat-faced analogue.

[edit] Curved spacefilling polyhedra

Two important types are:

Bubbles in froths and foams, such as Weaire-Phelan bubbles. Spacefilling forms used in architecture. See for example Pearce (1978).

[edit] 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.

[edit] 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.

[edit] Non-geometric polyhedra

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

[edit] 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.

[edit] Abstract polyhedra

An abstract polyhedron is a partially ordered set (poset) of elements whose partial ordering obeys certain rules. 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.

[edit] 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.

[edit] History

[edit] 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[citation needed].

[edit] 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, which came to be known as the Platonic solids.

Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.

[edit] 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.

[edit] Renaissance

As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Renaissance. Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.

[edit] Star polyhedra

For almost 2,000 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 .

[edit] Polyhedra in nature

For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.

Irregular polyhedra appear in nature as crystals.

[edit] See also

Wikimedia Commons has media related to: Polyhedron

Look up polyhedron in Wiktionary, the free dictionary.

Antiprism Archimedean solid Bipyramid Conway polyhedron notation (a notation for describing construction of polyhedra) Defect Deltahedron Deltohedron Escher Flexible polyhedra Johnson solid Kepler-Poinsot polyhedra List of polyhedral images Near-miss Johnson solid Net (polyhedron) Platonic solid Polychoron (4 dimensional analogues to polyhedra) Polyhedral compound Polyhedron models Prism Semiregular polyhedron Schlegel diagram Spidron Tessellation Trapezohedron Uniform polyhedron Waterman polyhedron Zonohedron

[edit] 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. (pdf)

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

[edit] Books on polyhedra

[edit] 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

0442264968, ISBN 978-0442264963. Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of

Topology. Princeton University Press. Senechal, M. & Fleck, G.; Shaping Space a Polyhedral Approch, Birhauser (1988), ISBN

0817633510 Tarquin publications : books of cut-out and make card models. Wenninger, Magnus ; 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.

[edit] 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).

[edit] Design and architecture bias

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

Williams, R.; The Geometry of Natural Structure (40th Anniversary Edition), San Francisco: Eudaemon Press (2009).ISBN 978-0-9823465-1-8

[edit] 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.

[edit] Historic books

Brückner, M. (1900). Vielecke und Vielflache: Theorie und Geschichte. Leipzig: B.G. Treubner. ISBN 978-1418165901. (German)

WorldCat English: Polygons and Polyhedra: Theory and History.

Fejes Toth, L.; Kepler, J.; De harmonices Mundi (Latin. Available in English translation). Pacioli, L.;

[edit] External links

[edit] General theory

Weisstein, Eric W. , "Polyhedron" from MathWorld. Polyhedra Pages uniform solution for uniform polyhedra by Dr. Zvi Har'El Symmetry, Crystals and Polyhedra Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons

[edit] Lists and databases of polyhedra

The Uniform Polyhedra Virtual Reality Polyhedra - The Encyclopedia of Polyhedra

o Polyhedra and Pyramids Interactive 3D polyhedra in Java Electronic Geometry Models - Contains a peer reviewed selection of polyhedra with

unusual properties. Origami Polyhedra - Models made with Modular Origami Polyhedra Collection - Various virtual and physical polyhedra models. Polyhedra plaited with paper strips - Polyhedra models constructed without use of glue. Rotatable polyhedron models - that work right in your web browser

Rotatable non convex self intersecting polyhedra these ones also work right in your browser

Polyhedron Models - Virtual polyhedra

[edit] Software

A Plethora of Polyhedra An interactive and free collection of polyhedra in JAVA. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.

Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.

World of Polyhedra - Comprehensive polyhedra in flash applet, showing vertices and edges (but not shaded faces)

Hyperspace Star Polytope Slicer - Explorer java applet, includes a variety of 3d viewer options.

HEDRON Polyhedron modelling software Uniform Polyhedra Java Applets with sources

[edit] Resources for making models, and models for sale

Making Polyhedra Paper Models of Polyhedra Free nets of polyhedra Paper Models of Uniform (and other) Polyhedra Polyhedra software, die-cast models, & posters Simple instructions for building over 30 paper polyhedra Kits to print out, cut, and fold into various polyhedra

Miscellaneous

PictureSpice - A site that lets you make polyhedra with your own uploaded pictures.

[hide] v • d • e

Polyhedron navigatorPlatonic solids (regular)

tetrahedron · cube · octahedron · dodecahedron · icosahedron

Archimedean solids(Semiregular/Uniform)

truncated tetrahedron · cuboctahedron · truncated cube · truncated octahedron · rhombicuboctahedron · truncated cuboctahedron · snub cube · icosidodecahedron · truncated dodecahedron · truncated icosahedron · rhombicosidodecahedron · truncated icosidodecahedron · snub dodecahedron

Catalan solids(Dual semiregular)

triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedron

Dihedral regular dihedron · hosohedron

Dihedral uniform prisms · antiprisms

Duals of dihedral uniform

bipyramids · trapezohedra

Dihedral otherspyramids · truncated trapezohedra  · gyroelongated bipyramid  · cupola  · bicupola  · pyramidal frusta

Degenerate polyhedra are in italics.

Autodesk 3ds Max, formerly 3D Studio MAX, is a modeling, animation and rendering package developed by Autodesk Media and Entertainment. 3ds Max is the largest selling 3D computer animation program in the world according to Autodesk internal reports. In addition, it is the third most widely-used off the shelf content creation tool used by professionals according to the Roncarelli report.[1] It has strong modeling capabilities, a flexible plugin architecture and a long heritage on the Microsoft Windows platform. It is mostly used by video game developers, TV commercial studios and architectural visualization studios. It is also used for movie effects and movie pre-visualization.

In addition to its modeling and animation tools, the latest version of 3ds Max also features advanced shaders (such as ambient occlusion and subsurface scattering), dynamic simulation, particle systems, radiosity, normal map creation and rendering, global illumination, an intuitive and fully-customizable user interface, and its own scripting language.[2] A plethora of specialized third-party renderer plugins, such as V-Ray, Brazil r/s , Maxwell Render, and finalRender, may be purchased separately.

Contents

[hide]

1 Early history & Releases 2 Modeling

o 2.1 Polygon modeling o 2.2 NURBS or Non Uniform Rational B-Spline o 2.3 Surface tool/Editable patch object

3 Predefined primitives o 3.1 Predefined Standard Primitives list o 3.2 Predefined Extended Primitives list

4 Rendering 5 Features 6 Film Use 7 Licensing 8 Notes 9 See also 10 External links

[edit] Early history & Releases

Further information: 3ds Max release history

The original 3D Studio product was created for the DOS platform by the Yost Group and published by Autodesk. After 3D Studio Release 4, the product was rewritten for the Windows NT platform, and re-named "3D Studio MAX." This version was also originally created by the Yost Group. It was released by Kinetix, which was at that time Autodesk's division of media and entertainment. Autodesk purchased the product at the second release mark of the 3D Studio MAX version and internalized development entirely over the next two releases. Later, the product name was changed to "3ds max" (all lower case) to better comply with the naming conventions of Discreet, a Montreal-based software company which Autodesk had purchased. At release 8, the product was again branded with the Autodesk logo, and the name was again changed to "3ds Max" (upper and lower case). At release 2009, the product name changed to "Autodesk 3ds Max".

[edit] Modeling

[edit] Polygon modeling

Main article: Polygon modeling

Polygon modeling is more common with game design than any other modeling technique as the very specific control over individual polygons allows for extreme optimization. Usually, the modeller begins with one of the 3ds max primitives, and using such tools as bevel and extrude, adds detail to and refines the model. Versions 4 and up feature the Editable Polygon object, which simplifies most mesh editing operations, and provides subdivision smoothing at customizable levels.

Version 7 introduced the edit poly modifier, which allows the use of the tools available in the editable polygon object to be used higher in the modifier stack (i.e., on top of other modifications).

[edit] NURBS or Non Uniform Rational B-Spline

A more advanced alternative to polygons, it gives a smoothed out surface that eliminates the straight edges of a polygon model. NURBS is a mathematically exact representation of freeform surfaces like those used for car bodies and ship hulls, which can be exactly reproduced at any resolution whenever needed. With NURBS, a smooth sphere can be created with only one face.

The non-uniform property of NURBS brings up an important point. Because they are generated mathematically, NURBS objects have a parameter space in addition to the 3D geometric space in which they are displayed. Specifically, an array of values called knots specifies the extent of influence of each control vertex (CV) on the curve or surface. Knots are invisible in 3D space and you can't manipulate them directly, but occasionally their behavior affects the visible appearance of the NURBS object. This topic mentions those situations. Parameter space is one-dimensional for curves, which have only a single U dimension topologically, even though they exist geometrically in 3D space. Surfaces have two dimensions in parameter space, called U and V.[citation needed]

NURBS curves and surfaces have the important properties of not changing under the standard geometric affine transformations (Transforms), or under perspective projections. The CVs have local control of the object: moving a CV or changing its weight does not affect any part of the object beyond the neighboring CVs. (You can override this property by using the Soft Selection controls.) Also, the control lattice that connects CVs surrounds the surface. This is known as the convex hull property.[citation needed]

[edit] Surface tool/Editable patch object

Surface tool was originally a 3rd party plugin, but Kinetix acquired and included this feature since version 3.0.[citation needed] The surface tool is for creating common 3ds max's splines, and then applying a modifier called "surface." This modifier makes a surface from every 3 or 4 vertices in a grid. This is often seen as an alternative to 'Mesh' or 'Nurbs' modeling, as it enables a user to interpolate curved sections with straight geometry (for example a hole through a box shape). Although the surface tool is a useful way to generate parametrically accurate geometry, it lacks the 'surface properties' found in the similar Edit Patch modifier, which enables a user to maintain the original parametric geometry whilst being able to adjust "smoothing groups" between faces.[citation needed]

[edit] Predefined primitives

This is a basic method, in which one models something using only boxes, spheres, cones, cylinders and other predefined objects from the list of Predefined Standard Primitives or a list of Predefined Extended Primitives. One may also apply boolean operations, including subtract, cut and connect. For example, one can make two spheres which will work as blobs that will connect with each other. These are called metaballs.[citation needed]

Some of the 3ds Max Standard Primitives as they appear in the wireframe view of 3ds Max 9

Some of the 3ds Max Extended Primitives as they appear in the wireframe view of 3ds Max 9

[edit] Predefined Standard Primitives list

Box — box produces a rectangular prism. An alternative variation of box is available — entitled cube — which proportionally constrains the length, width and height of the box.

Cylinder — cylinder produces a cylinder. Torus — torus produces a torus — or a ring — with a circular cross section, sometimes

referred to as a doughnut. Teapot — teapot produces the Utah teapot. Since the teapot is a parametric object, the

user can choose which parts of the teapot to display after creation. These parts include the body, handle, spout and lid.

Cone — cone produces round cones — either upright or inverted. Sphere — sphere produces a full sphere, hemisphere, or other portion of a sphere. Tube — tube can produce both round and prismatic tubes. The tube is similar to the

cylinder with a hole in it. Pyramid — The pyramid primitive has a square or rectangular base and triangular sides. Plane — The plane object is a special type of flat polygon mesh that can be enlarged by

any amount at render time. The user can specify factors to magnify the size or number of segments, or both. Modifiers such as displace can be added to a plane to simulate a hilly terrain.

Geosphere — GeoSphere produces spheres and hemispheres based on three classes of regular polyhedrons.

[edit] Predefined Extended Primitives list

Hedra — produces objects from several families of polyhedra. ChamferBox — creates a box with beveled or rounded edges. OilTank — creates a cylinder with convex caps. Spindle — creates a cylinder with conical caps.

Gengon — creates an extruded, regular-sided polygon with optionally filleted side edges. Prism — Creates a three-sided prism with independently segmented sides. Torus knot — creates a complex or knotted torus by drawing 2D curves in the normal

planes around a 3D curve. The 3D curve (called the Base Curve) can be either a circle or a torus knot. It can be converted from a torus knot object to a NURBS surface.

ChamferCyl — creates a cylinder with beveled or rounded cap edges. Capsule — creates a cylinder with hemispherical caps. L-Ext — creates an extruded L-shaped object. C-Ext — creates an extruded C-shaped object. Hose — a flexible object, similar to a spring.

[edit] Rendering

Scanline renderingThe default rendering method in 3DS Max is scanline rendering. Several advanced features have been added to the scanliner over the years, such as global illumination, radiosity, and ray tracing.

mental raymental ray is a production quality renderer developed by mental images. It is integrated into the later versions of 3ds Max, and is a powerful raytracing renderer with bucket rendering, a technique that allows distributing the rendering task for a single image between several computers efficiently, using TCP network protocol.

RenderManA third party connection tool to RenderMan pipelines is also available for those that need to integrate Max into Renderman render farms.

V-RayA third-party render engine plug-in for 3D Studio MAX. It is widely used, frequently substituting the standard and mental ray renderers which are included bundled with 3ds Max. V-Ray continues to be compatible with older versions of 3ds Max.

Brazil R/SA third-party high-quality photorealistic rendering system created by SplutterFish, LLC capable of fast ray tracing and global illumination.

FinalRenderAnother third-party raytracing render engine created by Cebas. Capable of simulating a wide range of real-world physical phenomena.

Indigo RendererA third-party photorealistic renderer with plugins for 3ds max.

Maxwell RenderA third-party photorealistic rendering system created by Next Limit Technologies providing robust materials and highly accurate unbiased rendering.

BIGrender 3.0Another third-party rendering plugin. Capable of overcoming 3DS rendering memory limitations with rendering huge pictures.

[edit] Features

MAXScriptMAXScript is a built-in scripting language, and can be used to automate repetitive tasks, combine existing functionality in new ways, develop new tools and user interfaces and much more. Plugin modules can be created entirely in MAXscript.

Character StudioCharacter Studio was a plugin which since version 4 of Max is now integrated in 3D Studio Max, helping user to animate virtual characters. The system works using a character rig or "Biped" which is pre-made and allows the user to adjust the rig to fit the character they will be animating. Dedicated curve editors and motion capture data import tools make Character Studio ideal for character animation. "Biped" objects have other useful features that automated the production of walk cycles and movement paths, as well as secondary motion.

Scene ExplorerScene Explorer, a tool that provides a hierarchical view of scene data and analysis, facilitates working with more complex scenes. Scene Explorer has the ability to sort, filter, and search a scene by any object type or property (including metadata). Added in 3ds Max 2008, it was the first component to facilitate .NET managed code in 3ds Max outside of MAXScript.

DWG Import3ds Max supports both import and linking of DWG files. Improved memory management in 3ds Max 2008 enables larger scenes to be imported with multiple objects.

Texture Assignment/Editing3ds Max offers operations for creative texture and planar mapping, including tiling, mirroring, decals, angle, rotate, blur, UV stretching, and relaxation; Remove Distortion; Preserve UV; and UV template image export. The texture workflow includes the ability to combine an unlimited number of textures, a material/map browser with support for drag-and-drop assignment, and hierarchies with thumbnails. UV workflow features include Pelt mapping, which defines custom seams and enables users to unfold UVs according to those seams; copy/paste materials, maps and colors; and access to quick mapping types (box, cylindrical, spherical).

General KeyframingTwo keying modes — set key and auto key — offer support for different keyframing workflows.Fast and intuitive controls for keyframing — including cut, copy, and paste — let the user create animations with ease. Animation trajectories may be viewed and edited directly in the viewport.

Constrained AnimationObjects can be animated along curves with controls for alignment, banking, velocity, smoothness, and looping, and along surfaces with controls for alignment. Weight path-controlled animation between multiple curves, and animate the weight. Objects can be constrained to animate with other objects in many ways — including look at, orientation in different coordinate spaces, and linking at different points in time. These constraints also support animated weighting between more than one target.All resulting constrained animation can be collapsed into standard keyframes for further editing.

Skinning

Either the Skin or Physique modifier may be used to achieve precise control of skeletal deformation, so the character deforms smoothly as joints are moved, even in the most challenging areas, such as shoulders. Skin deformation can be controlled using direct vertex weights, volumes of vertices defined by envelopes, or both.Capabilities such as weight tables, paintable weights, and saving and loading of weights offer easy editing and proximity-based transfer between models, providing the accuracy and flexibility needed for complicated characters.The rigid bind skinning option is useful for animating low-polygon models or as a diagnostic tool for regular skeleton animation.Additional modifiers, such as Skin Wrap and Skin Morph, can be used to drive meshes with other meshes and make targeted weighting adjustments in tricky areas.

Skeletons and Inverse Kinematics (IK)Characters can be rigged with custom skeletons using 3ds Max bones, IK solvers, and rigging tools.All animation tools — including expressions, scripts, list controllers, and wiring — can be used along with a set of utilities specific to bones to build rigs of any structure and with custom controls, so animators see only the UI necessary to get their characters animated.Four plug-in IK solvers ship with 3ds Max: history-independent solver, history-dependent solver, limb solver, and spline IK solver. These powerful solvers reduce the time it takes to create high-quality character animation. The history-independent solver delivers smooth blending between IK and FK animation and uses preferred angles to give animators more control over the positioning of affected bones.The history-dependent solver can solve within joint limits and is used for machine-like animation. IK limb is a lightweight two-bone solver, optimized for real-time interactivity, ideal for working with a character arm or leg. Spline IK solver provides a flexible animation system with nodes that can be moved anywhere in 3D space. It allows for efficient animation of skeletal chains, such as a character’s spine or tail, and includes easy-to-use twist and roll controls.

Integrated Cloth SolverIn addition to reactor’s cloth modifier, 3ds Max software has an integrated cloth-simulation engine that enables the user to turn almost any 3D object into clothing, or build garments from scratch. Collision solving is fast and accurate even in complex simulations.(image.3ds max.jpg)Local simulation lets artists drape cloth in real time to set up an initial clothing state before setting animation keys.Cloth simulations can be used in conjunction with other 3ds Max dynamic forces, such as Space Warps. Multiple independent cloth systems can be animated with their own objects and forces. Cloth deformation data can be cached to the hard drive to allow for nondestructive iterations and to improve playback performance.

Integration with Autodesk VaultAutodesk Vault plug-in, which ships with 3ds Max, consolidates users’ 3ds Max assets in a single location, enabling them to automatically track files and manage work in progress. Users can easily and safely share, find, and reuse 3ds Max (and design) assets in a large-scale production or visualization environment.

[edit] Film Use

See List of films made with Autodesk 3ds Max

Many recent films have made use of 3ds Max, or previous versions of the program under previous names, in CGI animation, such as I, Robot and X-Men, which contain computer generated graphics from 3ds Max alongside live-action acting.

[edit] Licensing

Earlier versions (up to and including 3D Studio Max R3.1) required a special copy protection device (called a dongle) to be plugged into the parallel port while the program was run, but later versions incorporated software based copy prevention methods instead. Current versions require online registration.

Due to the high price of the commercial version of the program, Autodesk also offers a heavily discounted student version, which explicitly states that it is to be used for "educational purposes only." The student version has identical features to the full version.

Torus knot 

Wikipedia: Torus knot 

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A (3,7)-3D torus knot rendered by Apple Grapher.

In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. The (p,q)-torus knot winds p times around a circle inside the torus, which goes all the way around the torus, and q times around a line through the hole in the torus, which passes once through the hole, (usually drawn as an axis of symmetry). If p and q are not relatively prime, then we have a torus link with more than one component.

The (p,q)-torus knot can be given by the parameterization

where 0 < φ < 2pπ. This lies on the surface of the torus given by (r − 2)2 + z2 = 1 (in cylindrical coordinates).

the (2,3)-torus knot, also known as the trefoil knot

A torus knot is trivial if and only if either p or q is equal to 1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

Properties

Diagram of a (3,8)-torus knot.

Each torus knot is prime and chiral. Any (p,q)-torus knot can be made from a closed braid with p strands. The appropriate braid word is

The crossing number of a torus knot is given by

c = min((p−1)q, (q−1)p).

The genus of a torus knot is

The Alexander polynomial of a torus knot is

The Jones polynomial of a (right-handed) torus knot is given by

The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres.

Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle. The knot complement of the (p, q)-torus knot deformation retracts to the space X. Therefore, the knot group of a torus knot has the presentation

Torus knots are the only knots whose knot groups have non-trivial center (which is infinite cyclic, generated by the element xp = yq in the presentation above).

See also

Alternating knot    Cinquefoil knot    Prime knot    Trefoil knot   

External links

Weisstein, Eric W.   , "Torus Knot" from MathWorld. Torus knot renderer in Actionscript   

Knot theory 

Sci-Tech Dictionary: knot theory 

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(′nät ′thē·ə·rē) 

(mathematics) The topological and algebraic study of knots emphasizing their classification and how one may be continuously deformed into another.

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Knot theory

5min Related Video: Knot theory 

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Britannica Concise Encyclopedia: knot theory 

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Mathematical theory of closed curves in three-dimensional space. The number of times and the manner in which a curve crosses itself distinguish different knots. The fewest possible crossings is three, for the overhand (trefoil) knot, which occurs in two mirror versions according to the directions in which the curve crosses itself. Knot theory has been used to understand both atomic and molecular structures (protein folding).

For more information on knot theory, visit Britannica.com.

Wikipedia: Knot theory 

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A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

0 WA NotLgd http://w iki.answ e http://w w w .answ

A knot diagram of the trefoil knot

In mathematics, knot theory is the area of topology that studies mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together to prevent it from becoming undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher dimensional knots are n -dimensional spheres in m-dimensional Euclidean space.

Contents

[hide]

1     History    2     Knot equivalence    3     Knot diagrams    

o 3.1      Reidemeister moves    4     Knot invariants    

o 4.1      Knot polynomials   o 4.2      Hyperbolic invariants   

5     Higher dimensions    6     Adding knots    7     Tabulating knots    

o 7.1      Alexander–Briggs notation   o 7.2      Dowker notation   o 7.3      Conway notation   

8     See also    9     References    10      Further reading    

o 10.1      Introductory textbooks   o 10.2      Surveys   o 10.3      Knot Theory in Popular Fiction   

11      External links    o 11.1      History   o 11.2      Knot tables and software   

History

Main article: History of knot theory

Intricate Celtic knotwork in the 1200 year old Book of Kells

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan

Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

Mathematical studies of knots began in the 19th century with Gauss, who defined the linking integral (Silver 2006). In the 1860s, Lord Kelvin's theory that atoms were knots in the aether led to Peter Guthrie Tait's creation of the first knot tables. Tabulation motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.

These topologists in the early part of the 20th century—Max Dehn, J. W. Alexander, and others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.

The first knot tabulator, Peter Guthrie Tait

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 (Sossinsky 2002, p. 71–89), and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.

In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not (Simon 1986). Tangles, strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA (Flapan 2000). Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation (Collins 2006).

Knot equivalence

A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (Adams 2004)(Sossinsky 2002). When topologists consider knots and other entanglements such as links and braids, they consider the space surrounding the knot as a viscous fluid. If the knot can be pushed about smoothly in the fluid, without intersecting itself, to coincide with another knot, the two knots are considered equivalent. The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots are equivalent if one can be transformed into the other via a type of deformation of R3 upon itself, known as an ambient isotopy.

(Left) The unknot, and a knot equivalent to it. (Right) It is more difficult to determine whether complex knots such as this are equivalent to the unknot.

The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s (Hass 1998). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (Hass 1998). The special case of recognizing the unknot, called the unknotting problem, is of particular interest (Hoste 2005).

Knot diagrams

A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath.

Reidemeister moves

Main article: Reidemeister move

In 1927, working with this diagrammatic form of knots, J.W. Alexander and G. B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves, are:

I. Twist and untwist in either direction.II. Move one strand completely over another.III. Move a strand completely over or under a crossing.

Reidemeister moves

Type I Type II

Type III

The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A

close inspection will show that complicated events can be eliminated, leaving only the simplest events: 1) a "kink" forming or being straightened out 2) two strands becoming tangent at a point and passing through 3) three strands crossing at a point. These are precisely the Reidemeister moves (Sossinsky 2002, ch. 3) (Lickorish 1997, ch. 1).

Knot invariants

Main article: knot invariant

A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2004)(Lickorish 1997)(Rolfsen 1976). For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability.

"Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (Lickorish 1997)(Rolfsen 1976). In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

Knot polynomials

Main article: knot polynomial

A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexander-Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish 1997).

The Alexander-Conway polynomial is actually defined in terms of links, which consist of one or more knots entangled with each other. The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.

Suppose there is a link diagram which is oriented, i.e. every component of the link has a preferred direction indicated by an arrow. Also suppose L + ,L − ,L0 are oriented link diagrams resulting from changing the diagram at a specified crossing of the diagram, as indicated in the figure:

Then the Alexander-Conway polynomial, C(z), is recursively defined according to the rules:

C(O) = 1 (where O is any diagram of the unknot) C(L + ) = C(L − ) + zC(L0)

The second rule is what is often referred to as a skein relation. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.

The following is an example of a typical computation using a skein relation. It computes the Alexander-Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied.

C( )=C( ) + z C( )

gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated,

C( ) = C( ) + z C( )

gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes a bit of sneakiness:

C( ) = C( )+ z C( )

which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.

Putting all this together will show:

C(trefoil) = 1 + z (0 + z) = 1 + z2

Since the Alexander-Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the

The left handed trefoil knot.The right handed trefoil knot.

mirror image). These are not equivalent to each other! This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn 1914). But the Alexander-Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left and right-handed trefoil knots (Lickorish 1997).

Hyperbolic invariants

William Thurston proved many knots are hyperbolic knots, meaning that the knot complement, i.e. the points of 3-space not on the knot, admit a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (Adams 2004).

The Borromean rings are a link with the property that removing one ring unlinks the others.

SnapPea's cusp view: the Borromean rings complement from the perspective of an inhabitant living near the red component.

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental paralleogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task (Adams, Hildebrand & Weeks 1991).

Higher dimensions

In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. This necessary deformation can be achieved in two steps. The first step is to "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. The second step is changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two dimensional sphere embedded in a four dimensional sphere. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n- spheres form knots only in (n+2)-space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted 4k-1-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere (Haefliger 1962)(Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere in an n-sphere with 2n-3k-3 > 0 is unknotted. The notion of a knot has further generalisations in mathematics, see: knot (mathematics).

Adding knots

Main article: Knot sum

Adding two knots

Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined as follows (Adams 2004): consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots. Depending on how this is done, two different knots (but no more) may result. This ambiguity in the sum can be eliminated regarding

the knots as oriented, i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle.

The knot sum of oriented knots is commutative and associative. There is also a prime decomposition for a knot which allows a prime or composite knot to be defined, analogous to prime and composite numbers (Schubert 1949). For oriented knots, this decomposition is also unique. Higher dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.

Tabulating knots

A table of prime knots up to seven crossings. The knots are labeled with Alexander–Briggs notation

Traditionally, knots have been catalogued in terms of crossing number. Knot tables generally include only prime knots and only one entry for a knot and its mirror image (even if they are different) (Hoste, Thistlethwaite & Weeks 1998). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (Hoste 2005, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (Hoste 2005, p. 28). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705... (sequence A002863 in OEIS). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (Adams 2004).

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the Dowker notation. Different notations have been invented for knots which allow more efficient tabulation (Hoste 2005).

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (Hoste, Thistlethwaite & Weeks 1998). The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by John Horton Conway, who not only developed a new notation but also the Alexander–Conway polynomial (Conway 1970)(Doll & Hoste 1991). This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait–Little tables; however he missed the duplicates called the Perko pair, which would only be noticed in 1974 by Kenneth Perko (Perko 1974). This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (Hoste, Thistlethwaite & Weeks 1998). In 2003 Rankin, Flint, and Schermann, tabulated the alternating knots through 22 crossings (Hoste 2005).

Alexander–Briggs notation

This is the most traditional notation, due to the 1927 paper of J. W. Alexander and G. Briggs and later extended by Dale Rolfsen in his knot table. The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance.

Dowker notation

Main article: Dowker notation

A knot diagram with crossings labelled for a Dowker sequence

The Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in the figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker notation for this labelling is the sequence: 6 −12 2 8 −4 −10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker notation.

Conway notation

Main article: Conway notation (knot theory)

The Conway notation for knots and links, named after John Horton Conway, is based on the theory of tangles (Conway 1970). The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedron. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an algebraic tangle substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or − signs.

An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a rational tangle. One inserts this tangle at the vertex of the basic polyhedron 1*.

A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

See also

Contact geometry#Legendrian submanifolds and knots    Knots and graphs    List of knot theory topics    Molecular knot    Quantum topology   

References

Adams, Colin    (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0821836781

Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey (1991), "Hyperbolic invariants of knots and links", Transactions of the American Mathemathical Society 326 (1): 1–56

Bar-Natan, Dror (1995), "On the Vassiliev knot invariants", Topology 34 (2): 423–472 Collins, Graham (April 2006), "Computing with Quantum Knots", Scientific American Conway, John    (1970), "An enumeration of knots and links, and some of their algebraic 

properties", Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, pp. 329–358

Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement.", Math. Comp. 57 (196): 747–761

Flapan, Erica (2000), "When topology meets chemistry: A topological look at molecular chirality", Outlooks (Cambridge University Press, Cambridge; Mathematical Association of America, Washington, DC), ISBN 0-521-66254-0

Haefliger, André (1962), "Knotted (4k − 1)-spheres in 6k-space", Annals of Mathematics (2) 75: 452–466

Hass, Joel (1998), "Algorithms for recognizing knots and 3-mainifolds", Chaos, Solitons and Fractals (Elsevier) 9: 569–581arΧiv:math/9712269

Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeffrey (1998), "The first 1,701,935 knots", Math. Intelligencer (Springer) 20: 33–48

Hoste, Jim (2005), "The enumeration and classification of knots and links", Handbook of Knot Theory, Amsterdam: Elsevier, http://pzacad.pitzer.edu/~jhoste/HosteWebPages/downloads/Enumeration.pdf

Levine, Jerome (1965), "A classification of differentiable knots", Annals of Mathematics (2) 1982: 15–50

Kontsevich, Maxim    (1993), "Vassiliev's knot invariants", I. M. Gelfand Seminar, Adv. Soviet Math. (Providence, RI: Amer. Math. Soc.) 16: 137–150

Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer-Verlag, ISBN 0-387-98254-X

Perko, Kenneth (1974), "On the classification of knots", Proceedings of the American Mathematical Society 45: 262–266

Rolfsen, Dale (1976), Knots and Links, Publish or Perish, ISBN 0-914098-16-0 Schubert, Horst (1949), "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten", 

Heidelberger Akad. Wiss. Math.-Nat. Kl. (3): 57–104 Silver, Dan (2006), "Knot theory's odd origins", American Scientist 94 (2): 158–165, 

http://www.southalabama.edu/mathstat/personal_pages/silver/scottish.pdf

Simon, Jonathan (1986), "Topological chirality of certain molecules", Topology 25: 229–235 Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press, ISBN 0-674-

00944-4 Turaev, V. G. (1994), "Quantum invariants of knots and 3-manifolds", De Gruyter Studies in

Mathematics (Berlin: Walter de Gruyter & Co.) 18, ISBN 3-11-013704-6 Witten, Edward    (1989), "Quantum field theory and the Jones polynomial", Comm. Math. Phys. 

121 (3): 351–399 Zeeman, E. C.    (1963), "Unknotting combinatorial balls", Annals of Mathematics (2) 78: 501–526

Further reading

Introductory textbooks

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is Rolfsen (1976), given in the references. Other good texts from the references are Adams (2001) and Lickorish (1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.

Richard H. Crowell and Ralph Fox,Introduction to Knot Theory, 1977, ISBN 0-387-90272-4 Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, 1985, Walter 

de Gruyter, ISBN 3-11-008675-1 Louis H. Kauffman   , On Knots, 1987, ISBN 0-691-08435-1

Surveys

William W. Menasco and Morwen Thistlethwaite (editors), Handbook of Knot Theory, Amsterdam : Elsevier, 2005. ISBN 0-444-51452-X 

o Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.

Knot Theory in Popular Fiction

Felix Culp, "A Frayed Knot", 2009, Incongruous Press, ISBN 978-144-866-4290 o "A Frayed Knot" introduces Tyler Trefoil as a fraying Bowline knot and amateur 

Topologist investigating the brutal untying of a Clove Hitch named Tether Marlingspike. Trefoil's investigation of the citizen knots of Tide-on-the-Bight leads him to consider a dangerous knot surgery and confront a pathological Wild Knot.