Cyclic symmetry in three dimensions
In three dimensional geometry, there are four infinite series of point groups in three dimensions with n-fold rotational or reflectional symmetry about one axis that does not change the object. They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used; the terms horizontal and vertical imply the existence and direction of reflections with respect to a vertical axis of symmetry. Shown are Coxeter notation in brackets, and, in parentheses, orbifold notation. Chiral Cn, +, of order n - n-fold rotational symmetry - acro-n-gonal group, it has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis. Cnv, of order 2n - pyramidal symmetry or full acro-n-gonal group. For n=1 we have again Cs, it has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid. S2n, of order 2n - gyro-n-gonal group. Thus, like Dnd, it contains a number of improper rotations without containing the corresponding rotations. for n=1 we have S2 denoted by Ci.
C2h, C2v, of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C2v applies e.g. for a rectangular tile with its top side different from its bottom side. In the limit these four groups represent Euclidean plane frieze groups as C∞, C∞h, C∞v, S∞. Rotations become translations in the limit. Portions of the infinite plane can be cut and connected into an infinite cylinder. Dihedral symmetry in three dimensions Sands, Donald E.. "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3. On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 N.
W. Johnson: Geometries and Transformations, ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups
Ball (association football)
A football, soccer ball, or association football ball is the ball used in the sport of association football. The name of the ball varies according to whether the sport is called "football", "soccer", or "association football"; the ball's spherical shape, as well as its size and material composition, are specified by Law 2 of the Laws of the Game maintained by the International Football Association Board. Additional, more stringent, standards are specified by FIFA and subordinate governing bodies for the balls used in the competitions they sanction. Early footballs began as animal bladders or stomachs that would fall apart if kicked too much. Improvements became possible in the 19th century with the introduction of rubber and discoveries of vulcanization by Charles Goodyear; the modern 32-panel ball design was developed in 1962 by Eigil Nielsen, technological research continues today to develop footballs with improved performance. The 32-panel ball design was soon overcome by 24-panel balls as well as 42-panel balls, both of which improved performance compared to before, in 2007.
A black-and-white patterned truncated icosahedron design, brought to prominence by the Adidas Telstar, has become an icon of the sport. Many different designs of balls exist, varying both in physical characteristics. In the year 1863, the first specifications for footballs were laid down by the Football Association. Previous to this, footballs were made out of inflated leather, with leather coverings to help footballs maintain their shapes. In 1872 the specifications were revised, these rules have been left unchanged as defined by the International Football Association Board. Differences in footballs created since this rule came into effect have been to do with the material used in their creation. Footballs have gone through a dramatic change over time. During medieval times balls were made from an outer shell of leather filled with cork shavings. Another method of creating a ball was using animal bladders for the inside of the ball making it inflatable. However, these two styles of creating footballs made it easy for the ball to puncture and were inadequate for kicking.
It was not until the 19th century. In 1838, Charles Goodyear introduced vulcanized rubber, which improved the football. Vulcanisation is the treatment of rubber to give it certain qualities such as strength and resistance to solvents. Vulcanisation of rubber helps the football resist moderate heat and cold. Vulcanisation helped create inflatable bladders that pressurize the outer panel arrangement of the football. Charles Goodyear's innovation made it easier to kick. Most balls of this time had tanned leather with eighteen sections stitched together; these were arranged in six panels of three strips each. During the 1900s, footballs were made out of leather with a lace of the same material used to stitch the panels. Although leather was perfect for bouncing and kicking the ball, when heading the football it was painful; this problem was most due to water absorption of the leather from rain, which caused a considerable increase in weight, causing head or neck injury. By around 2017, this had been associated with dementia in former players.
Another problem of early footballs was that they deteriorated as the leather used in manufacturing the footballs varied in thickness and in quality. The ball without the leather lace was developed and patented by Romano Polo, Antonio Tossolini and Juan Valbonesi in 1931 in Argentina; this innovative ball would be adopted by the Argentine Football Association as the official ball for its competitions since 1932. Elements of the football that today are tested are the deformation of the football when it is kicked or when the ball hits a surface. Two styles of footballs have been tested by the Sports Technology Research Group of Wolfson School of Mechanical and Manufacturing Engineering in Loughborough University; the basic model considered the ball as being a spherical shell with isotropic material properties. The developed model utilised isotropic material properties but included an additional stiffer stitching seam region. Companies such as Umbro, Adidas, Nike and Puma are releasing footballs made out of new materials which are intended to provide more accurate flight and more power to be transferred to the football.
Today's footballs are more complex than past footballs. Most modern footballs consist of twelve regular pentagonal and twenty regular hexagonal panels positioned in a truncated icosahedron spherical geometry; some premium-grade 32-panel balls use non-regular polygons to give a closer approximation to sphericality. The inside of the football is made up of a latex bladder which enables the football to be pressurised; the ball's panel pairs are stitched along the edge. The size of a football is 22 cm in diameter for a regulation size 5 ball. Rules state. Averaging that to 69 cm and dividing by π gives about 22 cm for a diameter; the ball's weight must be in the range of 410 to 450 grams and inflated to a pressure of between 0.6 and 1.1 standard atmospheres at sea level. There are a number of different types of football balls depending on the match and turf including training footballs, match footballs, professional match footballs, beach footballs, street footballs, indoor footballs, turf balls, futsa
Crystal system
In crystallography, the terms crystal system, crystal family, lattice system each refer to one of several classes of space groups, point groups, or crystals. Informally, two crystals are in the same crystal system if they have similar symmetries, although there are many exceptions to this. Crystal systems, crystal families and lattice systems are similar but different, there is widespread confusion between them: in particular the trigonal crystal system is confused with the rhombohedral lattice system, the term "crystal system" is sometimes used to mean "lattice system" or "crystal family". Space groups and crystals are divided into seven crystal systems according to their point groups, into seven lattice systems according to their Bravais lattices. Five of the crystal systems are the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems; the six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, orthorhombic, rhombohedral and cubic. In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems and hexagonal, because both exhibit threefold rotational symmetry; these point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, orthorhombic, trigonal and cubic. A crystal family is determined by lattices and point groups, it is formed by combining crystal systems which have space groups assigned to a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which are combined into one hexagonal crystal family.
In total there are six crystal families: triclinic, orthorhombic, tetragonal and cubic. Spaces with less than three dimensions have the same number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system. In 2D space, there are four crystal systems: oblique, rectangular and hexagonal; the relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table: Note: there is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used; the 7 crystal systems consist of 32 crystal classes as shown in the following table: The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, reflect them all through a single point, so that becomes; this is the'inverted structure'. If the original structure and inverted structure are identical the structure is centrosymmetric. Otherwise it is non-centrosymmetric.
Still in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic. A direction is called polar if its two directional senses are physically different. A symmetry direction of a crystal, polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis; some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent; the crystal structures of chiral biological molecules can only occur in the 65 enantiomorphic space groups.
The distribution of the 14 Bravais lattices into lattice systems and crystal families is given in the following table. In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups in three directions; such symmetry groups consist of translations by vectors of the form R = n1a1 + n2a2 + n3a3,where n1, n2, n3 are integers and a1, a2, a3 are three non-coplanar vectors, called primitive vectors. These lattices are classified by the space group of the lattice itself, viewed as a collection of points, they represent the maximum symmetry. All crystalline materials must, by definition, fit into one of these arrangements. For convenience a Bravais lattice is depicted by a unit cell, a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48; the Bravais lattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there we
William Rowan Hamilton
Sir William Rowan Hamilton MRIA was an Irish mathematician. While still an undergraduate he was appointed Andrews professor of Astronomy and Royal Astronomer of Ireland, lived at Dunsink Observatory, he made important contributions to classical mechanics and algebra. Although Hamilton was not a physicist–he regarded himself as a pure mathematician–his work was of major importance to physics his reformulation of Newtonian mechanics, now called Hamiltonian mechanics; this work has proven central to the modern study of classical field theories such as electromagnetism, to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. Hamilton is said to have shown immense talent at a early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton,'This young man, I do not say will be, but is, the first mathematician of his age.' William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions, solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions, linear operators on quaternions and proving a result for linear operators on the space of quaternions.
Hamilton invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex once. Hamilton was the fourth of nine children born to Sarah Hutton and Archibald Hamilton, who lived in Dublin at 29 Dominick Street renumbered to 36. Hamilton's father, from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath, his uncle soon discovered that Hamilton had a remarkable ability to learn languages, from a young age, had displayed an uncanny ability to acquire them. At the age of seven, he had made considerable progress in Hebrew, before he was thirteen he had acquired, under the care of his uncle as many languages as he had years of age; these included the classical and modern European languages, Persian, Hindustani and Marathi and Malay. He retained much of his knowledge of languages to the end of his life reading Persian and Arabic in his spare time, although he had long since stopped studying languages, used them just for relaxation.
In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton; the two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor. In reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18; the college awarded him off-the-chart grades. He studied both classics and mathematics, was appointed Professor of Astronomy just prior to his graduation, he took up residence at Dunsink Observatory where he spent the rest of his life. While attending Trinity College, Hamilton proposed to his friend's sister. Hamilton, being a sensitive young man, became sick and depressed, committed suicide, he was rejected again in 1831 by Aubrey De Vere. Luckily, Hamilton found a woman, she was Helen Marie Bayly, a country preacher's daughter, they married in 1833.
Hamilton had three children with Bayly: William Edwin Hamilton, Archibald Henry, Helen Elizabeth. Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy and chronically ill. Hamilton made important contributions to classical mechanics, his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of "Caustics" in 1824 to the Royal Irish Academy, it was referred as usual to a committee. While their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size by the additional details that the committee had suggested, but it became more intelligible, the features of the new method were now seen. Until this period Hamilton himself seems not to have understood either the nature or importance of optics, as he intended to apply his method to dynamics. In 1827, Hamilton presented a theory of a single function, now known as Hamilton's principal function, that brings together mechanics and mathematics, which helped to establish the wave theory of light.
He proposed it when he first predicted its existence in the third supplement to his "Systems of Rays", read in 1832. The Royal Irish Academy paper was entitled "Theory of Systems of Rays", the first part was printed in 1828 in the Transactions of the Royal Irish Academy; the more important contents of the second and third parts appeared in the three voluminous supplements (to the first
Truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. It has 20 regular hexagonal faces, 60 vertices and 90 edges, it is 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs patterned with white hexagons and black pentagons. Geodesic domes such as those whose architecture Buckminster Fuller pioneered are based on this structure, it corresponds to the geometry of the fullerene C60 molecule. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb; this polyhedron can be constructed from an icosahedron with the 12 vertices truncated such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, leaves the original 20 triangle faces as regular hexagons, thus the length of the edges is one third of that of the original edges. Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all permutations of: where φ = 1 + √5/2 is the golden mean.
The circumradius is √9φ + 10 ≈ 4.956 and the edges have length 2. The truncated icosahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, two types of faces: hexagonal and pentagonal; the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection; this projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere is: r u = a 2 1 + 9 φ 2 = a 4 58 + 18 5 ≈ 2.478 018 66 a where φ is the golden ratio. This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron as the starting point for our considerations; the angle between the segments joining the center and the vertices connected by shared edge is 23.281446°. The area A and the volume V of the truncated icosahedron of edge length a are: A = a 2 ≈ 72.607 253 a 2 V = 125 + 43 5 4 a 3 ≈ 55.287 7308 a 3.
With unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron demonstrates the Euler characteristic: 32 + 60 − 90 = 2; the balls used in association football and team handball are the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball; this ball type was introduced to the World Cup in 1970. Geodesic domes are based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the honeycomb wheels used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix; this shape was the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs.
The truncated icosahedron can be described as a model of the Buckminsterfullerene, or "buckyball," molecule, an allotrope of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and about 0.71 nm hence the size ratio is ≈31,000,000:1. In popular craft culture, large sparkleballs can be made using a icosahedron pattern and plastic, styrofoam or paper cups. A truncated icosahedron with "solid edges" by Leonardo da Vinci appears as an illustration in Luca Pacioli's book De divina proportione; these uniform star-polyhedra, one icosahedral stellation have nonuniform truncated icosa
Cyclic group
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group with order |G| = n, this means G =, where e is the identity element and gj = gk whenever j ≡ k modulo n. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.
Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z,with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.
If p is a prime Z/pZ is a finite field, is denoted Fp or GF. For every positive integer n, the set of the integers modulo n that are prime to n is written as ×; this group is not always cyclic, bu
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form