1.
Point groups in three dimensions
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In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups, accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more fixed points. We choose the origin as one of them, the rotation group of an object is equal to its full symmetry group if and only if the object is chiral. Finite Coxeter groups are a set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram, Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E+, which consists of direct isometries, i. e. isometries preserving orientation, it contains those that leave the origin fixed. O is the product of SO and the group generated by inversion. An example would be C4 for H and S4 for M, Thus M is obtained from H by inverting the isometries in H ∖ L. This is clarifying when categorizing isometry groups, see below, in 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of rotations about that axis is a normal subgroup of the group of all rotations about that axis. e. See also the similar overview including translations, when comparing the symmetry type of two objects, the origin is chosen for each separately, i. e. they need not have the same center. Moreover, two objects are considered to be of the symmetry type if their symmetry groups are conjugate subgroups of O. The conjugacy definition would allow a mirror image of the structure, but this is not needed. For example, if a symmetry group contains a 3-fold axis of rotation, there are many infinite isometry groups, for example, the cyclic group generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical abelian groups by adding more rotations around the same axis, there are also non-abelian groups generated by rotations around different axes. They will be infinite unless the rotations are specially chosen, all the infinite groups mentioned so far are not closed as topological subgroups of O
2.
List of spherical symmetry groups
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Spherical symmetry groups are also called point groups in three dimensions, however, this article is limited to the finite symmetries. There are five fundamental symmetry classes which have triangular fundamental domains, dihedral, cyclic, tetrahedral, octahedral and this article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups quaternion algebraic structure, the group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, ±, prefix, which implies a central inversion. The crystallography groups,32 in total, are a subset with element orders 2,3,4 and 6, there are four involutional groups, no symmetry, reflection symmetry, 2-fold rotational symmetry, and central point symmetry. There are four infinite cyclic symmetry families, with n=2 or higher, there are three infinite dihedral symmetry families, with n as 2 or higher. There are three types of symmetry, tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries. Crystallographic point group Triangle group List of planar symmetry groups Point groups in two dimensions Peter R. Cromwell, Polyhedra, Appendix I Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, N. W, johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Finite spherical symmetry groups Weisstein, Eric W. Schoenflies symbol. Weisstein, Eric W. Crystallographic point groups, simplest Canonical Polyhedra of Each Symmetry Type, by David I
3.
Cyclic symmetry in three dimensions
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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 does not change the object. They are the symmetry groups on a cone. For n = ∞ they correspond to four frieze groups, the terms horizontal and vertical imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses and 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. This is the group for a regular n-sided pyramid. Thus, like Dnd, it contains a number of improper rotations without containing the corresponding rotations. for n=1 we have S2, also denoted by Ci, this is inversion symmetry. C2h, and 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, portions of the infinite plane can also be cut and connected into an infinite cylinder. Dihedral symmetry in three dimensions Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups
4.
Polyhedral group
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In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three groups, The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. The reflection symmetries have 6,9, and 15 mirrors respectively, the octahedral symmetry, can be seen as the union of 6 tetrahedral symmetry mirrors, and 3 mirrors of dihedral symmetry Dih2. Pyritohedral symmetry is another doubling of tetrahedral symmetry, S. M. Regular Polytopes, 3rd ed
5.
Tetrahedral symmetry
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A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4, chiral and full are discrete point symmetries. They are among the point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these meet at order 2 and 3 gyration points. T,332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry, there are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the group on 4 elements, in fact it is the group of even permutations of the four 3-fold axes. The three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry and this group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes, td and O are isomorphic as abstract groups, they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion, see also the isometries of the regular tetrahedron. This group has the same axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 axes, and there is an inversion symmetry. Th is isomorphic to T × Z2, every element of Th is either an element of T, apart from these two normal subgroups, there is also a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the product of the normal subgroup of T with Ci. The quotient group is the same as above, of type Z3, the three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals
6.
Octahedral symmetry
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A regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the set of symmetries, since it is the dual of an octahedron. Chiral and full octahedral symmetry are the point symmetries with the largest symmetry groups compatible with translational symmetry. They are among the point groups of the cubic crystal system. But as it is also the direct product S4 × S2, one can identify the elements of S4 as a ∈ [0,4. ). So e. g. the identity is represented as 0, the pairs can be seen in the six files below. Each file is denoted by the m ∈, and the position of each permutation in the file corresponds to the n ∈. A rotoreflection is a combination of rotation and reflection,7 ′ ∘4 =19 ′,7 ′ ∘22 =17 ′, The reflection 7 ′ applied on the 90° rotation 22 gives the 90° rotoreflection 17 ′. O,432, or + of order 24, is chiral octahedral symmetry or rotational octahedral symmetry. This group is like chiral tetrahedral symmetry T, but the C2 axes are now C4 axes, Td and O are isomorphic as abstract groups, they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion, O is the rotation group of the cube and the regular octahedron. Oh, *432, or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry and this group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4. C4, and is the symmetry group of the cube. It is the group for n =3. See also the isometries of the cube, with the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ x ≤ y ≤ z. An object with symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z =1. Ax + by + cz =1 gives a polyhedron with 48 faces, faces are 8-by-8 combined to larger faces for a = b =0 and 6-by-6 for a = b = c. The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6, representing in two orthogonal subsymmetries, D2h, and Td, D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations
7.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves
8.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
9.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
10.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
11.
Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
12.
Orbifold notation
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Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, and their analogues on the hyperbolic plane. e. All translations which occur are assumed to form a subgroup of the group symmetries being described. The symbol ×, which is called a miracle and represents a topological crosscap where a pattern repeats as an image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space, a string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. By abuse of language, we say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way, the exceptional symbol o indicates that there are precisely two linearly independent translations. An orbifold symbol is called if it is not one of the following, p, pq, *p, *pq, for p, q>=2. An object is chiral if its symmetry group contains no reflections, the corresponding orbifold is orientable in the chiral case and non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value, n without or before an asterisk counts as n −1 n n after an asterisk counts as n −12 n asterisk, subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the values is 2, the order is infinite. Indeed, Conways Magic Theorem indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2, otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic, 1* and *1122 and 221 *22 and *221 2* and this is because 1-fold rotation is the empty rotation. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a dimension to the object which does not add or spoil symmetry. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point, thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object, on Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry,42, 475-507,2001, J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups, structural Chemistry,13, 247-257, August 2002
13.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
14.
Rotation
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A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
15.
Frieze group
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A frieze group is a mathematical concept used to classify designs on two-dimensional surfaces that are repetitive in one direction, according to the symmetries of the pattern. Such patterns occur frequently in architecture and decorative art, the mathematical study of such patterns reveals that exactly seven types of symmetry can occur. Frieze groups are two-dimensional line groups, having repetition in only one direction, formally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip, hence a class of groups of isometries of the plane, or of a strip. There are seven frieze groups, listed in the summary table, many authors present the frieze groups in a different order. Thus there are two degrees of freedom for group 1, three for groups 2,3, and 4, and four for groups 5,6, and 7. For two of the seven groups the symmetry groups are singly generated, for four they have a pair of generators. A symmetry group in frieze group 1,2,3, a symmetry group in frieze group 4 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. This last frieze group contains the groups of the simplest periodic patterns in the strip. Therefore, in a way, this frieze group contains the largest symmetry groups, the inclusion of the discrete condition is to exclude the group containing all translations, and groups containing arbitrarily small translations. Even apart from scaling and shifting, there are many cases. The inclusion of the condition is to exclude groups that have no translations. The group consisting of the identity and reflection in the horizontal axis, each of these subgroups is the symmetry group of a frieze pattern, and sample patterns are shown in Fig.1. The seven different groups correspond to the 7 infinite series of point groups in three dimensions, with n = ∞. As we have seen, up to isomorphism, there are four groups, the groups can be classified by their type of two-dimensional grid or lattice. The lattice being oblique means that the second direction need not be orthogonal to the direction of repeat, symmetry groups in one dimension Line group Rod group Wallpaper group Space group There exist software graphic tools that create 2D patterns using frieze groups. Usually, the pattern is updated automatically in response to edits of the original strip. Kali, a free and open source application for wallpaper, frieze. Kali, free downloadable Kali for Windows and Mac Classic, tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings
16.
Pyramidal symmetry
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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 does not change the object. They are the symmetry groups on a cone. For n = ∞ they correspond to four frieze groups, the terms horizontal and vertical imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses and 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. This is the group for a regular n-sided pyramid. Thus, like Dnd, it contains a number of improper rotations without containing the corresponding rotations. for n=1 we have S2, also denoted by Ci, this is inversion symmetry. C2h, and 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, portions of the infinite plane can also be cut and connected into an infinite cylinder. Dihedral symmetry in three dimensions Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups
17.
Reflection symmetry
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Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry, in 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its image is called mirror symmetric. The set of operations that preserve a property of the object form a group. Two objects are symmetric to each other with respect to a group of operations if one is obtained from the other by some of the operations. Another way to think about the function is that if the shape were to be folded in half over the axis. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match, a circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles, quadrilaterals with reflection symmetry are kites, deltoids, rhombuses, and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges, for an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between 2/3 and 1 for any convex shape, for each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically C2. The fundamental domain is a half-plane or half-space, in certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry, in contexts in modern physics the term parity or P-symmetry is used for both. For more general types of reflection there are more general types of reflection symmetry. For example, with respect to a non-isometric affine involution with respect to circle inversion, most animals are bilaterally symmetric, likely because this supports forward movement and streamlining. Mirror symmetry is used in architecture, as in the facade of Santa Maria Novella. It is also found in the design of ancient structures such as Stonehenge, Symmetry was a core element in some styles of architecture, such as Palladianism. Patterns in nature Point reflection symmetry Stewart, Ian, weidenfeld & Nicolson. is potty Weyl, Hermann. Mapping with symmetry - source in Delphi Reflection Symmetry Examples from Math Is Fun
18.
Rotoreflection
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In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis. Therefore it is called a rotoinversion or rotary inversion. A three-dimensional symmetry that has one fixed point is necessarily an improper rotation. In both cases the operations commute, rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection. An improper rotation of an object produces a rotation of its mirror image. The axis is called the rotation-reflection axis and this is called an n-fold improper rotation if the angle of rotation is 360°/n. The notation Sn denotes the group generated by an n-fold improper rotation. The notation n ¯ is used for n-fold rotoinversion, i. e. rotation by an angle of rotation of 360°/n with inversion, the Coxeter notation for S2n is, and orbifold notation is n×, order 2n. The direct subgroup, index 2, is Cn, +, order n, S2n for odd n contain inversion, with S2 = Ci is the group generated by inversion. S2n contain indirect isometries but not inversion for even n, in general, if odd p is a divisor of n, then S2n/p is a subgroup of S2n. For example S4 is a subgroup of S12, in a wider sense, an improper rotation may be defined as any indirect isometry, i. e. an element of E\E+, thus it can also be a pure reflection in a plane, or have a glide plane. An indirect isometry is a transformation with an orthogonal matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a rotation is defined as a direct isometry, i. e. an element of E+, it can also be the identity. A direct isometry is a transformation with an orthogonal matrix that has a determinant of 1. In either the narrower or the senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation
19.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
20.
Bipyramid
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An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices, the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points above and below the centroid of its base, nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a concave bipyramid has a concave interior polygon. The face-transitive regular bipyramids are the dual polyhedra of the uniform prisms, a bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh, the volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore, only three kinds of bipyramids can have all edges of the same length, the triangular, tetragonal, and pentagonal bipyramids. The rotation group is Dn of order 2n, except in the case of an octahedron, which has the larger symmetry group O of order 24. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of symmetry in three dimensions, Dnh, order 4n. The reflection domains can be shown as alternately colored triangles as mirror images, a scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. In one type the 2n vertices around the center alternate in rings above, in the other type, the 2n vertices are on the same plane, but alternate in two radii. The first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, in crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra, the second has symmetry Dn, order 2n. The smallest scalenohedron has 8 faces and is identical to the regular octahedron. The second type is a rhombic bipyramid, the first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z =0, and becoming a disphenoid with merged coplanar faces at z =1. For z >1, it becomes concave, self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points
21.
Antiprism
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In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a type of snub polyhedra, Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular n-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained when the line connecting the centers is perpendicular to the base planes. As faces, it has the two bases and, connecting those bases, 2n isosceles triangles. A uniform antiprism has, apart from the faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form a series of vertex-uniform polyhedra. For n =2 we have as degenerate case the regular tetrahedron as a digonal antiprism, the dual polyhedra of the antiprisms are the trapezohedra. Let a be the edge-length of a uniform antiprism, then the volume is V = n 4 cos 2 π2 n −1 sin 3 π2 n 12 sin 2 π n a 3 and the surface area is A = n 2 a 2. There are a set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron. These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower form of the icosahedron. The symmetry group contains inversion if and only if n is odd, uniform star antiprisms are named by their star polygon bases, and exist in prograde and retrograde solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/ instead of p/q, in the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry. Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, star antiprism compounds also can be constructed where p and q have common factors, thus a 10/4 antiprism is the compound of two 5/2 star antiprisms. Prism Apeirogonal antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, California, University of California Press Berkeley. Chapter 2, Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Antiprism, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms
22.
Trapezohedron
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The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites, the n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces, an n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism. These figures, sometimes called deltohedra, must not be confused with deltahedra, in texts describing the crystal habits of minerals, the word trapezohedron is often used for the polyhedron properly known as a deltoidal icositetrahedron. In the case of the dual of a triangular antiprism the kites are rhombi and they are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces, a special case of a rhombohedron is one in the which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra, a degenerate form, n =2, form a geometric tetrahedron with 6 vertices,8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a form of antiprism, also a tetrahedron. The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the symmetry group Od of order 48. The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, if the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n. Crystal arrangements of atoms can repeat in space with trapezohedral cells, the pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired, two dice of different colors are typically used for the two digits to represent numbers from 00 to 99. Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to two points. Diminished trapezohedron Rhombic dodecahedron Rhombic triacontahedron Bipyramid Conway polyhedron notation Anthony Pugh, California, University of California Press Berkeley. Chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <3> <4> <5> <6> <7> <8> <9> <10> Conway Notation for Polyhedra Try, dAn, where n=3,4,5. Example dA5 is a pentagonal trapezohedron
23.
Klein four-group
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In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884, with four elements, the Klein four-group is the smallest non-cyclic group, and the cyclic group of order 4 and the Klein four-group are, up to isomorphism, the only groups of order 4. The smallest non-abelian group is the group of degree 3. The Klein groups Cayley table is given by, The Klein four-group is also defined by the group presentation V = ⟨ a, b ∣ a 2 = b 2 =2 = e ⟩. All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation, the Klein four-group is the smallest non-cyclic group. It is however a group, and isomorphic to the dihedral group of order 4, Dih2, other than the group of order 2. The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the pairs under component-wise addition modulo 2, the Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. Another numerical construction of the Klein four-group is the set, with the operation being multiplication modulo 8, here a is 3, b is 5, and c = ab is 3 ×5 =15 ≡7. The three elements of two in the Klein four-group are interchangeable, the automorphism group of V is the group of permutations of these three elements. In fact, it is the kernel of a group homomorphism from S4 to S3. In the construction of finite rings, eight of the rings with four elements have the Klein four-group as their additive substructure. The quotient group / is isomorphic to the Klein four-group, in a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group. The Klein four-group as a subgroup of the alternating group A4 is not the group of any simple graph. It is, however, the group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. A. Armstrong Groups and Symmetry, Springer Verlag, page 53, W. E. Barnes Introduction to Abstract Algebra, D. C
24.
Cuboid
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In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
25.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
26.
Basketball
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Basketball is a non-contact team sport played on a rectangular court by two teams of five players each. The objective is to shoot a ball through a hoop 18 inches in diameter and 10 feet high that is mounted to a backboard at each end of the court. The game was invented in 1891 by Dr. James Naismith, a team can score a field goal by shooting the ball through the basket being defended by the opposition team during regular play. A field goal scores three points for the team if the player shoots from behind the three-point line. A team can also score via free throws, which are worth one point, the team with the most points at the end of the game wins, but additional time is mandated when the score is tied at the end of regulation. The ball can be advanced on the court by passing it to a teammate and it is a violation to lift, or drag, ones pivot foot without dribbling the ball, to carry it, or to hold the ball with both hands then resume dribbling. The game has many techniques for displaying skill—ball-handling, shooting, passing, dribbling, dunking, shot-blocking. The point guard directs the on court action of the team, implementing the coachs game plan, Basketball is one of the worlds most popular and widely viewed sports. Outside North America, the top clubs from national leagues qualify to continental championships such as the Euroleague, the FIBA Basketball World Cup attracts the top national teams from around the world. Each continent hosts regional competitions for teams, like EuroBasket. The FIBA Womens Basketball World Cup features the top womens basketball teams from continental championships. The main North American league is the WNBA, whereas the EuroLeague Women has been dominated by teams from the Russian Womens Basketball Premier League, in early December 1891, Canadian Dr. He sought a vigorous indoor game to keep his students occupied, after rejecting other ideas as either too rough or poorly suited to walled-in gymnasiums, he wrote the basic rules and nailed a peach basket onto a 10-foot elevated track. Basketball was originally played with a soccer ball and these laces could cause bounce passes and dribbling to be unpredictable. Eventually a lace-free ball construction method was invented, and this change to the game was endorsed by Naismith, dribbling was not part of the original game except for the bounce pass to teammates. Passing the ball was the means of ball movement. Dribbling was eventually introduced but limited by the shape of early balls. Dribbling only became a part of the game around the 1950s
27.
Baseball
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Baseball is a bat-and-ball game played between two teams of nine players each, who take turns batting and fielding. A run is scored when a player advances around the bases, Players on the batting team take turns hitting against the pitcher of the fielding team, which tries to prevent runs by getting hitters out in any of several ways. A player on the team who reaches a base safely can later attempt to advance to subsequent bases during teammates turns batting. The teams switch between batting and fielding whenever the team records three outs. One turn batting for both teams, beginning with the team, constitutes an inning. A game is composed of nine innings, and the team with the number of runs at the end of the game wins. Baseball has no clock, although almost all games end in the ninth inning. Baseball evolved from older bat-and-ball games already being played in England by the mid-18th century and this game was brought by immigrants to North America, where the modern version developed. By the late 19th century, baseball was widely recognized as the sport of the United States. Baseball is now popular in North America and parts of Central and South America, the Caribbean, in the United States and Canada, professional Major League Baseball teams are divided into the National League and American League, each with three divisions, East, West, and Central. The major league champion is determined by playoffs that culminate in the World Series, the top level of play is similarly split in Japan between the Central and Pacific Leagues and in Cuba between the West League and East League. The evolution of baseball from older bat-and-ball games is difficult to trace with precision, a French manuscript from 1344 contains an illustration of clerics playing a game, possibly la soule, with similarities to baseball. Other old French games such as thèque, la balle au bâton, consensus once held that todays baseball is a North American development from the older game rounders, popular in Great Britain and Ireland. Baseball Before We Knew It, A Search for the Roots of the Game, by David Block, suggests that the game originated in England, recently uncovered historical evidence supports this position. Block argues that rounders and early baseball were actually regional variants of other. It has long believed that cricket also descended from such games. The earliest known reference to baseball is in a 1744 British publication, A Little Pretty Pocket-Book, David Block discovered that the first recorded game of Bass-Ball took place in 1749 in Surrey, and featured the Prince of Wales as a player. William Bray, an English lawyer, recorded a game of baseball on Easter Monday 1755 in Guildford and this early form of the game was apparently brought to Canada by English immigrants
28.
Pentagrammic prism
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In geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. This polyhedron is identified with the indexed name U78 as a uniform polyhedron and it is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Note that the face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior depending on how interior is defined, one definition of interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the perimeter. In either case, it is best to show the boundary line to distinguish it from a concave decagon
29.
Pentagrammic antiprism
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In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. This polyhedron is identified with the indexed name U79 as a uniform polyhedron, an alternative representation with hollow centers to the pentagrams. Net, Prismatic uniform polyhedron Pentagrammic prism Pentagrammic crossed-antiprism Weisstein, Eric W. Pentagrammic antiprism
30.
Snub square antiprism
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In geometry, the snub square antiprism is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform and they were named by Norman Johnson, who first listed these polyhedra in 1966. The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, similarly constructed the ss is a snub triangular antiprism, and result as a regular icosahedron. A snub pentagonal antiprism, ss, or higher n-antiprisms can be similar constructed, the preceding Johnson solid, the snub disphenoid also fits constructionally as ss, but you have to retain two degenerate digonal faces in the digonal antiprism. Eric W. Weisstein, Snub square antiprism at MathWorld
31.
Pentagonal antiprism
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In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two joined to each other by a ring of 10 triangles for a total of 12 faces. Hence, it is a non-regular dodecahedron, if the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. The two pentagonal faces of either shape can be augmented with pyramids to form the icosahedron, the pentagonal antiprism occurs as a constituent element in some higher-dimensional polytopes. Two rings of 10 pentagonal antiprisms each bound the hypersurface of the 4-dimensional grand antiprism, if these antiprisms are augmented with pentagonal prism pyramids and linked with rings of 5 tetrahedra each, the 600-cell is obtained. The pentagonal antiprism can be truncated and alternated to form a snub antiprism, Weisstein, pentagonal Antiprism, Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A5
32.
Pentagrammic crossed-antiprism
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In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It differs from the pentagrammic antiprism by having opposite orientations on the two pentagrams and this polyhedron is identified with the indexed name U80 as a uniform polyhedron. The pentagrammic crossed-antiprism may be inscribed within an icosahedron, and has ten triangular faces in common with the great icosahedron and it has the same vertex arrangement as the pentagonal antiprism. In fact, it may be considered as a great icosahedron. Prismatic uniform polyhedron Weisstein, Eric W. Pentagrammic crossed antiprism
33.
Pentagonal trapezohedron
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The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces which are congruent kites and it can be decomposed into two pentagonal pyramids and a pentagonal antiprism in the middle. It can also be decomposed into two pentagonal pyramids and a dodecahedron in the middle, the pentagonal trapezohedron was patented for use as a gaming die in 1906. Subsequent patents on ten-sided dice have made minor refinements to the design by rounding or truncating the edges. This enables the die to tumble so that the outcome is less predictable, one such refinement became notorious at the 1980 Gen Con when the patent was incorrectly thought to cover ten-sided dice in general. Ten-sided dice are commonly numbered from 0 to 9, as this allows two to be rolled in order to obtain a percentile result. Where one die represents the tens, the other represents units therefore a result of 7 on the former and 0 on the latter would be combined to produce 70, a result of double-zero is commonly interpreted as 100. Ten-sided dice may also be numbered 1 to 10 for use in games where a number in this range is desirable. A fairly consistent arrangement of the faces on ten-digit dice has been observed, the even and odd digits are divided among the two opposing caps of the die, and each pair of opposite faces adds to nine. When casting a 10-sided die, if numbered from 0-9, two are used to obtain a percentage roll. Rolling 2 of these are attributed in the results 00-99, where 00 can be viewed as a 100 as the result in some games. Alone casting a 0-9 ten sided dice, the 0 face is valued at 10, cundy H. M and Rollett, A. P. Mathematical models, 2nd Edn. Oxford University Press, p.117 Generalized formula of uniform polyhedron having 2n congruent right kite faces from Academia. edu Weisstein, virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, dA5
34.
Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
35.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
36.
Norman Johnson (mathematician)
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Norman Woodason Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. He earned his Ph. D. from the University of Toronto in 1966 with a title of The Theory of Uniform Polytopes. In 1966 he enumerated 92 convex non-uniform polyhedra with regular faces, victor Zalgaller later proved that Johnsons list was complete, and the set is now known as the Johnson solids. The theory of polytopes and honeycombs, Ph. D. Dissertation,1966 Hyperbolic Coxeter Groups, paper, convex polyhedra with regular faces, paper containing the original enumeration of the 92 Johnson solids and the conjecture that there are no others. Norman W. Johnson at the Mathematics Genealogy Project Norman W. Johnson Endowed Fund in Mathematics and Computer Science at Wheaton College
37.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers