1.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
2.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two
3.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph
4.
Conway polyhedron notation
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In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation defined by Kepler, the basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example tC represents a cube, and taC, parsed as t, is a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like a cube is an octahedron. Applied in a series, these allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology, while exact geometry is not constrained, the seed polyhedra are the Platonic solids, represented by the first letter of their name, the prisms for n-gonal forms, antiprisms, cupolae and pyramids. Any polyhedron can serve as a seed, as long as the operations can be executed on it, for example regular-faced Johnson solids can be referenced as Jn, for n=1.92. In general, it is difficult to predict the appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice becomes the same as the operation, aa=e, while a truncation after ambo produces bevel. There has been no general theory describing what polyhedra can be generated in by any set of operators, instead all results have been discovered empirically. Elements are given from the seed to the new forms, assuming seed is a polyhedron, An example image is given for each operation. The basic operations are sufficient to generate the reflective uniform polyhedra, some basic operations can be made as composites of others. Special forms The kis operator has a variation, kn, which only adds pyramids to n-sided faces, the truncate operator has a variation, tn, which only truncates order-n vertices. The operators are applied like functions from right to left, for example, a cuboctahedron is an ambo cube, i. e. t = aC, and a truncated cuboctahedron is t = t = taC. Chirality operator r – reflect – makes the image of the seed. Alternately an overline can be used for picking the other chiral form, the operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices. The first row generates the Archimedean solids and the row the Catalan solids. Comparing each new polyhedron with the cube, each operation can be visually understood, the truncated icosahedron, tI or zD, which is Goldberg polyhedron G, creates more polyhedra which are neither vertex nor face-transitive
5.
Wythoff symbol
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In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra, a Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators, with a slight extension, Wythoffs symbol can be applied to all uniform polyhedra. However, the methods do not lead to all uniform tilings in euclidean or hyperbolic space. In three dimensions, Wythoffs construction begins by choosing a point on the triangle. If the distance of this point from each of the sides is non-zero, a perpendicular line is then dropped between the generator point and every face that it does not lie on. The three numbers in Wythoffs symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, the triangle is also represented with the same numbers, written. In this notation the mirrors are labeled by the reflection-order of the opposite vertex, the p, q, r values are listed before the bar if the corresponding mirror is active. The one impossible symbol | p q r implies the point is on all mirrors. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, the resulting figure has rotational symmetry only. The generator point can either be on or off each mirror and this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. A node is circled if the point is not on the mirror. There are seven generator points with each set of p, q, r, | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isnt Wythoff-constructible, There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the many such patterns in the hyperbolic plane are also listed. The list of Schwarz triangles includes rational numbers, and determine the set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a domain, colored by even. Selected tilings created by the Wythoff construction are given below, for a more complete list, including cases where r ≠2, see List of uniform polyhedra by Schwarz triangle
6.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
7.
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
8.
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
9.
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
10.
Dihedral angle
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A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common, in solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes, a dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection. A torsion angle is an example of a dihedral angle. In stereochemistry every set of three atoms of a molecule defines a plane, when two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation, stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar. The synperiplanar conformation is also known as the syn- or cis-conformation, antiperiplanar as anti or trans, for example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with an angle of 60° is less stable than the anti-configuration with a dihedral angle of 180°. For macromolecular usage the symbols T, C, G+, G−, A+, a Ramachandran plot, originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure, the figure at right illustrates the definition of the φ and ψ backbone dihedral angles. In a protein chain three dihedral angles are defined as φ, ψ and ω, as shown in the diagram, the planarity of the peptide bond usually restricts ω to be 180° or 0°. The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, the cis isomer is mainly observed in Xaa–Pro peptide bonds. The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche+, the stability of certain sidechain dihedral angles is affected by the values φ and ψ. For instance, there are steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = − n A ⋅ n B | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can also be described by two non-collinear vectors lying in that plane, taking their cross product yields a vector to the plane
11.
Harold Scott MacDonald 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
12.
List of Wenninger polyhedron models
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This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building and it contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex regular and quasiregular polyhedra. This list was written to honor this early work from Wenninger. Models listed here can be cited as Wenninger Model Number N, the polyhedra are grouped in 5 tables, Regular, Semiregular, regular star polyhedra, Stellations and compounds, and uniform star polyhedra. The four regular polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings. List of uniform polyhedra The fifty nine icosahedra Wenninger, Magnus, errata In Wenninger, the vertex figure for W90 is incorrectly shown as having parallel edges. Magnus J. Wenninger Software used to generate images in this article, Stella, Polyhedron Navigator Stella - Can create and print nets for all of Wenningers polyhedron models
13.
Convex polyhedron
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -manifold, Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
14.
Quasiregular polyhedron
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In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive, there are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. These forms representing a pair of a figure and its dual can be given a vertical Schläfli symbol or r to represent their containing the faces of both the regular and dual regular. A quasiregular polyhedron with this symbol will have a vertex configuration p. q. p. q, more generally, a quasiregular figure can have a vertex configuration r, representing r instances of the faces around the vertex. Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling,2. Or more generally,2, with 1/p+1/q<1/2, a regular figure with Schläfli symbol can be quasiregular, with vertex configuration q/2, if q is even. The octahedron can be considered quasiregular as a tetratetrahedron,2, similarly the square tiling 2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces,3, Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r. In this form it is known as the tetratetrahedron. The remaining convex polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram Each of these forms the core of a dual pair of regular polyhedra. The names of two of these clues to the associated dual pair, respectively the cube + octahedron. The octahedron is the core of a pair of tetrahedra. Each of these quasiregular polyhedra can be constructed by an operation on either regular parent, truncating the edges fully. This sequence continues as the tiling, vertex figure 2 - a quasiregular tiling based on the triangular tiling. But not everybody uses this terminology and these duals are transitive on their edges and faces, they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above, The rhombic dodecahedron, the rhombic triacontahedron, with two types of alternating vertices,20 with three rhombic faces, and 12 with five rhombic faces. In addition, by duality with the octahedron, the cube and their face configuration are of the form V3. n.3. n, and Coxeter-Dynkin diagram These three quasiregular duals are also characterised by having rhombic faces
15.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
16.
Rhombic triacontahedron
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In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types and it is a Catalan solid, and the dual polyhedron of the icosidodecahedron. The ratio of the diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1 = tan−1. A rhombus so obtained is called a golden rhombus, being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids and it contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra, the plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance. Using one of the three golden rectangles drawn into the inscribed icosahedron we can easily deduce the distance between the center of the solid and the center of its rhombic face. The rhombic triacontahedron can be dissected into 20 golden rhombohedra,10 acute ones and 10 flat ones, danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron, the simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube. Roger von Oechs Ball of Whacks comes in the shape of a rhombic triacontahedron, the rhombic triacontahedron is used as the d30 thirty-sided die, sometimes useful in some roleplaying games or other places. The rhombic triacontahedron has three positions, two centered on vertices, and one mid-edge. Embedded in projection 10 are the fat rhombus and skinny rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling, the rhombic triacontahedron has over 227 stellations. This polyhedron is a part of a sequence of rhombic polyhedra, the cube can be seen as a rhombic hexahedron where the rhombi are also rectangles. The rhombic triacontahedron forms the hull of one projection of a 6-cube to 3 dimensions. Truncated rhombic triacontahedron Rhombille tiling Golden rhombus Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design
17.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also 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 classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. 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 often defined in terms of polar reciprocation about a concentric sphere. 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 reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, 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. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, 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 stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
18.
Net (polyhedron)
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In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are an aid to the study of polyhedra and solid geometry in general. Many different nets can exist for a polyhedron, depending on the choices of which edges are joined. Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded, additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra. Shephard asked whether every convex polyhedron has at least one net and this question, which is also known as Dürers conjecture, or Dürers unfolding problem, remains unanswered. There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron so that the set of subdivided faces has a net, in 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation. The shortest path over the surface between two points on the surface of a polyhedron corresponds to a line on a suitable net for the subset of faces touched by the path. The net has to be such that the line is fully within it. Other candidates for the shortest path are through the surface of a third face adjacent to both, and corresponding nets can be used to find the shortest path in each category, the geometric concept of a net can be extended to higher dimensions. The above net of the tesseract, the hypercube, is used prominently in a painting by Salvador Dalí. However, it is known to be possible for every convex uniform 4-polytope, Paper model Cardboard modeling UV mapping Weisstein, Eric W. Net. Regular 4d Polytope Foldouts Editable Printable Polyhedral Nets with an Interactive 3D View Paper Models of Polyhedra Unfolder for Blender Unfolding package for Mathematica
19.
Hoberman sphere
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Colorful plastic versions have become popular as childrens toys, several toy sizes exist, with the original design capable of expanding from 15 centimetres in diameter to 76 centimetres. A Hoberman sphere typically consists of six great circles corresponding to the edges of an icosidodecahedron, the Hoberman sphere can be unfolded by allowing certain members to spread apart. This can be accomplished by feeding out a string or cable in the larger models, the operation of each joint is linked to all the others in a manner conceptually similar to the extension arm on a wall-mounted shaving mirror. The largest existing Hoberman sphere is in the AHHAA Science Center in Tartu, fully expanded, it is 5.9 metres in diameter. The motorized sphere weighs 340 kilograms, is constructed of aircraft-grade aluminum, the sphere is suspended above the Centers Science Court and is actuated with a computer-based motion control system. This system opens and closes the sphere in a series of lyrical motions choreographed to music, lighting. An earlier, similar but slightly smaller Hoberman sphere is in the atrium of Liberty Science Center in Jersey City, the 700-pound sphere, when fully expanded, measures 18 feet in diameter. 1993 a second geodesic sphere was installed at the Swiss Science Center Technorama in Winterthur, Hoberman mechanism Hoberman. com - Welcome to the World of Magical Transformation Radial expansion/retraction truss structures
20.
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
21.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
22.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations
23.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
24.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
25.
Pentagonal rotunda
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In geometry, the pentagonal rotunda is one of the Johnson solids. It can be seen as half an icosidodecahedron, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, the following formulae for volume, surface area, circumradius and height can be used if all faces are regular, with edge length a, V = a 3 ≈6.91776. A3 A = a 2 = a 2 ≈22.3472, a The dual of the pentagonal rotunda has 20 faces,10 triangular,5 rhombic, and 5 kites. Eric W. Weisstein, Pentagonal rotunda at MathWorld
26.
Johnson solid
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In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, an example of a Johnson solid is the square-based pyramid with equilateral sides, it has 1 square face and 4 triangular faces. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees, since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid is an example that actually has a degree-5 vertex. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3,4,5,6,8. In 1966, Norman Johnson published a list which included all 92 solids and he did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnsons list was complete, however, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid. The naming of Johnson Solids follows a flexible & precise descriptive formula, from there, a series of prefixes are attached to the word to indicate additions, rotations and transformations, Bi- indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that like either faces or unlike faces meet, using this nomenclature, an octahedron can be described as a square bipyramid, a cuboctahedron as a triangular gyrobicupola, and an icosidodecahedron as a pentagonal gyrobirotunda. Elongated indicates a prism is joined to the base of the solid in question, a rhombicuboctahedron can thus be described as an elongated square orthobicupola. Gyroelongated indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids, an icosahedron can thus be described as a gyroelongated pentagonal bipyramid. Augmented indicates a pyramid or cupola is joined to one or more faces of the solid in question, diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question. Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, the last three operations — augmentation, diminution, and gyration — can be performed multiple times certain large solids. Bi- & Tri- indicate a double and treble operation respectively, for example, a bigyrate solid has two rotated cupolae, and a tridiminished solid has three removed pyramids or cupolae. In in certain solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, for example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had 2 oblique faces gyrated. The last few Johnson solids have names based on certain polygon complexes from which they are assembled and these names are defined by Johnson with the following nomenclature, A lune is a complex of two triangles attached to opposite sides of a square. Spheno- indicates a complex formed by two adjacent lunes
27.
Pentagonal orthobirotunda
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In geometry, the pentagonal orthobirotunda is one of the Johnson solids. It can be constructed by joining two pentagonal rotundae along their faces, matching like faces. 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. Eric W. Weisstein, Pentagonal orthobirotunda at MathWorld
28.
Dihedral symmetry
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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
29.
Wire-frame figure
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A wire-frame model is a visual presentation of a 3-dimensional or physical object used in 3D computer graphics. It is created by specifying each edge of the object where two mathematically continuous smooth surfaces meet, or by connecting an objects constituent vertices using straight lines or curves. The object is projected into space by drawing lines at the location of each edge. The term wire frame comes from designers using metal wire to represent the shape of solid objects. 3D wire frame allows to construct and manipulate solids and solid surfaces, the 3D solid modeling technique efficiently draws higher quality representations of solids than the conventional line drawing. Using a wire-frame model allows visualization of the design structure of a 3D model. Traditional two-dimensional views and drawings can be created by appropriate rotation of the object, since wire-frame renderings are relatively simple and fast to calculate, they are often used in cases where a high screen frame rate is needed. When greater graphical detail is desired, surface textures can be added automatically after completion of the rendering of the wire frame. This allows the designer to quickly review solids or rotate the object to new desired views without long delays associated with more realistic rendering, the wire frame format is also well suited and widely used in programming tool paths for direct numerical control machine tools. Hand-drawn wire-frame-like illustrations date back as far as the Italian Renaissance, wire-frame models are also used as the input for computer-aided manufacturing. There are mainly three types of 3D CAD models, wire frame is one of them and it is the most abstract and least realistic. Other types of 3D CAD models are surface and solid and this method of modelling consists of only lines, points and curves defining the edges of an object. Wireframing is one of the used in geometric modelling systems. A wireframe model represents the shape of an object with its characteristic lines. There are two types of modelling, Pros and Cons. In Pros user gives a simple input to create a shape and it is useful in developing systems. While in Cons wireframe model, it does not include information about inside and outside boundary surfaces, today, wireframe models are used to define complex solid objects. The designer makes a model of a solid object, and then the CAD operator reconstructs the object
30.
Decagon
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In geometry, a decagon is a ten-sided polygon or 10-gon. A regular decagon has all sides of length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a pentagon, t. By simple trigonometry, d =2 a, and it can be written algebraically as d = a 5 +25. The side of a regular decagon inscribed in a circle is −1 +52 =1 ϕ. As 10 =2 ×5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon. An alternative method is as follows, Construct a pentagon in a circle by one of the shown in constructing a pentagon. Extend a line from each vertex of the pentagon through the center of the circle to the side of that same circle. Where each line cuts the circle is a vertex of the decagon, the five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon, both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element. In the construction with given circumcircle the circular arc around G with radius GE3 produces the segment AH, a M ¯ M H ¯ = A H ¯ A M ¯ =1 +52 = Φ ≈1.618. In the construction with side length the circular arc around D with radius DA produces the segment E10F. E1 E10 ¯ E1 F ¯ = E10 F ¯ E1 E10 ¯ = R a =1 +52 = Φ ≈1.618, the regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries, Dih5, Dih2, and Dih1, and 4 cyclic group symmetries, Z10, Z5, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, john Conway labels these by a letter and group order. Full symmetry of the form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g10 subgroup has no degrees of freedom but can seen as directed edges
31.
Great circle
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A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant
32.
Buckminster Fuller
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Richard Buckminster Bucky Fuller was an American architect, systems theorist, author, designer, and inventor. Fuller published more than 30 books, coining or popularizing terms such as Spaceship Earth, ephemeralization and he also developed numerous inventions, mainly architectural designs, and popularized the widely known geodesic dome. Carbon molecules known as fullerenes were later named by scientists for their structural and mathematical resemblance to geodesic spheres, Fuller was the second World President of Mensa from 1974 to 1983. Fuller was born on July 12,1895, in Milton, Massachusetts, the son of Richard Buckminster Fuller and Caroline Wolcott Andrews and he spent much of his youth on Bear Island, in Penobscot Bay off the coast of Maine. He often made items from materials he found in the woods and he experimented with designing a new apparatus for human propulsion of small boats. Later in life, Fuller took exception to the term invention, Fuller earned a machinists certification, and knew how to use the press brake, stretch press, and other tools and equipment used in the sheet metal trade. Fuller attended Milton Academy in Massachusetts, and after that studying at Harvard College. He was expelled from Harvard twice, first for spending all his money partying with a vaudeville troupe, by his own appraisal, he was a non-conforming misfit in the fraternity environment. Between his sessions at Harvard, Fuller worked in Canada as a mechanic in a textile mill, and later as a laborer in the meat-packing industry. He also served in the U. S. Navy in World War I, as a radio operator, as an editor of a publication. After discharge, he worked again in the packing industry. In 1917, he married Anne Hewlett, Buckminster Fuller recalled 1927 as a pivotal year of his life. His daughter Alexandra had died in 1922 of complications from polio, Fuller dwelled on her death, suspecting that it was connected with the Fullers damp and drafty living conditions. This provided motivation for Fullers involvement in Stockade Building Systems, a business which aimed to provide affordable, in 1927, at age 32, Fuller lost his job as president of Stockade. The Fuller family had no savings, and the birth of their daughter Allegra in 1927 added to the financial challenges, Fuller drank heavily and reflected upon the solution to his familys struggles on long walks around Chicago. During the autumn of 1927, Fuller contemplated suicide, so that his family could benefit from an insurance payment. Fuller said that he had experienced a profound incident which would provide direction and he felt as though he was suspended several feet above the ground enclosed in a white sphere of light. A voice spoke directly to Fuller, and declared, From now on you need never await temporal attestation to your thought and you do not have the right to eliminate yourself
33.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
34.
Even permutation
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In mathematics, when X is a finite set of at least two elements, the permutations of X fall into two classes of equal size, the even permutations and the odd permutations. The sign or signature of a permutation σ is denoted sgn and defined as +1 if σ is even, the signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol, which is defined for all maps from X to X, the sign of a permutation can be explicitly expressed as sgn = N where N is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as sgn = m where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, consider the permutation σ of the set which turns the initial arrangement 12345 into 34521. It can be obtained by three transpositions, first exchange the places of 1 and 3, then exchange the places of 2 and 4 and this shows that the given permutation σ is odd. Using the notation explained in the Permutation article, we can write σ = = =. There are many ways of writing σ as a composition of transpositions, for instance σ = . The identity permutation is an even permutation, furthermore, we see that the even permutations form a subgroup of Sn. This is the group on n letters, denoted by An. It is the kernel of the homomorphism sgn, the odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of An. If n > 1 , then there are just as many even permutations in Sn as there are odd ones, consequently, a cycle is even if and only if its length is odd. This follows from formulas like = In practice, in order to determine whether a given permutation is even or odd, the permutation is odd if and only if this factorization contains an odd number of even-length cycles. Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix, the value of the determinant is the same as the parity of the permutation. Every permutation of odd order must be even, the permutation in A4 shows that the converse is not true in general. Since we cannot be left with just a single element in an incorrect position, given a permutation σ, we can write it as a product of transpositions in many different ways. We want to show that all of those decompositions have an even number of transpositions. Suppose we have two such decompositions, σ = T1 T2 and we want to show that k and m are either both even, or both odd
35.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
36.
Orthogonal projection
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, though abstract, this definition of projection formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on an object by examining the effect of the projection on points in the object. For example, the function maps the point in three-dimensional space R3 to the point is an orthogonal projection onto the x–y plane. This function is represented by the matrix P =, the action of this matrix on an arbitrary vector is P =. To see that P is indeed a projection, i. e. P = P2, a simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P2 = = = P. proving that P is indeed a projection, the projection P is orthogonal if and only if α =0. Let W be a finite dimensional space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively, then P has the following properties, By definition, P is idempotent. P is the identity operator I on U ∀ x ∈ U, P x = x and we have a direct sum W = U ⊕ V. Every vector x ∈ W may be decomposed uniquely as x = u + v with u = P x and v = x − P x = x, the range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and we say P is a projection along V onto U and Q is a projection along U onto V. In infinite dimensional spaces, the spectrum of a projection is contained in as −1 =1 λ I +1 λ P. Only 0 or 1 can be an eigenvalue of a projection, the corresponding eigenspaces are the kernel and range of the projection. Decomposition of a space into direct sums is not unique in general. Therefore, given a subspace V, there may be many projections whose range is V, if a projection is nontrivial it has minimal polynomial x 2 − x = x, which factors into distinct roots, and thus P is diagonalizable. The product of projections is not, in general, a projection, if projections commute, then their product is a projection. When the vector space W has a product and is complete the concept of orthogonality can be used
37.
Coxeter plane
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections
38.
Spherical tiling
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In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
39.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =
40.
Conformal map
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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
41.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
42.
Rectification (geometry)
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In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the facets of the original polytope. A rectification operator is denoted by the symbol r, for example, r is the rectified cube. Conway polyhedron notation uses ambo for this operator, in graph theory this operation creates a medial graph. Rectification is the point of a truncation process. The highest degree of rectification creates the dual polytope, a rectification truncates edges to points. A birectification truncates faces to points, a trirectification truncates cells to points, and so on. New vertices are placed at the center of the edges of the original polygon, each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual, the rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron, a rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling, examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. The resulting medial graph remains polyhedral, so by Steinitzs theorem it can be represented as a polyhedron, the Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conways expand operation, e, which is the same as Johnsons cantellation operation, t0,2 generated from regular polyhedral, each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. Its rectification will have two types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however, a further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. Examples A first rectification truncates edges down to points, If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1 or r. A second rectification, or birectification, truncates faces down to points, If regular it has notation t2 or 2r. For polyhedra, a birectification creates a dual polyhedron, higher degree rectifications can be constructed for higher dimensional polytopes
43.
Truncated dodecahedron
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In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces,20 regular triangular faces,60 vertices and 90 edges and this polyhedron can be formed from a dodecahedron by truncating the corners so the pentagon faces become decagons and the corners become triangles. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb, the truncated dodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces, hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes, the truncated dodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. Schlegel diagrams are similar, with a projection and straight edges. In the mathematical field of theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron. It has 60 vertices and 90 edges, and is a cubic Archimedean graph, spinning truncated cube Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, Eric W. Weisstein, Truncated dodecahedron at MathWorld. Weisstein, Eric W. Truncated dodecahedral graph, 3D convex uniform polyhedra o3x5x - tid. Editable printable net of a dodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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Truncated icosahedron
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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 12 regular pentagonal faces,20 regular hexagonal faces,60 vertices and 90 edges and it is the Goldberg polyhedron GPV or 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs typically patterned with white hexagons, geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene C60 molecule and 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 and this creates 12 new pentagon faces, and 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 even permutations of, where φ =1 + √5/2 is the golden mean. Using φ2 = φ +1 one verifies that all vertices are on a sphere, centered at the origin, with the radius equal to √9φ +10. Permutations, X axis Y axis Z axis The truncated icosahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane and 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 approximately 23. 281446°. The area A and the volume V of the truncated icosahedron of edge length a are, with unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron easily demonstrates the Euler characteristic,32 +60 −90 =2, the balls used in association football and team handball are perhaps 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 due to the pressure of the air inside. This ball type was introduced to the World Cup in 1970, geodesic domes are typically 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 wheels used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix. This shape was also the configuration of the used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs. The truncated icosahedron can also be described as a model of the Buckminsterfullerene, or buckyball, molecule, an allotrope of elemental carbon, discovered in 1985
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Rhombicosidodecahedron
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It has 20 regular triangular faces,30 square faces,12 regular pentagonal faces,60 vertices and 120 edges. The name rhombicosidodecahedron refers to the fact that the 30 square faces lie in the planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. It can also be called an expanded or cantellated dodecahedron or icosahedron, therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either. The rhombicosidodecahedron shares the vertex arrangement with the stellated truncated dodecahedron. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors, the balls are expanded rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles, eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae. Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are all permutations of. The rhombicosidodecahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The rhombicosidodecahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure and these vertex-transitive figures have reflectional symmetry. It also shares its vertex arrangement with the compounds of six or twelve pentagrammic prisms. In the mathematical field of theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph, the Geometrical Foundation of Natural Structure, A Source Book of Design. Eric W. Weisstein, Small Rhombicosidodecahedron at MathWorld, 3D convex uniform polyhedra x3o5x - srid
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Truncated icosidodecahedron
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In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. It has 30 square faces,20 regular hexagonal faces,12 regular decagonal faces,120 vertices and 180 edges – more than any other convex nonprismatic uniform polyhedron, since each of its faces has point symmetry, the truncated icosidodecahedron is a zonohedron. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure, however, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular. One unfortunate point of confusion is there is a nonconvex uniform polyhedron of the same name. The surface area A and the volume V of the truncated icosidodecahedron of edge length a are, V = a 3 ≈206.803399 a 3. If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ −2, centered at the origin, are all the permutations of, and. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosidodecahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, schlegel diagrams are similar, with a perspective projection and straight edges. Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces, the truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases. In the mathematical field of theory, a truncated icosidodecahedral graph is the graph of vertices and edges of the truncated icosidodecahedron. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph and this polyhedron can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Wenninger, Magnus, Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR0467493 Cromwell, the Geometrical Foundation of Natural Structure, A Source Book of Design. Cromwell, P. Polyhedra, CUP hbk, pbk, eric W. Weisstein, GreatRhombicosidodecahedron at MathWorld. 3D convex uniform polyhedra x3x5x - grid, editable printable net of a truncated icosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra