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.
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
7.
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
8.
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
9.
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
10.
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
11.
Chirality (mathematics)
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In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral, in 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane, a chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ, the hand, the most familiar chiral object, a non-chiral figure is called achiral or amphichiral. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. A right shoe is different from a left shoe only for being mirror images of each other, in contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped tetrominoes of the video game Tetris also exhibit chirality. Individually they contain no mirror symmetry in the plane, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In three dimensions, every figure that possesses a plane of symmetry S1, an inversion center of symmetry S2. Note, however, that there are achiral figures lacking both plane and center of symmetry, an example is the figure F0 = which is invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry. The figure F1 = also is achiral as the origin is a center of symmetry, note also that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry and its symmetry group is a frieze group generated by a single glide reflection. A knot is called if it can be continuously deformed into its mirror image. For example the unknot and the knot are achiral, whereas the trefoil knot is chiral. Chirality in Metric Spaces, Symmetry, Culture and Science 21, pp. 27–36 Chiral Polyhedra by Eric W. Weisstein, when Topology Meets Chemistry by Erica Flapan. Chiral manifold at the Manifold Atlas
12.
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
13.
Pentagonal hexecontahedron
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In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two forms, which are mirror images of each other. It is also well-known to be the Catalan solid with the most vertices, among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices. The faces are irregular pentagons with two edges and three short edges. The ratio of lengths is about 1,1.7489525667362. The faces have four angles of about 118.13662 degrees each. The dihedral angle is about 153.178732558 degrees between all faces, consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron. The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual, pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron, the result is the pentagonal hexecontahedron. The pentagonal hexecontahedron has three positions, two on vertices, and one mid-edge. This polyhedron is topologically related as a part of sequence of polyhedra and these face-transitive figures have rotational symmetry. The Geometrical Foundation of Natural Structure, A Source Book of Design, pentagonal Hexecontrahedron – Interactive Polyhedron Model
14.
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
15.
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
16.
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
17.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
18.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
19.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
20.
Pentagon
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In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, a self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°, a regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2
21.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
22.
Mirror image
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A mirror image is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of such as a mirror or water. It is also a concept in geometry and can be used as a process for 3-D structures. Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside out. If we look at an object that is effectively two-dimensional and then turn it towards a mirror, in this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror, then we compare the object with its reflection by turning ourselves 180 degrees, towards the mirror. Again we perceive a left-right reversal due to a change in orientation, so, in these examples the mirror does not actually cause the observed reversals. The concept of reflection can be extended to three-dimensional objects, including the inside parts, the term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics, in chemistry, two versions of a molecule, one a mirror image of the other, are called enantiomers if they are not superposable on each other. That is an example of chirality, in general, an object and its mirror image are called enantiomorphs. If a point of an object has coordinates then the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirrors surface. In everyday use, a mirror does not reverse right and left, however, there is often a perception of left-right reversal, probably because the left and right of an object are defined by its top and front. Reflection in a mirror does result in a change in chirality, as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror, then this gives a reversal of the third axis. Similarly, if you stand side-on to a mirror your left and its important to realise there are only two enantiomorphs, the object and its image. So, no matter how the object is oriented towards the mirror, all the images are fundamentally identical. In the photograph of the urn and mirror, the urn is fairly symmetrical front-back, so, its not surprising that no obvious reversal of the urn can be seen in the mirror image. A mirror image appears more obviously three-dimensional if the observer moves and this is because the relative position of objects changes as the observers perspective changes, or is different viewed with each eye
23.
Convex hull
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In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space is the smallest convex set that contains X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces, they may also be generalized further, to oriented matroids. The algorithmic problem of finding the convex hull of a set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry. A set of points is defined to be if it contains the line segments connecting each pair of its points. The convex hull of a given set X may be defined as The minimal convex set containing X The intersection of all convex sets containing X The set of all combinations of points in X. The union of all simplices with vertices in X and it is not obvious that the first definition makes sense, why should there exist a unique minimal convex set containing X, for every X. Thus, it is exactly the unique minimal convex set containing X. Each convex set containing X must contain all convex combinations of points in X, in fact, according to Carathéodorys theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N +1 points are sufficient in the definition above. If the convex hull of X is a set, then it is the intersection of all closed half-spaces containing X. The hyperplane separation theorem proves that in case, each point not in the convex hull can be separated from the convex hull by a half-space. However, there exist convex sets, and convex hulls of sets, more abstractly, the convex-hull operator Conv has the characteristic properties of a closure operator, It is extensive, meaning that the convex hull of every set X is a superset of X. It is non-decreasing, meaning that, for two sets X and Y with X ⊆ Y, the convex hull of X is a subset of the convex hull of Y. It is idempotent, meaning that for every X, the hull of the convex hull of X is the same as the convex hull of X. The convex hull of a point set S is the set of all convex combinations of its points. For each choice of coefficients, the convex combination is a point in the convex hull. Expressing this as a formula, the convex hull is the set. The convex hull of a point set S ⊊ R n forms a convex polygon when n =2. Each point x i in S that is not in the hull of the other points is called a vertex of Conv . In fact, every convex polytope in R n is the hull of its vertices
24.
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
25.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
26.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
27.
Harmonices Mundi
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Harmonices Mundi is a book by Johannes Kepler. In the work Kepler discusses harmony and congruence in geometrical forms, the final section of the work relates his discovery of the so-called third law of planetary motion. Thus Kepler could reason that his relationships gave evidence for God acting as a geometer, rather than a Pythagorean numerologist. The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler. Musica Universalis was a philosophical metaphor that was taught in the quadrivium. He notes musical harmony as being a product of man, derived from angles, in turn, this allowed Kepler to claim the Earth has a soul because it is subjected to astrological harmony. Chapters 1 and 2 of The Harmony of the World contain most of Kepler’s contributions concerning polyhedra and he is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather and he returns to this concept later in Harmonices Mundi with relation to astronomical explanations. He describes polyhedra in terms of their faces, which is similar to the used in Plato’s Timaeus to describe the formation of Platonic solids in terms of basic triangles. While medieval philosophers spoke metaphorically of the music of the spheres and he found that the difference between the maximum and minimum angular speeds of a planet in its orbit approximates a harmonic proportion. For instance, the angular speed of the Earth as measured from the Sun varies by a semitone, from mi to fa. Venus only varies by a tiny 25,24 interval, Kepler explains the reason for the Earths small harmonic range, The Earth sings Mi, Fa, Mi, you may infer even from the syllables that in this our home misery and famine hold sway. The celestial choir Kepler formed was made up of a tenor, at very rare intervals all of the planets would sing together in perfect concord, Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring orbits approximate musical harmonies within a margin of error of less than a diesis. The orbits of Mars and Jupiter produce the one exception to this rule, in fact, the cause of Keplers dissonance might be explained by the fact that the asteroid belt separates those two planetary orbits, as discovered in 1801,150 years after Keplers death. Keplers previous book Astronomia nova related the discovery of the first two of the principles that we know today as Keplers laws, a copy of the 1619 edition was stolen from the National Library of Sweden in the 1990s. A small number of recent compositions either make reference to or are based on the concepts of Harmonices Mundi or Harmony of the Spheres, the most notable of these are, Laurie Spiegel, Keplers Harmony of the Worlds. An excerpt of the piece was selected by Carl Sagan for inclusion on the Voyager Golden Record, mike Oldfield, Music of the Spheres
28.
H. S. M. 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
29.
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
30.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
31.
Cube root
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In mathematics, a cube root of a number x is a number such that a3 = x. All real numbers have one real cube root and a pair of complex conjugate cube roots. For example, the cube root of 8, denoted 3√8, is 2, because 23 =8, while the other cube roots of 8 are −1 + √3i. The three cube roots of −27i are 3 i,332 −32 i, the cube root operation is not associative or distributive with addition or subtraction. In some contexts, particularly when the number whose root is to be taken is a real number, one of the cube roots is referred to as the principal cube root. The cube roots of a x are the numbers y which satisfy the equation y 3 = x. For any real number y, there is one real number x such that x3 = y, the cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one, then we can define an inverse function that is also one-to-one. For real numbers, we can define a cube root of all real numbers. If this definition is used, the root of a negative number is a negative number. If x and y are allowed to be complex, then there are three solutions and so x has three cube roots, a real number has one real cube root and two further cube roots which form a complex conjugate pair. This can lead to interesting results. For instance, the roots of the number one are,13 = {1 −12 +32 i −12 −32 i. The last two of these lead to a relationship between all roots of any real or complex number. If a number is one root of any real or complex number. For complex numbers, the cube root is usually defined as the cube root that has the largest real part, or, equivalently. It is related to the value of the natural logarithm by the formula x 13 = exp . This means that in polar coordinates, we are taking the root of the radius
32.
Parity of a 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
33.
Great snub icosidodecahedron
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In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It can be represented by a Schläfli symbol sr, and Coxeter-Dynkin diagram and this polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.5488772. Taking the odd permutations of the coordinates with an odd number of plus signs gives another form. The great pentagonal hexecontahedron is a nonconvex isohedral polyhedron and dual to the great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces,120 edges, and 92 vertices
34.
Great inverted snub icosidodecahedron
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In geometry, the great inverted snub icosidodecahedron is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr.2224727, taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. The great inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron and it is composed of 60 self-intersecting pentagonal faces,150 edges and 92 vertices. It is the dual of the great inverted snub icosidodecahedron
35.
Great retrosnub icosidodecahedron
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In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It is given a Schläfli symbol s.3264046, taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. List of uniform polyhedra Great snub icosidodecahedron Great inverted snub icosidodecahedron Weisstein, Eric W
36.
Point reflection
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Not to be confused with inversive geometry, in which inversion is through a circle instead of a point. In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space, point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has one fixed point. It is equivalent to a transformation with scale factor equal to -1. The point of inversion is called homothetic center. The term reflection is loose, and considered by some an abuse of language, with preferred, however. Such maps are involutions, meaning that they have order 2 – they are their own inverse, in dimension 1 these coincide, as a point is a hyperplane in the line. In terms of algebra, assuming the origin is fixed. Reflection in a hyperplane has a single −1 eigenvalue, while point reflection has only the −1 eigenvalue. The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across a plane perpendicular to the axis of rotation, in dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is R e f p =2 p − a, in the case where p is the origin, point reflection is simply the negation of the vector a. In Euclidean geometry, the inversion of a point X with respect to a point P is a point X* such that P is the midpoint of the segment with endpoints X. In other words, the vector from X to P is the same as the vector from P to X*, the formula for the inversion in P is x*=2a−x where a, x and x* are the position vectors of P, X and X* respectively. This mapping is an isometric involutive affine transformation which has one fixed point. When the inversion point P coincides with the origin, point reflection is equivalent to a case of uniform scaling. This is an example of linear transformation, when P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation, homothety with homothetic center coinciding with P, and scale factor = -1. This is an example of non-linear affine transformation), the composition of two point reflections is a translation
37.
Projection (linear algebra)
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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
38.
Coxeter element
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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
39.
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
40.
Expansion (geometry)
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In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position, the expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation. For polyhedra, a polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron. According to Coxeter, this term was defined by Alicia Boole Stott for creating new polytopes. The expansion operation is symmetric with respect to a regular polytope, the resulting figure contains the facets of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements. It has somewhat different meanings by dimension, in a Wythoff construction, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, by dimension, A regular polygon expands into a regular 2n-gon. The operation is identical to truncation for polygons, e = e1 = t0,1 = t and has Coxeter-Dynkin diagram, a regular polyhedron expands into a polyhedron with vertex figure p.4. q.4. This operation for polyhedra is also called cantellation, e = e2 = t0,2 = rr, for example, a rhombicuboctahedron can be called an expanded cube, expanded octahedron, as well as a cantellated cube or cantellated octahedron. This operation for 4-polytopes is also called runcination, e = e3 = t0,3, similarly a regular 5-polytope expands into a new 5-polytope with facets, × prisms, × prisms, and × duoprisms. This operation is called sterication, e = e4 = t0,4 = 2r2r and has Coxeter diagram, the general operator for expansion of a regular n-polytope is t0, n-1. New regular facets are added at each vertex, and new prismatic polytopes are added at each divided edge, ridge, etc. Conway polyhedron notation Weisstein, Eric W. Expansion. Coxeter, H. S. M. Regular Polytopes, norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
41.
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
42.
Circumscribed circle
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
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Alternation (geometry)
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In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M
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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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Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron 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. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids