1.
Archimedean solid
–
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
–
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
–
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
–
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
–
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
–
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
–
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
–
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
–
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
–
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.
Vertex figure
–
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
12.
Pentakis dodecahedron
–
In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face, that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name, there are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include, The usual Catalan pentakis dodecahedron, a convex hexecontahedron with sixty isosceles triangular faces illustrated in the sidebar figure and it is a Catalan solid, dual to the truncated icosahedron, an Archimedean solid. As the heights of the pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi. As the height is raised further, the shape becomes non-convex, other more non-convex geometric variants include, The small stellated dodecahedron. Great pentakis dodecahedron Wenningers third stellation of icosahedron, if one affixes pentagrammic pyramids into Wenningers third stellation of icosahedron one obtains the great icosahedron. The pentakis dodecahedron in a model of buckminsterfullerene, each surface segment represents a carbon atom, equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom. The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses and these have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron. The pentakis dodecahedron has three positions, two on vertices, and one on a midedge, The Spaceship Earth structure at Walt Disney Worlds Epcot is a derivative of a pentakis dodecahedron. The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https, in Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron and these Domes also appear whenever the player transforms on a dome in the Hypno Ray level. Some Geodomes in which play on are Pentakis Dodecahedra. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Pentakis dodecahedron at MathWorld. Pentakis Dodecahedron – Interactive Polyhedron Model
13.
Dual polyhedron
–
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
14.
Net (polyhedron)
–
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
15.
Geometry
–
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
16.
Icosahedron
–
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
17.
Isogonal figure
–
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.
Face (geometry)
–
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
19.
Regular polygon
–
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
20.
Pentagon
–
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.
Hexagon
–
In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
22.
Goldberg polyhedron
–
A Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described by Michael Goldberg in 1937 and they are defined by three properties, each face is either a pentagon or hexagon, exactly three faces meet at each vertex, they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric, e. g. GP and GP are enantiomorphs of each other, a consequence of Eulers polyhedron formula is that there will be exactly twelve pentagons. Icosahedral symmetry ensures that the pentagons are always regular, although many of the hexagons may not be, typically all of the vertices lie on a sphere, but they can also be computed as equilateral. It is a polyhedron of a geodesic sphere, with all triangle faces and 6 triangles per vertex. Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron, other forms can be described by taking a chess knight move from one pentagon to the next, first take m steps in one direction, then turn 60° to the left and take n steps. Such a polyhedron is denoted GP, a dodecahedron is GP and a truncated icosahedron is GP. A similar technique can be applied to polyhedra with tetrahedral symmetry. These polyhedra will have triangles or squares rather than pentagons and these variations are given Roman numeral subscripts, GPIII, GPIV, and GPV. The chamfer operator, c, replaces all edges by hexagons, transforming GP to GP, the truncated kis operator, y=tk, generates GP, transforming GP to GP, with a T multiplier of 9. For class 2 forms, the dual kis operator, z=dk, transforms GP into GP, for class 3 forms, the whirl operator, w, generates GP, with a T multiplier of 7. A clockwise and counterclockwise whirl generator, ww=wrw generates GP in class 1, in general, a whirl can transform a GP into GP for a>b and the same chiral direction. If chiral directions are reversed, GP becomes GP if a>=2b, capsid Geodesic sphere Fullerene#Other buckyballs Conway polyhedron notation Goldberg, Michael. Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture Hart, George
23.
Ball (association football)
–
A football, soccer ball, or association football ball is the ball used in the sport of association football. The name of the ball according to whether the sport is called football, soccer. The balls spherical shape, as well as its size, weight, additional, more stringent, standards are specified by FIFA and subordinate governing bodies for the balls used in the competitions they sanction. Early footballs began as animal bladders or stomachs that would fall apart if kicked too much. Improvements became possible in the 19th century with the introduction of rubber, the modern 32-panel ball design was developed in 1962 by Eigil Nielsen, and technological research continues today to develop footballs with improved performance. In 1863, the first specifications for footballs were laid down by the Football Association, previous to this, footballs were made out of inflated leather, with later leather coverings to help footballs maintain their shapes. In 1872 the specifications were revised, and these rules have been essentially unchanged as defined by the International Football Association Board. Differences in footballs created since this rule came into effect have been to do with the used in their creation. Footballs have gone through a change over time. During medieval times balls were made from an outer shell of leather filled with cork shavings. Another method of creating a ball was using animal bladders for the inside of the making it inflatable. However, these two styles of creating footballs made it easy for the ball to puncture and were inadequate for kicking and it was not until the 19th century that footballs developed into what a football looks like today. In 1838, Charles Goodyear introduced the use of rubber and their discoveries of vulcanisation, Vulcanization is the treatment of rubber to give it certain qualities such as strength, elasticity, and resistance to solvents. Vulcanization of rubber also helps the football resist moderate heat and cold, Vulcanization helped create inflatable bladders that pressurise the outer panel arrangement of the football. Charles Goodyears innovation increased the ability of the ball and made it easier to kick. Most of the balls of this time had tanned leather with eighteen sections stitched together and these were arranged in six panels of three strips each. During the 1900s, footballs were made out of rubber and leather which was perfect for bouncing and kicking the ball, however, when heading the football it was usually painful. This problem was most probably due to absorption of the leather from rain
24.
Geodesic dome
–
A geodesic dome is a hemispherical thin-shell structure based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the stress throughout the structure. A first, small dome was patented, constructed by the firm of Dykerhoff and Wydmann on the roof of the Zeiss plant in Jena, a larger dome, called The Wonder of Jena opened to the public in July 1926. Some 20 years later, R. Buckminster Fuller named the dome geodesic from field experiments with artist Kenneth Snelson at Black Mountain College in 1948 and 1949. Although Fuller was not the inventor, he is credited with the U. S. popularization of the idea for which he received U. S. patent 2,682,23529 June 1954. Howard of Synergetics, Inc. and specialty buildings like the Kaiser Aluminum domes, auditoriums, weather observatories, the dome was soon breaking records for covered surface, enclosed volume, and construction speed. Marines experimented with helicopter-deliverable geodesic domes, a 30-foot wood and plastic geodesic dome was lifted and carried by helicopter at 50 knots without damage, leading to the manufacture of a standard magnesium dome by Magnesium Products of Milwaukee. The dome was introduced to an audience as a pavilion for the 1964 Worlds Fair in New York City designed by Thomas C. Howard of Synergetics. This dome is now used as an aviary by the Queens Zoo in Flushing Meadows Corona Park after it was redesigned by TC Howard of Synergetics, another dome is from Expo 67 at the Montreal Worlds Fair, where it was part of the American Pavilion. The structures covering later burned, but the structure still stands and, under the name Biosphère. In the 1970s, Zomeworks licensed plans for structures based on other geometric solids, such as the Johnson solids, Archimedean solids and these structures may have some faces that are not triangular, being squares or other polygons. In 1975, a dome was constructed at the South Pole, on October 1,1982, one of the most famous geodesic domes, Spaceship Earth at the EPCOT Center in Walt Disney World, opened. The building is Epcots icon, and is included in the parks logo. In the year 2000 the worlds first fully sustainable geodesic dome hotel, the hotels dome design is key to resisting the regions strong winds and is based on the dwellings of the indigenous Kaweskar people. Wooden domes have a hole drilled in the width of a strut, a stainless steel band locks the struts hole to a steel pipe. With this method, the struts may be cut to the length needed. Triangles of exterior plywood are then nailed to the struts, the dome is wrapped from the bottom to the top with several stapled layers of tar paper, in order to shed water, and finished with shingles. This type of dome is called a hub-and-strut dome because of the use of steel hubs to tie the struts together
25.
Buckminster Fuller
–
Richard Buckminster Bucky Fuller was an American architect, systems theorist, author, designer, and inventor. Fuller published more than 30 books, coining or popularizing terms such as Spaceship Earth, ephemeralization and he also developed numerous inventions, mainly architectural designs, and popularized the widely known geodesic dome. Carbon molecules known as fullerenes were later named by scientists for their structural and mathematical resemblance to geodesic spheres, Fuller was the second World President of Mensa from 1974 to 1983. Fuller was born on July 12,1895, in Milton, Massachusetts, the son of Richard Buckminster Fuller and Caroline Wolcott Andrews and he spent much of his youth on Bear Island, in Penobscot Bay off the coast of Maine. He often made items from materials he found in the woods and he experimented with designing a new apparatus for human propulsion of small boats. Later in life, Fuller took exception to the term invention, Fuller earned a machinists certification, and knew how to use the press brake, stretch press, and other tools and equipment used in the sheet metal trade. Fuller attended Milton Academy in Massachusetts, and after that studying at Harvard College. He was expelled from Harvard twice, first for spending all his money partying with a vaudeville troupe, by his own appraisal, he was a non-conforming misfit in the fraternity environment. Between his sessions at Harvard, Fuller worked in Canada as a mechanic in a textile mill, and later as a laborer in the meat-packing industry. He also served in the U. S. Navy in World War I, as a radio operator, as an editor of a publication. After discharge, he worked again in the packing industry. In 1917, he married Anne Hewlett, Buckminster Fuller recalled 1927 as a pivotal year of his life. His daughter Alexandra had died in 1922 of complications from polio, Fuller dwelled on her death, suspecting that it was connected with the Fullers damp and drafty living conditions. This provided motivation for Fullers involvement in Stockade Building Systems, a business which aimed to provide affordable, in 1927, at age 32, Fuller lost his job as president of Stockade. The Fuller family had no savings, and the birth of their daughter Allegra in 1927 added to the financial challenges, Fuller drank heavily and reflected upon the solution to his familys struggles on long walks around Chicago. During the autumn of 1927, Fuller contemplated suicide, so that his family could benefit from an insurance payment. Fuller said that he had experienced a profound incident which would provide direction and he felt as though he was suspended several feet above the ground enclosed in a white sphere of light. A voice spoke directly to Fuller, and declared, From now on you need never await temporal attestation to your thought and you do not have the right to eliminate yourself
26.
Buckminsterfullerene
–
Buckminsterfullerene is a spherical fullerene molecule with the formula C60. It was first generated in 1985 by Harold Kroto, James R. Heath, Sean OBrien, Robert Curl, Kroto, Curl and Smalley were awarded the 1996 Nobel Prize in Chemistry for their roles in the discovery of buckminsterfullerene and the related class of molecules, the fullerenes. The name is a reference to Buckminster Fuller, as C60 resembles his trademark geodesic domes, Buckminsterfullerene is the most common naturally occurring fullerene molecule, as it can be found in small quantities in soot. Solid and gaseous forms of the molecule have been detected in deep space, Buckminsterfullerene is one of the largest objects to have been shown to exhibit wave–particle duality, as stated in the theory every object exhibits this behavior. Its discovery led to the exploration of a new field of chemistry, Buckminsterfullerene derives from the name of the noted futurist and inventor Buckminster Fuller. One of his designs of a dome structure bears great resemblance to C60, as a result. The general public, however, sometimes refers to buckminsterfullerene, and even Fullers dome structure, the structure associated with fullerenes was described by Leonardo da Vinci. Albrecht Dürer also reproduced a similar icosahedron containing 12 pentagonal and 20 hexagonal faces, theoretical predictions of buckyball molecules appeared in the late 1960s – early 1970s, but they went largely unnoticed. In the early 1970s, the chemistry of unsaturated carbon configurations was studied by a group at the University of Sussex, led by Harry Kroto, in the 1980s a technique was developed by Richard Smalley and Robert Curl at Rice University, Texas to isolate these substances. They used laser vaporization of a target to produce clusters of atoms. Kroto realized that by using a target, any carbon chains formed could be studied. Another interesting fact is that, at the time, astrophysicists were working along with spectroscopists to study infrared emissions from giant red carbon stars. Smalley and team were able to use a laser vaporization technique to create carbon clusters which could potentially emit infrared at the wavelength as had been emitted by the red carbon star. Hence, the inspiration came to Smalley and team to use the technique on graphite to create the first fullerene molecule. C60 was discovered in 1985 by Robert Curl, Harold Kroto, using laser evaporation of graphite they found Cn clusters of which the most common were C60 and C70. A solid rotating graphite disk was used as the surface from which carbon was vaporized using a laser beam creating hot plasma that was passed through a stream of high-density helium gas. The carbon species were subsequently cooled and ionized resulting in the formation of clusters, clusters ranged in molecular masses but Kroto and Smalley found predominance in a C60 cluster that could be enhanced further by letting the plasma react longer. They also discovered that the C60 molecule formed a cage-like structure, for this discovery they were awarded the 1996 Nobel Prize in Chemistry
27.
Isohedral figure
–
In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
28.
Bitruncated order-5 dodecahedral honeycomb
–
The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol, it has five dodecahedral cells around each edge and its vertex figure is an icosahedron. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells and it is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The dihedral angle of a Euclidean regular dodecahedron is ~116. 6°, the dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°. There are four regular compact honeycombs in 3D hyperbolic space, There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb and these honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space, with three dodecahedra on each edge. Lastly the dodecahedral ditope, exists on a 3-sphere, with 2 hemispherical cells, There are nine uniform honeycombs in the Coxeter group family, including this regular form. Also the bitruncated form, t1,2, of this honeycomb has all truncated icosahedron cells, the Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb. There are four rectified compact regular honeycombs, The truncated order-5 dodecahedral honeycomb, has icosahedron and truncated dodecahedron cells, the bitruncated order-5 dodecahedral honeycomb, has truncated icosahedron cells, with a disphenoid vertex figure. The cantellated order-5 dodecahedral honeycomb, has alternating rhombicosidodecahedron and icosidodecahedron cells, the cantitruncated order-5 dodecahedral honeycomb, has truncated icosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure. The runcinated order-5 dodecahedral honeycomb, has dodecahedron and pentagonal prism cells, the runcitruncated order-5 dodecahedral honeycomb, has truncated dodecahedron, icosidodecahedron and pentagonal prism cells, with a distorted square pyramid vertex figure. The omnitruncated order-5 dodecahedral honeycomb, has truncated icosidodecahedron and decagonal prism cells, convex uniform honeycombs in hyperbolic space List of regular polytopes 57-cell - An abstract regular polychoron which shared the symbol. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999 ISBN 0-486-40919-8 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 N. W, Johnson, Geometries and Transformations, Chapter 13, Hyperbolic Coxeter groups
29.
Truncation (geometry)
–
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Keplers names for the Archimedean solids, in general any polyhedron can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, there are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation, for example, the icosidodecahedron, represented as Schläfli symbols r or, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the adjacent to the ringed node. A truncated n-sided polygon will have 2n sides, a regular polygon uniformly truncated will become another regular polygon, t is. A complete truncation, r, is another regular polygon in its dual position, a regular polygon can also be represented by its Coxeter-Dynkin diagram, and its uniform truncation, and its complete truncation. Star polygons can also be truncated, a truncated pentagram will look like a pentagon, but is actually a double-covered decagon with two sets of overlapping vertices and edges. A truncated great heptagram gives a tetradecagram and this sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron, the middle image is the uniform truncated cube. It is represented by a Schläfli symbol t, a bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. The truncated octahedron is a cube, 2t is an example. A complete bitruncation is called a birectification that reduces original faces to points, for polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube, = 2r is an example, another type of truncation is called cantellation, cuts edge and vertices, removing original edges and replacing them with rectangles. Higher dimensional polytopes have higher truncations, runcination cuts faces, edges, in 5-dimensions sterication cuts cells, faces, and edges. Edge-truncation is a beveling or chamfer for polyhedra, similar to cantellation but retains original vertices, in 4-polytopes edge-truncation replaces edges with elongated bipyramid cells. Alternation or partial truncation only removes some of the original vertices, a partial truncation or alternation - Half of the vertices and connecting edges are completely removed. The operation only applies to polytopes with even-sided faces, faces are reduced to half as many sides, and square faces degenerate into edges
30.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
31.
Parity of a permutation
–
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
32.
Golden ratio
–
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
33.
Projection (linear algebra)
–
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
34.
Coxeter element
–
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
35.
Spherical tiling
–
In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
36.
Stereographic projection
–
In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =
37.
Conformal map
–
In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
38.
Handball
–
Handball is a team sport in which two teams of seven players each pass a ball using their hands with the aim of throwing it into the goal of the other team. A standard match consists of two periods of 30 minutes, and the team scores more goals wins. Modern handball is played on a court 40 by 20 metres, the goals are surrounded by a 6-meter zone where only the defending goalkeeper is allowed, goals must be scored by throwing the ball from outside the zone or while diving into it. The sport is played indoors, but outdoor variants exist in the forms of field handball and Czech handball. The game is fast and high-scoring, professional teams now typically score between 20 and 35 goals each, though lower scores were not uncommon until a few decades ago, body contact is permitted by the defenders trying to stop the attackers from approaching the goal. The game was codified at the end of the 19th century in northern Europe, the modern set of rules was published in 1917 in Germany, and had several revisions since. The first international games were played under rules for men in 1925. Mens handball was first played at the 1936 Summer Olympics in Berlin as outdoors, and the time at the 1972 Summer Olympics in Munich as indoors. Womens team handball was added at the 1976 Summer Olympics, the International Handball Federation was formed in 1946 and, as of 2016, has 197 member federations. The sport is most popular in the countries of continental Europe, in the womens world championships, only two non-European countries have won the title, South Korea and Brazil. The game also enjoys popularity in the Far East, North Africa, There is evidence of ancient Roman women playing a version of handball called expulsim ludere. There are records of games in medieval France, and among the Inuit in Greenland. By the 19th century, there existed similar games of håndbold from Denmark, házená in the Czech Republic, handbol in Ukraine, the team handball game of today was codified at the end of the 19th century in northern Europe—primarily in Denmark, Germany, Norway and Sweden. The first written set of team handball rules was published in 1906 by the Danish gym teacher, lieutenant and Olympic medalist Holger Nielsen from Ordrup grammar school, north of Copenhagen. The modern set of rules was published on 29 October 1917 by Max Heiser, Karl Schelenz, after 1919 these rules were improved by Karl Schelenz. The first international games were played under rules, between Germany and Belgium by men in 1925 and between Germany and Austria by women in 1930. In 1926, the Congress of the International Amateur Athletics Federation nominated a committee to draw up rules for field handball. The International Amateur Handball Federation was formed in 1928 and later the International Handball Federation was formed in 1946, Mens field handball was played at the 1936 Summer Olympics in Berlin
39.
Spherical polyhedron
–
In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
40.
1970 FIFA World Cup
–
The 1970 FIFA World Cup was the ninth FIFA World Cup, the quadrennial international football championship for mens national teams. Held from 31 May to 21 June in Mexico, it was the first World Cup tournament staged in North America, Teams representing 75 nations from all six populated continents entered the competition, and its qualification rounds began in May 1968. Fourteen teams qualified from this process to join host nation Mexico, El Salvador, Israel, and Morocco made their first appearances at the final stage, and Peru their first since 1930. The tournament was won by Brazil, which defeated another former champion, Italy. The win gave Brazil its third World Cup title, which allowed them to keep the Jules Rimet Trophy. The victorious team, led by Carlos Alberto and featuring players such as Pelé, Gérson, Jairzinho, Rivellino and they achieved a perfect record of wins in all six games in the finals, as well as winning all their qualifying fixtures. The World Cup returned to Mexico just sixteen years later in 1986, Argentina, Australia, Colombia, Japan, Mexico and Peru were all considered to host the 1970 FIFA World Cup. Mexico was chosen as the host nation in 1964 through a vote at FIFAs congress in Tokyo on 8 October, a total of 75 teams entered the 1970 FIFA World Cup, and 73 were required to qualify. Due to rejected entries and withdrawals,68 teams eventually participated in the qualifying stages, Mexico as the host nation and England as reigning World Cup champions were granted automatic qualification, with the remaining fourteen finals places divided among the continental confederations. Eight places were available to teams from UEFA, three for CONMEBOL, one for CAF, one for an AFC/OFC team, and one for CONCACAF. The draw for the stages was conducted on 1 February 1968 in Casablanca, Morocco, with matches beginning in May 1968. North Korea, quarter-finalists at the tournament, were disqualified during the process after refusing to play in Israel for political reasons. El Salvador qualified for the finals after beating Honduras in a play-off match and those who failed to qualify included Argentina, France, Hungary, Portugal and Spain. Five stadia in five different cities were selected to host the World Cup matches, alternative venues in Hidalgo state and the port city of Veracruz were also considered. Each group was based solely in one city with exception of Group 2, aside from the Estadio Luis Dosal, all the stadia had only been constructed during the 1960s, as Mexico prepared to host both the World Cup and the 1968 Summer Olympics. The altitude of the varied and the importance of acclimatisation was strongly considered by all the participating teams. As a result, in contrast to the tournament staged in England. Some teams had already experienced the conditions when competing in the football competition at 1968 Summer Olympics
41.
2006 FIFA World Cup
–
The 2006 FIFA World Cup was the 18th FIFA World Cup, the quadrennial international football world championship tournament. It was held from 9 June to 9 July 2006 in Germany, teams representing 198 national football associations from all six populated continents participated in the qualification process which began in September 2003. Thirty-one teams qualified from this process, along with the host nation, Germany and it was the second time that Germany staged the competition, and the tenth time that it was held in Europe. Italy won the tournament, claiming their fourth World Cup title and they defeated France 5–3 in a penalty shootout in the final, after extra time had finished in a 1–1 draw. Germany defeated Portugal 3–1 to finish in third place, Angola, Ghana, Ivory Coast, Serbia and Montenegro, Trinidad and Tobago, and Togo made their first appearances in the finals. The 2006 World Cup stands as one of the most watched events in history, garnering an estimated 26.29 billion non-unique viewers. The final attracted an audience of 715.1 million people. The 2006 World Cup ranks fourth in non-unique viewers, behind the World Cup in 1994,2002, as the winner, Italy represented the World in the 2009 FIFA Confederations Cup. The vote to choose the hosts of the 2006 tournament was held in July 2000 in Zürich and it involved four bidding nations after Brazil had withdrawn three days earlier, Germany, South Africa, England and Morocco. Three rounds of voting were required, each round eliminating the nation with the least votes, the first two rounds were held on 6 July 2000, and the final round was held on 7 July 2000, which Germany won over South Africa. Accusations of bribery and corruption have marred the success of Germanys bid from the very beginning, on the very day of the vote, a hoax bribery affair was made public, leading to calls for a re-vote. Oceania delegate Charlie Dempsey, who had initially backed England, had then been instructed to support South Africa following Englands elimination and he abstained, citing intolerable pressure on the eve of the vote. Just a week before the vote, the German government under Chancellor Gerhard Schröder lifted their arms embargo on Saudi Arabia, daimlerChrysler invested several hundred million Euro in Hyundai, while one of the sons of the companys founders was a member of FIFAs executive committee. Both Volkswagen and Bayer announced investments in Thailand and South Korea, whose respective delegates Worawi Makudi, the sum of 6.7 million Euro was later demanded back by Dreyfus. In order to retrieve the money, the Organizing Committee paid an aquivalent sum to the FIFA, allegedly as a German share for the cost of a closing ceremony, the DFB announced they would consider seeking legal action against Der Spiegel. During a press conference on 22 October 2015, Nierbach repeated his stance, according to Niersbach, the payment had been agreed upon during a meeting between Franz Beckenbauer and FIFA president Blatter, with the money being provided by Dreyfus. On the same day, FIFA contradicted Niersbachs statement, saying, By our current state of knowledge, the following day, former DFB president Theo Zwanziger publicly accused Niersbach of lying, saying, It is evident that there was a slush fund for the German World Cup application. According to Zwanziger, the 6.7 million Euros went to Mohamed Bin Hammam, on 22 March 2016 it was announced that the FIFA Ethics Committee was opening proceedings into the bid
42.
Pontiac
–
Pontiac was a brand of automobile manufactured and sold by General Motors between 1926 and 2010. Introduced as a make for GMs more expensive line of Oakland automobiles, Pontiac overtook Oakland in popularity. Sold in the United States, Canada, and Mexico by GM, the last Pontiac badged cars were built in December 2009, with one final vehicle in January,2010. Franchise agreements for Pontiac dealers expired October 31,2010, leaving GM to focus on its four remaining North American brands, Chevrolet, Buick, Cadillac, and GMC. The Pontiac brand was introduced by General Motors in 1926 as the marque to GMs Oakland division. It was named after the famous Ottawa chief who had given his name to the city of Pontiac. Within months of its introduction, Pontiac was outselling Oakland, which was essentially a 1920s Chevrolet with a six-cylinder engine installed. Body styles offered included a sedan with two and four doors, Landau Coupe, with the Sport Phaeton, Sport Landau Sedan, Sport Cabriolet. As a result of Pontiacs rising sales, versus Oaklands declining sales, Pontiac became the companion marque to survive its parent. It was also manufactured from knock-down kits at GMs short-lived Japanese factory at Osaka Assembly in Osaka, Japan from 1927-1941. Pontiac produced cars offering 40 hp 186.7 cu in L-head straight 6-cylinder engines in the Pontiac Chief of 1927, the Chief sold 39,000 units within six months of its appearance at the 1926 New York Auto Salon, hitting 76,742 at twelve months. The next year, it became the six in the U. S. ranking seventh in overall sales. By 1933, it had moved up to producing the least expensive cars available with straight eight-cylinder engines, in the late 1930s, Pontiac used the so-called torpedo body of the Buick for one of its models, just prior to its being used by Chevrolet. This body style brought some attention to the marque, an unusual feature of the torpedo body exhibition car, was that with push of a button the front half of the car body would open showing the engine and the cars front seat interior. In 1937, the eight-cylinder had a 122-inch wheelbase, while the six-cylinder had a 117-inch wheelbase, for an extended period of time—prewar through the early 1950s—the Pontiac was a quiet, solid car, but not especially powerful. It came with a straight eight. Straight 8s were slightly less expensive to produce than the increasingly popular V8s, additionally, the long crankshaft suffered from excessive flex, restricting straight 8s to a relatively low compression ratio with a modest redline. However, in application, inexpensive flatheads were not a liability
43.
Pontiac Firebird
–
The Pontiac Firebird is an American automobile built by Pontiac from the 1967 to the 2002 model years. Designed as a car to compete with the Ford Mustang, it was introduced 23 February 1967. The first generation Firebird had characteristic Coke bottle styling shared with its cousin, unlike the Camaro, the Firebirds bumpers were integrated into the design of the front end. The Firebirds rear slit taillights were inspired by the 1966–1967 Pontiac GTO, both a two-door hardtop and a convertible were offered through the 1969 model year. Originally, the car was a prize for Pontiac, which had desired to produce a two-seat sports car based on its original Banshee concept car. However, GM feared this would cut into Chevrolet Corvette sales, the 1967 base model Firebird came equipped with the Chevrolet 230 cu in SOHC inline-six. Fitted with a carburetor, it was rated at 165 hp. The Sprint model six came with a carburetor, developing 215 hp. Most buyers opted for one of three V8s, the 326 cu in with a two-barrel carburetor producing 250 hp, the four-barrel HO326, producing 285 hp, all 1967–1968400 CI engines had throttle restrictors that blocked the carburetors second barrels from fully opening. A Ram Air option was available, providing functional hood scoops, higher flow heads with stronger valve springs. Power for the Ram Air package was the same as the conventional 400 HO, also for the 1968 model, the 326 CID engine was replaced by the Pontiac 350 cu in V8, which actually displaced 355 cu in, and produced 265 hp with a two-barrel carburetor. An HO version of the 350 CID with a revised cam was also offered starting in that year, power output of the other engines was increased marginally. There was an additional Ram Air IV option for the 400 CID engines during 1969, complementing the Ram Air III, the 350 CID HO engine was revised again with a different cam and cylinder heads resulting in 325 hp. During 1969 a special 303 cu in engine was designed for SCCA road racing applications that was not available in production cars, the front door vent-windows were replaced with a single pane of glass and Astro Ventilation, a fresh-air-inlet system. The 1969 model received a facelift with a new front end design but unlike the GTO. The instrument panel and steering wheel were revised, the ignition switch was moved from the dashboard to the steering column with the introduction of GMs new locking ignition switch/steering wheel. In March 1969, a $725 optional handling package called the Trans Am performance and appearance package, UPC WS4, of these first Trans Ams, only 689 hardtops and eight convertibles were made. By late spring of 1969, Pontiac had deleted all references on Firebird literature and promotional materials
44.
Pontiac Grand Prix
–
The Pontiac Grand Prix is an American autombile produced in seven generations in various sizes and degrees of performance and luxury by Pontiac from 1962 to 2008. The performance-minded John De Lorean, head of Advanced Engineering at Pontiac, early models were available with Pontiac performance options, including the factory-race Super Duty 421 powertrain installed in a handful of 1962 and 1963 cars. The full-size Catalina-based Grand Prix sold well through the 1960s, the first Grand Prix was a Catalina hardtop coupe trimmed to standards similar to the larger top-line Bonneville, with a distinctive grille and taillights. The bucket seats were upholstered in Morrokide vinyl, while nylon loop-blend carpeting covered the floor, the center console-mounted transmission shifter included a storage compartment and a tachometer. The rear bench seat included a center fold-down armrest and a grille that could be made functional with the extra-cost Bi-Phonic rear speaker. Included were an instrument panel, deluxe steering wheel, courtesy lights. The standard engine was the Bonnevilles 303 hp 389 cu in V8, Tri-Power carburation raised output to 318 hp. Two other high-performance 389s were offered, including a four-barrel version rated at 333 hp, late in the model year a street version of the race-oriented 421 Pontiac offered in 1961-62 became available, but only in a four-barrel form rated at 320 hp. Pontiac also offered the 421 cu in Super Duty with two carburetors, rated at 405 hp, as a US$2,250 option. A three-speed manual transmission was standard, with a Borg-Warner T-10 four-speed with Hurst shifter, also new was a Pontiac-trademark split grille with vertical headlights and round parking lights and hidden taillights out back. Aside from grillework, taillight covering and bumpers, chrome trim was limited to lower rocker panels, wheel arches, inside, the GP continued with luxurious interiors featuring real walnut trim on the instrument panel and bucket seats upholstered in Morrokide vinyl. The center console was now built into the instrument panel and featured a vacuum gauge to go along with a dash mounted tachometer, pedals received revised custom trim plates. New options this year included an AM/FM radio, cruise control, the 303 hp 389 four-barrel V8 remained the standard engine. The same selection of transmissions continued including the standard three-speed manual, optional four-speed manual, the 1964 Grand Prix received only minor appearance changes from the 1963 edition. Those included a grille with new GP logos and rear deck trim with new taillights, still hidden. Revised upholstery trims highlighted the interior, still featuring expanded Morrokide vinyl bucket seats, engine offerings were mostly unchanged from 1963 except that the standard 303 hp 389 four-barrel V8 gained three 3 hp, with the extra-cost Hydra-matic transmission. Grand Prixs and all other full-sized Pontiacs were completely restyled for 1965 featuring more rounded bodylines with Coke-bottle profiles, the old GM-X frame was replaced with a new box-frame with side perimeter rails. The standard bucket seats could be upholstered either in expanded Morrokide vinyl or a new cloth-and-Morrokide trim, new for 1965 was a no-cost bench seat option with center armrest available with either upholstery choice
45.
Trinity (nuclear test)
–
Trinity was the code name of the first detonation of a nuclear weapon. It was conducted by the United States Army at 5,29 am on July 16,1945, the test was conducted in the Jornada del Muerto desert about 35 miles southeast of Socorro, New Mexico, on what was then the USAAF Alamogordo Bombing and Gunnery Range. The only structures originally in the vicinity were the McDonald Ranch House and its ancillary buildings, a base camp was constructed, and there were 425 people present on the weekend of the test. The code name Trinity was assigned by J. Robert Oppenheimer, the test was of an implosion-design plutonium device, informally nicknamed The Gadget, of the same design as the Fat Man bomb later detonated over Nagasaki, Japan, on August 9,1945. The complexity of the design required a major effort from the Los Alamos Laboratory, the test was planned and directed by Kenneth Bainbridge. Fears of a fizzle led to the construction of a containment vessel called Jumbo that could contain the plutonium, allowing it to be recovered. A rehearsal was held on May 7,1945, in which 108 short tons of high explosive spiked with radioactive isotopes were detonated, the Gadgets detonation released the explosive energy of about 22 kilotons of TNT. Observers included Vannevar Bush, James Chadwick, James Conant, Thomas Farrell, Enrico Fermi, Richard Feynman, Leslie Groves, Robert Oppenheimer, Geoffrey Taylor, and Richard Tolman. The test site was declared a National Historic Landmark district in 1965, the creation of nuclear weapons arose from scientific and political developments of the 1930s. The decade saw many new discoveries about the nature of atoms, the concurrent rise of fascist governments in Europe led to a fear of a German nuclear weapon project, especially among scientists who were refugees from Nazi Germany and other fascist countries. When their calculations showed that nuclear weapons were theoretically feasible, the British and these efforts were transferred to the authority of the U. S. Army in June 1942, and became the Manhattan Project. Brigadier General Leslie R. Groves, Jr. was appointed its director in September 1942, the weapons development portion of this project was located at the Los Alamos Laboratory in northern New Mexico, under the directorship of physicist J. Robert Oppenheimer. The University of Chicago, Columbia University and the Radiation Laboratory at the University of California, production of the fissile isotopes uranium-235 and plutonium-239 were enormous undertakings given the technology of the 1940s, and accounted for 80% of the total costs of the project. Uranium enrichment was carried out at the Clinton Engineer Works near Oak Ridge, theoretically, enriching uranium was feasible through pre-existing techniques, but it proved difficult to scale to industrial levels and was extremely costly. Only 0.71 percent of natural uranium was uranium-235, and it was estimated that it would take 27,000 years to produce a gram of uranium with mass spectrometers, plutonium is a synthetic element with complicated physical, chemical and metallurgical properties. It is not found in nature in appreciable quantities, until mid-1944, the only plutonium that had been isolated had been produced in cyclotrons in microgram amounts, whereas weapons required kilograms. In April 1944, physicist Emilio Segrè, the head of the Los Alamos Laboratorys P-5 Group and he discovered that, in addition to the plutonium-239 isotope, it also contained significant amounts of plutonium-240. The Manhattan Project produced plutonium in nuclear reactors at the Hanford Engineer Works near Hanford and this meant that the Thin Man bomb design that the laboratory had developed would not work properly
