1.
Dispersive prism
–
In optics, a dispersive prism is a type of optical prism, usually having the shape of a geometrical triangular prism. It is the most widely known type of optical prism, although not the most common in actual use. Triangular prisms are used to light, that is, to break light up into its spectral components. A good mathematical description of dispersion is given by Born. Prism dispersion played an important role in understanding the nature of light, through experiments by Sir Isaac Newton, although the refractive index is dependent on the wavelength in every material, some materials have a much more powerful wavelength dependence than others. Crown glasses such as BK7 have a relatively small dispersion, while flint glasses have a much stronger dispersion, fused quartz is used in the ultraviolet as normal glasses lose their transparency there. The top angle of the prism can be chosen to influence the exact dispersion characteristics, however, it is typically chosen such that both the incoming and outgoing light rays hit the surface approximately at the Brewsters angle, so that reflection losses are minimized. An example is the use of type of prisms in prism compressors for generation of ultrafast laser pulses. Types of dispersive prism include, Triangular prism Abbe prism Pellin–Broca prism Amici prism Compound prism Diffraction gratings may be replicated onto prisms to form grating prisms, a transmission grism is a useful component in an astronomical telescope, allowing observation of stellar spectra. A reflection grating replicated onto a prism allows light to diffract inside the prism medium, the iconic graphic shows a coherent ray of white light entering the prism and beginning to disperse, and shows the spectrum leaving the prism. Multiple-prism dispersion theory Multiple-prism grating laser oscillators Light
2.
Prismatic uniform polyhedron
–
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids, because they are isogonal, their vertex arrangement uniquely corresponds to a symmetry group. Each has p reflection planes which contain the p-fold axis, the Dph symmetry group contains inversion if and only if p is even, while Dpd contains inversion symmetry if and only if p is odd. There are, prisms, for each rational number p/q >2, with symmetry group Dph, antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph if q is even. If p/q is an integer, i. e. if q =1, an antiprism with p/q <2 is crossed or retrograde, its vertex figure resembles a bowtie. If p/q ≤ 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality, Uniform polyhedron Prism Antiprism Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Philosophical Transactions of the Royal Society of London, P.175 Skilling, John, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society,79, 447–457, doi,10. 1017/S0305004100052440, MR0397554. Prisms and Antiprisms George W. Hart
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.
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
5.
Dihedral symmetry in three dimensions
–
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn. There are 3 types of symmetry in three dimensions, each shown below in 3 notation, Schönflies notation, Coxeter notation. For n = ∞ they correspond to three frieze groups, Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal is used with respect to an axis of rotation. In 2D the symmetry group Dn includes reflections in lines, in 3D the two operations are distinguished, the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, with reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh. Dnd, has vertical mirror planes between the rotation axes, not through them. As a result the vertical axis is a 2n-fold rotoreflection axis, Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the group for a regular n-sided antiprism. Dn is the group of a partially rotated prism. D2 +, of order 4 is one of the three symmetry group types with the Klein four-group as abstract group and it has three perpendicular 2-fold rotation axes. It is the group of a cuboid with an S written on two opposite faces, in the same orientation. D2h, of order 8 is the group of a cuboid D2d. For Dnh, order 4n Cnh, order 2n Cnv, order 2n Dn, +, order 2n For Dnd, order 4n S2n, order 2n Cnv, order 2n Dn, +, cS1 maint, Multiple names, authors list N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conway, John Horton, Huson, Daniel H
6.
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
7.
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
8.
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
9.
Triangular bipyramid
–
In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the prism with 6 isosceles triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face, although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four. The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, as a Johnson solid with all faces equilateral triangles, it is also a deltahedron. The dual polyhedron of the bipyramid is the triangular prism, with five faces. Although the triangular prism has a form that is a polyhedron, the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces. This polyhedron has 24 equilateral triangle faces, but it is not a Johnson solid because it has coplanar faces and it is a coplanar 24-triangle deltahedron. This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb, larger triangular polyhedra can be generated similarly, like 9,16 or 25 triangles per larger triangle face, seen as a section of a triangular tiling. The triangular bipyramid can form a tessellation of space with octahedra or with truncated tetrahedra, trigonal bipyramidal molecular geometry Eric W. Weisstein, Triangular dipyramid at MathWorld. Conway Notation for Polyhedra Try, dP3
10.
Convex set
–
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the hull of A. It is the smallest convex set containing A, a convex function is a real-valued function defined on an interval with the property that its epigraph is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets, the branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
11.
Vertex configuration
–
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
12.
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
13.
Prism (geometry)
–
In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
14.
Polyhedron
–
In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
15.
Triangle
–
A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
16.
Translation (geometry)
–
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
17.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
18.
Normal (geometry)
–
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a normal, or simply normal. The word normal is used as an adjective, a line normal to a plane, the normal component of a force. The concept of normality generalizes to orthogonality, the concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P, in the case of differential curves, the curvature vector is a normal vector of special interest. For a convex polygon, a surface normal can be calculated as the cross product of two edges of the polygon. For a plane given by the equation a x + b y + c z + d =0, the vector is a normal. For a hyperplane in n+1 dimensions, given by the equation r = a 0 + α1 a 1 + ⋯ + α n a n, where a0 is a point on the hyperplane and ai for i =1. N are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the space of A where A is given by A =. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. If a surface S is parameterized by a system of coordinates x, with s and t real variables. For a surface S given explicitly as a function f of the independent variables x, y, the first one is obtaining its implicit form F = z − f =0, from which the normal follows readily as the gradient ∇ F. The second way of obtaining the normal follows directly from the gradient of the form, ∇ f, by inspection, ∇ F = k ^ − ∇ f. Note that this is equal to ∇ F = k ^ − ∂ f ∂ x i ^ − ∂ f ∂ y j ^, if a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base, however, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous, a normal to a surface does not have a unique direction, the vector pointing in the opposite direction of a surface normal is also a surface normal. For an oriented surface, the normal is usually determined by the right-hand rule
19.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, by comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. Each diagonal divides the quadrilateral into two congruent triangles, the sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2, the sum of the distances from any interior point to the sides is independent of the location of the point. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms
20.
Semiregular polyhedron
–
The term semiregular polyhedron is used variously by different authors. In its original definition, it is a polyhedron with faces and a symmetry group which is transitive on its vertices. These polyhedra include, The thirteen Archimedean solids, an infinite series of convex prisms. An infinite series of convex antiprisms and these semiregular solids can be fully specified by a vertex configuration, a listing of the faces by number of sides in order as they occur around a vertex. For example,3.5.3.5, represents the icosidodecahedron which alternates two triangles and two pentagons around each vertex,3.3.3.5 in contrast is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive, since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte provided a definition which Coxeter found too artificial, Coxeter himself dubbed Gossets figures uniform, with only a quite restricted subset classified as semiregular. Yet others have taken the path, categorising more polyhedra as semiregular. These include, Three sets of polyhedra which meet Gossets definition. The duals of the above semiregular solids, arguing that since the polyhedra share the same symmetries as the originals. These duals include the Catalan solids, the convex dipyramids and antidipyramids or trapezohedra, a further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing. Gossets definition of semiregular includes figures of higher symmetry, the regular and quasiregular polyhedra and this naming system works well, and reconciles many of the confusions. Assuming that ones stated definition applies only to convex polyhedra is probably the most common failing, Coxeter, Cromwell and Cundy & Rollett are all guilty of such slips. In many works semiregular polyhedron is used as a synonym for Archimedean solid and we can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular and facially-transitive duals. Later, Coxeter would quote Gossets definition without comment, thus accepting it by implication, peter Cromwell writes in a footnote to Page 149 that, in current terminology, semiregular polyhedra refers to the Archimedean and Catalan solids. On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans, by implication this treats the Catalans as not semiregular, thus effectively contradicting the definition he provided in the earlier footnote. Semiregular polytope Regular polyhedron Weisstein, Eric W. Semiregular polyhedron
21.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
22.
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