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Cyclic Symmetries
In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object. They are the finite symmetry groups on a cone. For ''n'' = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation. Types ;Chiral: *''Cn'', sup>+, (''nn'') of order ''n'' - ''n''-fold rotational symmetry - acro-n-gonal group (abstract group ''Zn''); for ''n''=1: no symmetry ( trivial group) ;Achiral: *''Cnh'', +,2 (''n''*) of order 2''n'' - prismatic symmetry or ortho-n-gonal group (abstract group ''Zn'' × ''Dih1''); for ''n''=1 this is denoted by ''Cs'' (1*) and called reflection symmetry, also bila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a '' geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion In A Point
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniaxial S6
In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frieze Example P11g
In architecture, the frieze is the wide central section part of an entablature and may be plain in the Ionic or Doric order, or decorated with bas-reliefs. Paterae are also usually used to decorate friezes. Even when neither columns nor pilasters are expressed, on an astylar wall it lies upon the architrave ("main beam") and is capped by the moldings of the cornice. A frieze can be found on many Greek and Roman buildings, the Parthenon Frieze being the most famous, and perhaps the most elaborate. This style is typical for the Persians. In interiors, the frieze of a room is the section of wall above the picture rail and under the crown moldings or cornice. By extension, a frieze is a long stretch of painted, sculpted or even calligraphic decoration in such a position, normally above eye-level. Frieze decorations may depict scenes in a sequence of discrete panels. The material of which the frieze is made of may be plasterwork, carved wood or other decorative medium. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniaxial C6h
In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frieze Example P11m
In architecture, the frieze is the wide central section part of an entablature and may be plain in the Ionic or Doric order, or decorated with bas-reliefs. Paterae are also usually used to decorate friezes. Even when neither columns nor pilasters are expressed, on an astylar wall it lies upon the architrave ("main beam") and is capped by the moldings of the cornice. A frieze can be found on many Greek and Roman buildings, the Parthenon Frieze being the most famous, and perhaps the most elaborate. This style is typical for the Persians. In interiors, the frieze of a room is the section of wall above the picture rail and under the crown moldings or cornice. By extension, a frieze is a long stretch of painted, sculpted or even calligraphic decoration in such a position, normally above eye-level. Frieze decorations may depict scenes in a sequence of discrete panels. The material of which the frieze is made of may be plasterwork, carved wood or other decorative medium. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniaxial C6v
In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frieze Example P1m1
In architecture, the frieze is the wide central section part of an entablature and may be plain in the Ionic or Doric order, or decorated with bas-reliefs. Paterae are also usually used to decorate friezes. Even when neither columns nor pilasters are expressed, on an astylar wall it lies upon the architrave ("main beam") and is capped by the moldings of the cornice. A frieze can be found on many Greek and Roman buildings, the Parthenon Frieze being the most famous, and perhaps the most elaborate. This style is typical for the Persians. In interiors, the frieze of a room is the section of wall above the picture rail and under the crown moldings or cornice. By extension, a frieze is a long stretch of painted, sculpted or even calligraphic decoration in such a position, normally above eye-level. Frieze decorations may depict scenes in a sequence of discrete panels. The material of which the frieze is made of may be plasterwork, carved wood or other decorative medium. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniaxial C6
In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frieze Example P1
In architecture, the frieze is the wide central section part of an entablature and may be plain in the Ionic or Doric order, or decorated with bas-reliefs. Paterae are also usually used to decorate friezes. Even when neither columns nor pilasters are expressed, on an astylar wall it lies upon the architrave ("main beam") and is capped by the moldings of the cornice. A frieze can be found on many Greek and Roman buildings, the Parthenon Frieze being the most famous, and perhaps the most elaborate. This style is typical for the Persians. In interiors, the frieze of a room is the section of wall above the picture rail and under the crown moldings or cornice. By extension, a frieze is a long stretch of painted, sculpted or even calligraphic decoration in such a position, normally above eye-level. Frieze decorations may depict scenes in a sequence of discrete panels. The material of which the frieze is made of may be plasterwork, carved wood or other decorative medium. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation. Although Schoenflies notation without superscripts is a pure point group notation, optionally, superscripts can be added to further specify individual space groups. However, for space groups, the connection to the underlying symmetry elements is much more clear in Hermann–Mauguin notation, so the latter n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold Notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2). Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotation of finite order around a point * infinite rotation around a line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
