|
Polyhedral Group
In geometry, the polyhedral groups are the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The Tetrahedral symmetry, tetrahedral group of order 12, rotational symmetry group of the tetrahedron, regular tetrahedron. It is isomorphic to Alternating group, A4. ** The conjugacy classes of T are: *** identity *** 4 × rotation by 120°, order 3, cw *** 4 × rotation by 120°, order 3, ccw *** 3 × rotation by 180°, order 2 *The Octahedral symmetry, octahedral group of order 24, rotational symmetry group of the cube and the octahedron, regular octahedron. It is isomorphic to Symmetric group, S4. ** The conjugacy classes of O are: *** identity *** 6 × rotation by ±90° around vertices, order 4 *** 8 × rotation by ±120° around triangle centers, order 3 *** 3 × rotation by 180° around vertices, order 2 *** 6 × rotation by 180° around midpoints of edges, order 2 * The Icosahedral symmetry, icosahedral group of order 60, rotational symmetry group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Symmetry Group Th
A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrakis Hexahedron Stereographic D2 Gyrations
Tetrakis may refer to: *Tetrakis (Paphlagonia), an ancient Greek city *Tetrakis cuboctahedron, convex polyhedron with 32 triangular faces *Tetrakis hexahedron, an Archimedean dual solid or a Catalan solid *Tetrakis square tiling, a tiling of the Euclidean plane *Tetrakis(triphenylphosphine)palladium(0), a catalyst in organic chemistry See also *Tetracus *Tetrakis legomenon, a word that occurs only four times within a context *Tetricus (other) *Tetrix (other) *Truncated tetrakis cube, a convex polyhedron with 32 faces * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Symmetry Group T
A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhomb
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , romanized: , meaning "to turn round and round." The word was used both by Euclid and Archimedes, who used the term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purple Fire
Purple is a color similar in appearance to violet light. In the RYB color model historically used in the arts, purple is a secondary color created by combining red and blue pigments. In the CMYK color model used in modern printing, purple is made by combining magenta pigment with either cyan pigment, black pigment, or both. In the RGB color model used in computer and television screens, purple is created by mixing red and blue light in order to create colors that appear similar to violet light. According to color theory, purple is considered a cool color. Purple has long been associated with royalty, originally because Tyrian purple dye—made from the secretions of sea snails—was extremely expensive in antiquity. Purple was the color worn by Roman magistrates; it became the imperial color worn by the rulers of the Byzantine Empire and the Holy Roman Empire, and later by Roman Catholic bishops. Similarly in Japan, the color is traditionally associated with the emperor and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-fold Rotation Axis
In algebraic geometry, a 3-fold or threefold is a 3-dimensional algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, .... The Mori program showed that 3-folds have minimal models. References * * * {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Symmetry
image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating group, alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere). They are among the Crystal system#Overview of point groups by crystal system, crystallographic point gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
