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Uniform Polyhedron Compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices. The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. The prismatic compounds of prisms ( UC20 and UC21) exist only when , and when and are coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv .... The uniform prismatic compounds of antiprisms ( UC22, UC23, UC24 and UC25) exist only when , and when and are coprime. Furthermore, when , the antiprisms dege ... [...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] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation. ;Chiral: *''Dn'', [''n'',2]+, (22''n'') of order 2''n'' – dihedral symmetry or para-n-gonal group (abstract group: Dihedral group, ''Dihn''). ;Achiral: *''Dnh'', [''n'',2], (*22''n'') of order 4''n'' – prismatic symmetry or full ortho-n-gonal group (abstract group: ''Dihn'' × ''Z''2). *''Dnd'' (or ''Dnv''), [2''n'',2+], (2*''n'') of order 4''n'' – antiprismatic symmetry or full gyro-n-gonal group (abstract group: ''Dih''2''n''). For a given ''n'', all three have ''n''-fold rotational symmetry about one axis (rotation by an angle of 360°/''n'' does not change the object), and 2-fold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UC03-6 Tetrahedra
UC may refer to: Education In the United States * University of California system * University of Charleston, West Virginia * University of Chicago, Illinois * University of Cincinnati, Ohio * Upsala College, East Orange, New Jersey (''defunct since 1995'') * Utica College, Utica, New York * Harvard Undergraduate Council, Harvard College's student government body * University college In other countries * Pontifical Catholic University of Chile * University of Canberra, Australia * University of Cantabria, Spain * University of Canterbury, New Zealand * University of Cebu, Cebu City, Philippines * University of Coimbra, Portugal * University of the Cordilleras, Baguio, Philippines * Uva College, Badulla, Sri Lanka * Uxbridge College, England * University of Calgary, Canada Science, technology, and mathematics Biology and medicine * Ulcerative colitis, a type of inflammatory bowel disease * Umbilical cord * Unassisted childbirth, birth without the aid of professional bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Six Tetrahedra
The compound of six tetrahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating each octahedron in the compound of three octahedra. It is one of only five polyhedral compounds (along with the compound of two great dodecahedra, the compound of five great dodecahedra, the compound of two small stellated dodecahedra, and the compound of five small stellated dodecahedra) which is vertex-transitive and face-transitive but not edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea .... References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual polyhedron, dual to an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the Crystal system#Overview of point groups by crystal system, crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full octah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UC02-12 Tetrahedra
UC may refer to: Education In the United States * University of California system * University of Charleston, West Virginia * University of Chicago, Illinois * University of Cincinnati, Ohio * Upsala College, East Orange, New Jersey (''defunct since 1995'') * Utica College, Utica, New York * Harvard Undergraduate Council, Harvard College's student government body * University college In other countries * Pontifical Catholic University of Chile * University of Canberra, Australia * University of Cantabria, Spain * University of Canterbury, New Zealand * University of Cebu, Cebu City, Philippines * University of Coimbra, Portugal * University of the Cordilleras, Baguio, Philippines * Uva College, Badulla, Sri Lanka * Uxbridge College, England * University of Calgary, Canada Science, technology, and mathematics Biology and medicine * Ulcerative colitis, a type of inflammatory bowel disease * Umbilical cord * Unassisted childbirth, birth without the aid of professional bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Twelve Tetrahedra With Rotational Freedom
This uniform polyhedron compound is a symmetric arrangement of 12 tetrahedra, considered as antiprisms. It can be constructed by superimposing six identical copies of the stella octangula, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each stella octangula is rotated by an equal (and opposite, within a pair) angle θ. Equivalently, a stella octangula may be inscribed within each cube in the compound of six cubes with rotational freedom, which has the same vertices as this compound. When ''θ'' = 0, all six stella octangula coincide. When ''θ'' is 45 degrees, the stella octangula coincide in pairs yielding (two superimposed copies of) the compound of six tetrahedra. Except if θ = 0 or θ = 45, the compound of twelve tetrahedra with rotational freedom will have a convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 bilatera ... [...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] |
Tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such net (polyhedron), nets. For any tetrahedron there exists a sphere (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UC01-6 Tetrahedra
UC may refer to: Education In the United States * University of California system * University of Charleston, West Virginia * University of Chicago, Illinois * University of Cincinnati, Ohio * Upsala College, East Orange, New Jersey (''defunct since 1995'') * Utica College, Utica, New York * Harvard Undergraduate Council, Harvard College's student government body * University college In other countries * Pontifical Catholic University of Chile * University of Canberra, Australia * University of Cantabria, Spain * University of Canterbury, New Zealand * University of Cebu, Cebu City, Philippines * University of Coimbra, Portugal * University of the Cordilleras, Baguio, Philippines * Uva College, Badulla, Sri Lanka * Uxbridge College, England * University of Calgary, Canada Science, technology, and mathematics Biology and medicine * Ulcerative colitis, a type of inflammatory bowel disease * Umbilical cord * Unassisted childbirth, birth without the aid of professional bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
