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Icosahedral
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Icosahedron
In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol , or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the regular dodecahedron, which is represented by , having three pentagonal faces around each vertex. In most contexts, the unqualified use of the word "icosahedron" refers specifically to this figure. A regular icosahedron is a strictly convex deltahedron and a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes . The plural can be either "icosahedrons" or "icosahedra" (). Dimensions If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is r_u = \frac \sqrt = \frac \sqrt = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedron, dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 Rotational symmetry, rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a Reflection symmetry, reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters. Description Icosahedral symmetry is a mathematical property of objects indicating that an object has the same Symmetry, symmetries as a regular icosahedron. As point group Apart from the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol . Dimensions If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is :r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is :r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a while the midradius, which touches the middle of each edge, is :r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a These quantities may also be expressed as :r_u = a\, \frac \phi :r_i = a\, \frac :r_m = a\, \frac where ''ϕ'' is the golde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the five wounds of Jesus. Today the symbol is widely used by the Wiccans, witches, and pagans. The pentagram has magical associations. Many people who practice neopaganism wear jewelry incorporating the symbol. The word ''pentagram'' comes from the Greek word πεντάγραμμον (''pentagrammon''), from πέντε (''pente''), "five" + γραμμή (' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the '' Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyritohedral Symmetry
image:tetrahedron.jpg, 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 grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Spherical Symmetry Groups
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.Sands, 1993 Involutional symmetry There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Tensegrity Structure
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as 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 many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded (finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the origin as one of them. The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection of its full symmetry group with E+(3), which consists of all ''direct isometries'', i.e., isometries preserving orientation. For a bounded object, the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. 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 a polytope, a more general concept in any number of dimensions. Definition 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, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
