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Great Stellated Dodecahedron
In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. Shaving the triangular pyramids off results in an icosahedron. If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Polytope
In geometry, a pentagonal polytope is a regular polytope in ''n'' dimensions constructed from the H''n'' Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as (dodecahedral) or (icosahedral). Family members The family starts as 1-polytopes and ends with ''n'' = 5 as infinite tessellations of 4-dimensional hyperbolic space. There are two types of pentagonal polytopes; they may be termed the ''dodecahedral'' and ''icosahedral'' types, by their three-dimensional members. The two types are duals of each other. Dodecahedral The complete family of dodecahedral pentagonal polytopes are: # Line segment, # Pentagon, # Dodecahedron, (12 pentagonal faces) # 120-cell, (120 dodecahedral cells) # Order-3 120-cell honeycomb, (tessellates hyperbolic 4-space (∞ 120-cell facets) The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Great Stellated Dodecahedron
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron. A small complex icosidodecahedron can be Wythoff construction, constructed from a number of different vertex figures. A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ( --> ), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes. As a compound The small complex icosidodecahedron can be seen as a polyhedron compound, compound of the icosahedron and the great dodecahedron where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) 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, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. 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 \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosidodecahedron
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . Related polyhedra The figure is a rectification of the great icosahedron or the great stellated dodecahedron, much as the (small) icosidodecahedron is related to the (small) icosahedron and (small) dodecahedron, and the cuboctahedron to the cube and octahedron. It shares its vertex arrangement with the icosidodecahedron, which is its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead. It also shares its edge arrangement with the great icosihemidodecahedron (having the triangle faces in common), and with the great dodecahemidodecah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Dodecahedron Truncations
Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" * Artel Great (born 1981), American actor * Great Osobor (born 2002), Spanish-born British basketball player Other uses * ''Great'' (1975 film), a British animated short about Isambard Kingdom Brunel * ''Great'' (2013 film), a German short film * Great (supermarket), a supermarket in Hong Kong * GReAT, Graph Rewriting and Transformation, a Model Transformation Language * Gang Resistance Education and Training Gang Resistance Education And Training, abbreviated G.R.E.A.T., provides a school-based, police officer-instructed program in America that includes classroom instruction and a variety of learning activities. The program was originally adminis ..., or GREAT, a school-based and police officer-instructed program * Global Research and Analysis Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Net Of A Great Stellated Dodecahedron
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 '' 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. 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, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries. During th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Wenninger Polyhedron Models
This is an indexed list of the uniform and stellated polyhedra from the book ''Polyhedron Models'', by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic Uniform polyhedron, uniform polyhedra, as well as 44 Stellation, stellated forms of the convex regular and quasiregular polyhedra. Models listed here can be cited as "Wenninger Model Number ''N''", or ''W''''N'' for brevity. The polyhedra are grouped in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). ''The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.'' Platonic solids (regular convex polyhedra) W1 to W5 Archimedean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Third Stellation Of Dodecahedron Facets
Third or 3rd may refer to: Numbers * 3rd, the ordinal form of the cardinal number 3 * , a fraction of one third * 1⁄60 of a ''second'', i.e., the third in a series of fractional parts in a sexagesimal number system Places * 3rd Street (other) * Third Avenue (other) * Highway 3 Music Music theory *Interval number of three in a musical interval **Major third, a third spanning four semitones **Minor third, a third encompassing three half steps, or semitones **Neutral third, wider than a minor third but narrower than a major third **Augmented third, an interval of five semitones **Diminished third, produced by narrowing a minor third by a chromatic semitone *Third (chord), chord member a third above the root *Degree (music), three away from tonic **Mediant, third degree of the diatonic scale **Submediant, sixth degree of the diatonic scale – three steps below the tonic ** Chromatic mediant, chromatic relationship by thirds *Ladder of thirds, similar to the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Dodecahedron Net
Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" * Artel Great (born 1981), American actor * Great Osobor (born 2002), Spanish-born British basketball player Other uses * ''Great'' (1975 film), a British animated short about Isambard Kingdom Brunel * ''Great'' (2013 film), a German short film * Great (supermarket), a supermarket in Hong Kong * GReAT, Graph Rewriting and Transformation, a Model Transformation Language * Gang Resistance Education and Training Gang Resistance Education And Training, abbreviated G.R.E.A.T., provides a school-based, police officer-instructed program in America that includes classroom instruction and a variety of learning activities. The program was originally adminis ..., or GREAT, a school-based and police officer-instructed program * Global Research and Analysis Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (polyhedron)
In geometry, a net of a polyhedron is an arrangement of non-overlapping Edge (geometry), edge-joined polygons in the plane (geometry), plane which can be folded (along edges) to become the face (geometry), faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel. Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Tiling
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called ''spherical polygons''. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron. The most familiar spherical polyhedron is the Ball (association football), soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some #Improper_cases, "improper" polyhedra, such as hosohedron, hosohedra and their dual polyhedron, duals, dihedron, dihedra, exist as spherical polyhedra, but their flat-faced analogs are Degeneracy (mathematics), degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron. History During the 10th Century, the Islamic scholar Abū al-Wafā' Būz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
