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Disdyakis Triacontahedron
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron, kisrhombic triacontahedron or d120 is a Catalan solid with 120 faces and the dual to the Archimedean solid, Archimedean truncated icosidodecahedron. As such it is Isohedral figure, face-uniform but with Regular polygon, irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single Vertex (geometry), vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place. If the bipyramids, the gyroelongated bipyramids, and the trapezohedra are excluded, the disdyakis triacontahedron has the most ... [...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] |
Gyroelongated Bipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves. Forms Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the '' gyroelongated triangular bipyramid'' if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles. Use Pentagonal Gyroelongated Bipyramid * The Pentagonal gyroelongated bipyramid is commonly used in 3D modeling. It appears in various 3D programs such as Autodesk Maya, Blender (software) or Cinema 4D. In most programs, it's referred to as an ico-spere, or Icosahedron. It is also commonly used as a dice in Dungeons and Dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base Vertex (geometry), vertices are usually coplanar and a bipyramid is usually ''symmetric'', meaning the two pyramids are mirror images across their common base plane. When each apex (geometry), apex (, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a ''right'' bipyramid; otherwise it is ''oblique''. When the base is a regular polygon, the bipyramid is also called ''regular''. Definition and properties A bipyramid is a polyhedron constructed by fusing two Pyramid (geometry), pyramids which share the same polygonal base (geometry), base; a pyramid is in turn constructed by connecting each vertex of its base to a single new vertex (geometry), vertex (th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex Isogonal figure, isogonal nonprismatic solids constructed by two or more types of regular polygon Face (geometry), faces. The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or "Chirality (mathematics), enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron. Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol s \scriptstyle\begin 5 \\ 3 \end and flat Schläfli symbol Cartesian coord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . 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 , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Poinsot polyhedron, regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The #Pyritohedron, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling polyhedra, space-filling. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool in algebraic topology. Motivation The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: This substitution allows one to assign combinatorial invariants such as the Euler characteristic to the spaces. One can ask whether there is an analogous way to replace the continuous functions defined on the topological spaces with functions that are linear on the simplices and homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning that one replaces larger simplices with a union of smaller simplices. A standard way to c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleetope
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a pyramid. In some cases, the pyramid is chosen to have regular sides, often producing a non-convex polytope; alternatively, by using sufficiently shallow pyramids, the results may remain convex. Kleetopes are named after Victor Klee,. although the same concept was known under other names long before the work of Klee. Examples In each of these cases, the Kleetope is formed by attaching pyramids onto each face of the original polyhedron. These examples can be seen from the Platonic solids: * The triakis tetrahedron is the Kleetope of a tetrahedron, the triakis octahedron is the Kleetope of an octahedron, and the triakis icosahedron is the Kleetope of an icosahedron. These Kleetopes are formed by adding a triangular pyramid to each face of them. * The tetrakis hexahedron is the Kleetope of the cube, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (: vertices or vertexes), also called a corner, is a point (geometry), point where two or more curves, line (geometry), lines, or line segments Tangency, meet or Intersection (geometry), intersect. For example, the point where two lines meet to form an angle and the point where edge (geometry), edges of polygons and polyhedron, polyhedra meet are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Triacontahedron
The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex (geometry), vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron and can be seen as a elongated rhombic icosahedron. The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, , so that the Angle#Types of angles, acute angles on each face measure , or approximately 63.43°. A rhombus so obtained is called a ''golden rhombus''. Being the dual of an Archimedean solid, the rhombic triacontahedron is ''face-transitive'', meaning the symmetry group of the solid acts transitive action, transitively on the set of faces. This means that for any two faces, and , there is a rotation or reflection (mathematics), reflection of the solid that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
