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Rectified 120-cell
In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC120. There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself. The birectified 120-cell is more easily seen as a rectified 600-cell, and the trirectified 120-cell is the same as the dual 600-cell. Rectified 120-cell In geometry, the rectified 120-cell or rectified hecatonicosachoron is a convex uniform 4-polytope composed of 600 regular tetrahedra and 120 icosidodecahedra cells. Its vertex figure is a triangular prism, with three icosidodecahedra and two tetrahedra meeting at each vertex. Alternative names: * Rectified 120-cell ( Norman Johnson) * Rectified hecatonicosichoron / rectified dodecacontachoron / rectified polydodecahedron * Icosidodecahedral hexacosihecatonicosachoron * Rahi (Jonathan Bowers: for rectified hecatonicosachoron) * Ambohecatonicos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
120-cell T0 H3
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural number, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean space, Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a Recursive definition, recursive description, starting with \ for a p-sided regular polygon that is Convex set, convex. For example, is an equilateral triangle, is a Square (geometry), square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols \ contain irreducible fractions p/q, where p is the number of vertices, and q is their turning number. Equivalently, \ is created from the vertices of \, connected every q. For example, \ is a pentagram; \ is a pentagon. A regular pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 means that there is only one type of edge to the object: given two edges, there is a Translation (geometry), translation, Rotation (mathematics), rotation, and/or Reflection (mathematics), reflection that will move one edge to the other while leaving the region occupied by the object unchanged. Isotoxal polygons An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The Duality (mathematics)#Dimension-reversing dualities, duals of isotoxal polygons are isogonal polygons. Isotoxal 4n-gons are Central symmetry, centrally symmetric, thus are also zonogons. In general, a (non-regular) isotoxal 2n-gon has \mathrm_n, (^*nn) dihedral symmetry. For example, a (non-square) rhombus is an isotoxa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single '' symmetry orbit''. All vertices of a finite -dimensional isogonal figure exist on an -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates that simply asserting that "all vertices look ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Prism
In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangular prism may be both Semiregular polyhedron, semiregular and Uniform polyhedron, uniform. The triangular prism can be used in constructing another polyhedron. Examples are some of the Johnson solids, the truncated right triangular prism, and Schönhardt polyhedron. Properties A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its ''bases'', and the bases of a triangular prism are triangles. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 parallelograms as other faces. If the prism's edges are perpendicular to the base, the lateral faces are rectangles, and the prism is called a ''right triangular prism''. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 120-cell Verf
Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Rectifiable set, in mathematics Science * GHK flux equation#Rectification, in biology, a process in cell membranes Technology * Image rectification, adjustment of images to simplify stereo vision or to map images to a map coordinate system (GIS) * The function of a rectifier, a device that converts alternating electrical current to direct current * Rectified airspeed, a means of displaying the airspeed of high-speed aircraft * Rectification (chemical/process engineering), countercurrent distillation, a unit operation also used for the production of rectified spirit (see Distillation#Fractional distillation) Other uses * Rectification (law), an equitable legal remedy whereby a court orders a change in a written document to reflect what i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...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] |
