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Truncated 600-cell
In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 600-cell''. Truncated 120-cell The truncated 120-cell or truncated hecatonicosachoron is a uniform 4-polytope, constructed by a uniform truncation of the regular 120-cell 4-polytope. It is made of 120 truncated dodecahedral and 600 tetrahedral cells. It has 3120 faces: 2400 being triangles and 720 being decagons. There are 4800 edges of two types: 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron. Each vertex has 3 truncated dodecahedra and one tetrahedron around it. Its vertex figure is an equilateral triangular pyramid. Alternate names * Truncated 120-cell ( Norman W. Johnson) ** Tuncated hecatonicosachoron / Truncated dodecacontachoron / Truncated polydodecahedron * Truncated-icosahedral hexaco ... [...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] |
Schlegel Half-solid Truncated 120-cell
Schlegel is a German occupational surname. Notable people with the surname include: * Anthony Schlegel (born 1981), American football player * August Wilhelm Schlegel (1767–1845), German poet, brother of Friedrich * Brad Schlegel (born 1968), Canadian ice hockey player * Bernhard Schlegel (born 1951), German-Canadian chemist * Carmela Schlegel (born 1983), Swiss swimmer * Catharina von Schlegel (1697 – after 1768), German hymn writer * Dorothea von Schlegel (1764–1839), German novelist and translator, wife of Friedrich * Elfi Schlegel (born 1964), Canadian gymnast and sportscaster * Frits Schlegel (1896–1965), Danish architect * Gustaaf Schlegel (1840–1903), Dutch sinologist and field naturalist * Hans Schlegel (born 1951), German astronaut * Helmut Schlegel (born 1943), German Franciscan, priest, author, meditation instructor, songwriter * Hermann Schlegel (1804–1884), German ornithologist and herpetologist * Johan Frederik Schlegel (1817–1896), Danish civil servant ... [...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] |
Tetrakis 600-cell , a convex polyhedron with 32 faces
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Tetrakis may refer to: *Tetrakis (Paphlagonia), an ancient Greek city * Tetrakis cuboctahedron, convex polyhedron with 32 triangular faces *Tetrakis hexahedron, an Archimedean dual solid or a Catalan solid *Tetrakis square tiling, a tiling of the Euclidean plane *Tetrakis(triphenylphosphine)palladium(0), a catalyst in organic chemistry See also * Tetracus * Tetrakis legomenon, a word that occurs only four times within a context * Tetricus (other) * Tetrix (other) *Truncated tetrakis cube The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces: 24 sets of 3 bilateral symmetry pentagons arranged in an Octahedral symmetry, octahedral arrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Pyramid
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of 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 nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces. Reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 120-cell Verf
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb_+ to be truncated and n \in \mathbb_0, the number of elements to be kept behind the decimal point, the truncated value of x is :\operatorname(x,n) = \frac. However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the \operatorname function rounds towards negative infinity. For a given number x \in \mathbb_-, the function \operatorname is used instead :\operatorname(x,n) = \frac. Causes of truncation With computers, truncation can occur when a decimal number is typecast as an integer; it is truncated to zero decimal digits because integers cannot store non-integer real numbers. In algebra An analogue of truncation can be applied to polynomials. In t ... [...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] |
Decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. Regular decagon A '' regular decagon'' has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a truncated pentagon, t, a quasiregular decagon alternating two types of edges. Side length The picture shows a regular decagon with side length a and radius R of the circumscribed circle. * The triangle E_E_1M has two equally long legs with length R and a base with length a * The circle around E_1 with radius a intersects ]M\,E_ in a point P (not designated in the picture). * Now the triangle \; is an isosceles triangle">/math> in a point P (not designated in the picture). * Now the triangle \; is an isosceles triangle with vertex E_1 and with base angles m\angle E_1 E_ P = m\angle E_ P E_1 = 72 ... [...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] |
Truncated Dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. Construction The truncated dodecahedron is constructed from a regular dodecahedron by cutting all of its vertices off, a process known as truncation. Alternatively, the truncated dodecahedron can be constructed by expansion: pushing away the edges of a regular dodecahedron, forming the pentagonal faces into decagonal faces, as well as the vertices into triangles. Therefore, it has 32 faces, 90 edges, and 60 vertices. The truncated dodecahedron may also be constructed by using Cartesian coordinates. With an edge length 2\varphi - 2 centered at the origin, they are all even permutations of \left(0, \pm \frac, \pm (2 + \varphi) \right), \qquad \left(\pm \frac, \pm \varphi, \pm 2 \varphi \right), \qquad \left(\pm \varphi, \pm 2, \pm (\varphi + 1) \right), where \varphi = \frac is the golden ratio. Properties The surfac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including '' The Fifty-Nine Icosahedra'' (1938) and '' Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter ... [...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] |
