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Tetrakis Hexahedron
In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube[1]) is a Catalan solid
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Riemannian Symmetric Space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be made more precise, in either the language of Riemannian geometry
Riemannian geometry
or of Lie theory. The Riemannian definition is more geometric, and plays a deep role in the theory of holonomy. The Lie-theoretic definition is more algebraic. In Riemannian geometry, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M,g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space T p M displaystyle T_ p M of M at p by minus the identity
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Omnitruncated
In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a shortcut term which has a different meaning in progressively-higher-dimensional polytopes:Uniform polytope#Truncation_operatorsFor regular polygons: An ordinary truncation, t0,1 p = t p = 2p .Coxeter-Dynkin diagram For uniform polyhedra (3-polytopes): A cantitruncation, t0,1,2 p,q = tr p,q . (Application of both cantellation and truncation operations)Coxeter-Dynkin diagram: For Uniform 4-polytopes: A runcicantitruncation, t0,1,2,3 p,q,r . (Application of runcination, cantellation, and truncation operations)Coxeter-Dynkin diagram: , , For uniform polytera (5-polytopes): A steriruncicantitruncation, t0,1,2,3,4 p,q,r,s
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On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences
On-Line Encyclopedia of Integer Sequences
(OEIS), also cited simply as Sloane's, is an online database of integer sequences. It was created and maintained by Neil Sloane
Neil Sloane
while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009.[3] Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited
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Stereographic Projection
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography
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Great Circle
A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them
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Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k displaystyle u_ 0 ,dots ,u_ k in mathbb R ^ k are affinely independent, which means u 1 − u 0 , … , u k − u 0 displaystyle u_ 1 -u_ 0 ,dots ,u_ k -u_ 0 are linearly independent
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Special Linear Group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det : GL ⁡ ( n , F ) → F × . displaystyle det colon operatorname GL (n,F)to F^ times
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1][2] Groups share a fundamental kinship with the notion of symmetry
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Building (mathematics)
In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces
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Perspective Projection
Perspective (from Latin: perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye
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Gamer
A gamer is a person who plays interactive games, either video games, board games, skill-based card games or physical games, and plays for usually long periods of time. (In some countries, such as the UK, the term "gaming" can also refer to legalized gambling, which can take both traditional tabletop and digital forms.) There are many different gamer communities around the world
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Fluorite
Fluorite
Fluorite
(also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon. Mohs scale of mineral hardness, based on scratch Hardness comparison, defines value 4 as Fluorite. Fluorite
Fluorite
is a colorful mineral, both in visible and ultraviolet light, and the stone has ornamental and lapidary uses. Industrially, fluorite is used as a flux for smelting, and in the production of certain glasses and enamels. The purest grades of fluorite are a source of fluoride for hydrofluoric acid manufacture, which is the intermediate source of most fluorine-containing fine chemicals. Optically clear transparent fluorite lenses have low dispersion, so lenses made from it exhibit less chromatic aberration, making them valuable in microscopes and telescopes
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Copper
Copper
Copper
is a chemical element with symbol Cu (from Latin: cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a reddish-orange color. Copper
Copper
is used as a conductor of heat and electricity, as a building material, and as a constituent of various metal alloys, such as sterling silver used in jewelry, cupronickel used to make marine hardware and coins, and constantan used in strain gauges and thermocouples for temperature measurement. Copper
Copper
is one of the few metals that occur in nature in directly usable metallic form (native metals) as opposed to needing extraction from an ore. This led to very early human use, from c. 8000 BC. It was the first metal to be smelted from its ore, c. 5000 BC, the first metal to be cast into a shape in a mold, c
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Pythagorean Theorem
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry
Euclidean geometry
among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":[1] a 2 + b 2 = c 2 , displaystyle a^ 2 +b^ 2 =c^ 2 , where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. Although it is often argued that knowledge of the theorem predates him,[2][3] the theorem is named after the ancient Greek mathematician Pythagoras
Pythagoras
(c
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Crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.[1][2] In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification. The word crystal derives from the Ancient Greek
Ancient Greek
word κρύσταλλος (krustallos), meaning both "ice" and "rock crystal",[3] from κρύος (kruos), "icy cold, frost".[4][5] Examples of large crystals include snowflakes, diamonds, and table salt. Most inorganic solids are not crystals but polycrystals, i.e. many microscopic crystals fused together into a single solid
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