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122 relations: Abelian group, ADE classification, Alternating group, Antiprism, Binary cyclic group, Binary icosahedral group, Binary octahedral group, Binary tetrahedral group, Bipyramid, C2-Symmetric ligands, Chirality (mathematics), Circular symmetry, Closed set, Compound of five tetrahedra, Concanavalin A, Cone, Coordination complex, Correspondence theorem (group theory), Covalent bond, Covering space, Coxeter element, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Crystal system, Crystallographic restriction theorem, Crystallography, Cube, Cycle graph (algebra), Cyclic group, Cyclic symmetry in three dimensions, Cylinder, Dicyclic group, Dihedral group, Dihedral symmetry in three dimensions, Dihedron, Direct product of groups, Disdyakis triacontahedron, Dodecahedron, Du Val singularity, Euclidean group, Euclidean plane isometry, Euler characteristic, Free group, Frieze group, Fundamental domain, Galois connection, Geometry, Great circle, Group action, ..., Harold Scott MacDonald Coxeter, Hermann–Mauguin notation, Homotetramer, Hurwitz quaternion, Icosahedral symmetry, Icosahedron, Icosian, Improper rotation, Index of a subgroup, Involution (mathematics), Irrational number, Isometry, Isometry group, Isomorphism, Journal of Mathematics and the Arts, Kaleidoscope, Lens space, List of character tables for chemically important 3D point groups, List of finite spherical symmetry groups, List of small groups, Lune (geometry), Molecular orbital, Molecular symmetry, Molecule, Normal subgroup, Octahedral symmetry, Octahedron, Orbifold, Orbifold notation, Order (group theory), Orientation (vector space), Orthogonal group, Orthogonal matrix, Pin group, Point group, Point groups in four dimensions, Point groups in two dimensions, Point reflection, Polyhedral group, Polyhedron, Prism (geometry), Projective polyhedron, Projective space, Propeller, Pyramid (geometry), Quaternion, Quaternion group, Reflection symmetry, Rotation, Rotation group SO(3), Rotational symmetry, Schoenflies notation, Simply connected space, Space group, Sphere, Spherical trigonometry, Spin group, Spin representation, Subgroup, Symmetric group, Symmetry, Symmetry group, Symmetry number, Tetrahedral symmetry, Tetrahedron, Topological group, Torsion group, Translational symmetry, Trapezohedron, Unit (ring theory), Wallpaper group, Weyl group. Expand index (72 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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ADE classification
In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams.
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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.
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Binary cyclic group
In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, C_, thought of as an extension of the cyclic group C_n by a cyclic group of order 2.
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Binary icosahedral group
In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120.
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Binary octahedral group
In mathematics, the binary octahedral group, name as 2O or is a certain nonabelian group of order 48.
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Binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or 2,3,3 is a certain nonabelian group of order 24.
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Bipyramid
An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.
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C2-Symmetric ligands
In homogeneous catalysis, a C2-symmetric ligands usually describes bidentate ligands that are dyssymmetric but not asymmetric by virtue of their C2-symmetry.
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Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone.
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Circular symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Compound of five tetrahedra
The compound of five tetrahedra is one of the five regular polyhedral compounds.
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Concanavalin A
Concanavalin A (ConA) is a lectin (carbohydrate-binding protein) originally extracted from the jack-bean, Canavalia ensiformis.
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Cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
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Coordination complex
In chemistry, a coordination complex consists of a central atom or ion, which is usually metallic and is called the coordination centre, and a surrounding array of bound molecules or ions, that are in turn known as ligands or complexing agents.
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Correspondence theorem (group theory)
In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theoremSome authors use "fourth isomorphism theorem" to designate the Zassenhaus lemma; see for example by Alperin & Bell (p. 13) or Depending how one counts the isomorphism theorems, the correspondence theorem can also be called the 3rd isomorphism theorem; see for instance H.E. Rose (2009), p. 78.
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Covalent bond
A covalent bond, also called a molecular bond, is a chemical bond that involves the sharing of electron pairs between atoms.
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Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
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Coxeter element
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.
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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).
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Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.
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Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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Crystal system
In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals.
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Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold.
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Crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure).
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Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
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Cycle graph (algebra)
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Cyclic symmetry in three dimensions
In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) does not change the object.
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Cylinder
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
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Dicyclic group
In group theory, a dicyclic group (notation Dicn or Q4n) is a member of a class of non-abelian groups of order 4n (n > 1).
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Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.
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Dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn (n ≥ 2).
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Dihedron
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges.
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Direct product of groups
In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.
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Disdyakis triacontahedron
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron.
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Dodecahedron
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.
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Du Val singularity
In algebraic geometry, a du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type.
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Euclidean group
In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.
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Euclidean plane isometry
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.
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Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
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Free group
In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.
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Frieze group
In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.
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Fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.
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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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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.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
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Hermann–Mauguin notation
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups.
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Homotetramer
A homotetramer is a protein complex made up of four identical subunits which are associated but not covalently bound.
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Hurwitz quaternion
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded).
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Icosahedral symmetry
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
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Icosahedron
In geometry, an icosahedron is a polyhedron with 20 faces.
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Icosian
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell.
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Improper rotation
In geometry, an improper rotation,.
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Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
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Involution (mathematics)
In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.
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Irrational number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Isometry group
In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Journal of Mathematics and the Arts
The Journal of Mathematics and the Arts is a quarterly peer-reviewed academic journal that deals with relationship between mathematics and the arts.
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Kaleidoscope
A kaleidoscope is an optical instrument with two or more reflecting surfaces tilted to each other in an angle, so that one or more (parts of) objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection.
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Lens space
A lens space is an example of a topological space, considered in mathematics.
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List of character tables for chemically important 3D point groups
This lists the character tables for the more common molecular point groups used in the study of molecular symmetry.
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List of finite spherical symmetry groups
Finite spherical symmetry groups are also called point groups in three dimensions.
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List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
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Lune (geometry)
In plane geometry, a lune is the concave-convex area bounded by two circular arcs, while a convex-convex area is termed a lens.
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Molecular orbital
In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule.
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Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.
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Molecule
A molecule is an electrically neutral group of two or more atoms held together by chemical bonds.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.
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Octahedron
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.
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Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.
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Orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Orientation (vector space)
In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.
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Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
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Orthogonal matrix
In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.
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Pin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space.
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Point group
In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.
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Point groups in four dimensions
In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.
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Point groups in two dimensions
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane.
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Point reflection
In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.
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Polyhedral group
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.
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Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.
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Prism (geometry)
In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.
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Projective polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Propeller
A propeller is a type of fan that transmits power by converting rotational motion into thrust.
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Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.
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Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers.
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Quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.
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Reflection symmetry
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.
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Rotation
A rotation is a circular movement of an object around a center (or point) of rotation.
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Rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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Rotational symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.
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Schoenflies notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe point groups.
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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Space group
In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
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Spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.
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Spin group
In mathematics the spin group Spin(n) is the double cover of the special orthogonal group, such that there exists a short exact sequence of Lie groups (with) As a Lie group, Spin(n) therefore shares its dimension,, and its Lie algebra with the special orthogonal group.
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Spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups).
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.
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Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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Symmetry number
The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, i.e. the order of its symmetry group.
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Tetrahedral symmetry
A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order.
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Translational symmetry
In geometry, a translation "slides" a thing by a: Ta(p).
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Trapezohedron
The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Wallpaper group
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern.
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Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.
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References
[1] https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions
