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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. [1]

102 relations: Bauhinia × blakeana, Bravais lattice, Chirality, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Crystallographic point group, Crystallographic restriction theorem, Crystallography, Cube, Cuboid, David Hestenes, Demihypercube, Determinant, Digon, Dihedral group, Dodecahedron, Duoprism, E6 (mathematics), E7 (mathematics), E8 (mathematics), Equilateral triangle, Euclidean space, Factorial, Geometry, Group (mathematics), Harold Scott MacDonald Coxeter, Hexagon, Hexagonal prism, Hong Kong, Hosohedron, Hyperrectangle, Icosahedral prism, Icosahedron, Improper rotation, Isometry, Isomorphism, John Horton Conway, Lattice (group), Metal carbonyl, Molecular symmetry, Molecule, Norman Johnson (mathematician), Octahedron, Orbifold, Orbifold notation, Orthogonal group, Orthogonal matrix, Parity (mathematics), PDF, ... Expand index (52 more) »

Bauhinia × blakeana

Bauhinia blakeana commonly called the Hong Kong Orchid Tree is a legume tree of the genus Bauhinia, with large thick leaves and striking purplish red flowers.

Bravais lattice

In geometry and crystallography, a Bravais lattice, named after, is an infinite array of discrete points in three dimensional space generated by a set of discrete translation operations described by: where ni are any integers and ai are known as the primitive vectors which lie in different directions and span the lattice.

Chirality

Chirality is a property of asymmetry important in several branches of science.

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).

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.

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).

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind.

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.

Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure).

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.

Cuboid

In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.

David Hestenes

David Orlin Hestenes, Ph.D. (born May 21, 1933) is a theoretical physicist and science educator.

Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as h&gamma;n for being half of the hypercube family, &gamma;n.

Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

Digon

In geometry, a digon is a polygon with two sides (edges) and two vertices.

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

Duoprism

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher.

E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.

E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

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.

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.

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

Hexagon

In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six-sided polygon or 6-gon.

Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base.

Hong Kong

Hong Kong (Chinese: 香港), officially the Hong Kong Special Administrative Region of the People's Republic of China, is an autonomous territory of China on the eastern side of the Pearl River estuary in East Asia.

Hosohedron

In geometry, an ''n''-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

Hyperrectangle

In geometry, an n-orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.

Icosahedral prism

In geometry, an icosahedral prism is a convex uniform 4-polytope (four-dimensional polytope).

Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

Improper rotation

In geometry, an improper rotation,.

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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.

John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

Metal carbonyl

Metal carbonyls are coordination complexes of transition metals with carbon monoxide ligands.

Molecular symmetry

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.

Molecule

A molecule is an electrically neutral group of two or more atoms held together by chemical bonds.

Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

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.

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.

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.

Parity (mathematics)

In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.

PDF

The Portable Document Format (PDF) is a file format developed in the 1990s to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.

Pentagon

In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon or 5-gon.

Pentagonal prism

In geometry, the pentagonal prism is a prism with a pentagonal base.

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.

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.

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.

Polyhedral group

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

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.

Proprism

In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher.

Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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.

Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.

Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.

Stellated octahedron

The stellated octahedron is the only stellation of the octahedron.

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.

Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

Tetrahedral prism

In geometry, a tetrahedral prism is a convex uniform 4-polytope.

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.

Triangular prism

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

X-ray crystallography

X-ray crystallography is a technique used for determining the atomic and molecular structure of a crystal, in which the crystalline atoms cause a beam of incident X-rays to diffract into many specific directions.

Yin and yang

In Chinese philosophy, yin and yang (and; 陽 yīnyáng, lit. "dark-bright", "negative-positive") describes how seemingly opposite or contrary forces may actually be complementary, interconnected, and interdependent in the natural world, and how they may give rise to each other as they interrelate to one another.

1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

1 42 polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

2 41 polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.

4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.

5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

5-cube

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

5-demicube

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

5-orthoplex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

6-cube

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

6-demicube

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.

6-orthoplex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

600-cell

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

7-cube

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

7-demicube

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.

7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

8-cube

In geometry, an 8-cube is an eight-dimensional hypercube (8-cube).

8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed.

8-orthoplex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

8-simplex

In geometry, an 8-simplex is a self-dual regular 8-polytope.

References

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