Group of geometric symmetries with at least one fixed point
The Bauhinia blakeana flower on the Hong Kong region flag has C5 symmetry; the star on each petal has D5 symmetry.
The Yin and Yang symbol has C2 symmetry of geometry with inverted colors
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).
The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups.
In geometry, a pointgroup is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate...
In crystallography, a crystallographic pointgroup is a three dimensional pointgroup whose symmetry operations are compatible with a three dimensional...
spectroscopy. Spectroscopic notation is based on symmetry considerations. The pointgroup symmetry of a molecule is defined by the presence or absence of 5 types...
geometry, a pointgroup in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a...
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...
In geometry, a polar pointgroup is a pointgroup in which there is more than one point that every symmetry operation leaves unmoved. The unmoved points...
under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. A pointgroup including a point reflection...
The Elk PointGroup is a stratigraphic unit of Early to Middle Devonian age in the Western Canada and Williston sedimentary basins. It underlies a large...
group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete. Discrete symmetry groups come...
Schoenflies, is a notation primarily used to specify pointgroups in three dimensions. Because a pointgroup alone is completely adequate to describe the symmetry...
geometry, a pointgroup in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere...
corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane...
pointgroup. This ranges from 1 in the case of space group P1 to 192 for a space group like Fm3m, the NaCl structure. The elements of the space group...
The Six PointGroup was a British feminist campaign group founded by Lady Rhondda in 1921 to press for changes in the law of the United Kingdom in six...
isometries along the axis a discrete pointgroup, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction...
the preimage of a pointgroup (hence denoted 2G, for the pointgroup G), or is an index 2 subgroup of the preimage of a pointgroup which maps (isomorphically)...
two-dimensional pointgroup or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a...
three-fold rotoinversion (pointgroup D3d). Under this space group the two A positions are equivalent. If the space group is F43m then the three-fold...
In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood...
The Long PointGroup is a geologic group in Newfoundland and Labrador. It preserves fossils dating back to the Ediacaran period. Earth sciences portal...
crystallographic pointgroup. A crystal system is a set of pointgroups in which the pointgroups themselves and their corresponding space groups are assigned...
The Connecting PointGroup is a Late Neoproterozoic geological formation cropping out on the Avalon Peninsula of Newfoundland, dominated by deep marine...
a centrosymmetric pointgroup contains an inversion center as one of its symmetry elements. In such a pointgroup, for every point (x, y, z) in the unit...
set of pointgroups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified...
describe pointgroups. This notation is used in spectroscopy and is used here to specify a molecular pointgroup. There are two pointgroups for diatomic...
gluteus group (gluteus maximus, gluteus medius, and gluteus minimus). Often there is a heat differential in the local area of a trigger point.[citation...
trading day. In 2013, the company acquired the German retailer Runners PointGroup. After not meeting corporate expectations, Foot Locker planned to close...
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