(crystallography) A group consisting of the symmetry elements of an object having a single fixed point; 32 such groups are possible.
| Sci-Tech Dictionary: point group |
(crystallography) A group consisting of the symmetry elements of an object having a single fixed point; 32 such groups are possible.
| 5min Related Video: Point group |
| Military Dictionary: holding point |
(DOD, NATO) A geographically or electronically defined location used in stationing aircraft in flight in a predetermined pattern in accordance with air traffic control clearance. See also orbit point.
| Wikipedia: Point group |
In chemistry, a point group is a group of geometric symmetries (isometries) leaving a point fixed.
Contents |
Point groups can exist in a Euclidean space of any dimension. The discrete point groups in two dimensions, also called rosette groups, are used to describe the symmetries of an ornament. The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups.
There are infinitely many discrete point groups in each number of dimensions. However, the crystallographic restriction theorem demonstrates that only a finite number are compatible with translational symmetry. In 1D there are 2, in 2D 10, and in 3D 32 such groups, called crystallographic point groups.
Point groups in 2D fall into two distinct families, according to whether they consist of rotations only, or include reflections. The cyclic groups, Cn (abstract group type Zn), consist of rotations by 360°/n, and all integer multiples. For example, a four legged chair has symmetry group C4, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of dihedral groups, Dn (abstract group type Dihn), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S1 is distinct from Dih(S1) because the latter explicitly includes the reflections.
An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/√2, which does not include rotation by 180°. Depending on its application, homogeneity up to an arbitrarily fine level of detail in a transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored.
Cn and Dn for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups.
In any dimension d, the continuous group of all possible fixed point isometries is the orthogonal group, denoted by O(d); and its continuous subgroup of all possible rotations is the special orthogonal group, denoted by SO(d). This is not Schönflies notation, but the conventional names from Lie group theory.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
| Best of the Web: Point group |
Some good "Point group" pages on the web:
Math mathworld.wolfram.com |
| space group | |
| accidental point (graphic arts) | |
| Neumann's principle (crystallography) |
| What is the point group for al2cl6? Read answer... | |
| What is the point group of dichloromethane? Read answer... | |
| What is the point group symmetry of ammonia? Read answer... |
| What are the group trends of boiling point? | |
| What is the Point group of nitrogen dioxide? | |
| What are the Group 7 melting points? |
Copyrights:
![]() | Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more | |
![]() | Military Dictionary. US Department of Defense Dictionary of Military and Associated Words, 2003. Read more | |
![]() | Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Point group". Read more |
Mentioned in