In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O(d). Point groups can be realized as sets of orthogonal matrices M that transform point x into point y:
- y = Mx
where the origin is the fixed point. Point-group elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotation-reflections, or rotoreflections (determinant of M = −1). All point groups of rotations with dimension d are subgroups of the special orthogonal group SO(d).
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. These are the crystallographic point groups.
Read more about Point Group: One Dimension, Two Dimensions, Three Dimensions, Four Dimensions, Five Dimensions, Six Dimensions, Seven Dimensions, Eight Dimensions
Famous quotes containing the words point and/or group:
“Taking the child’s point of view demands good will, time, and effort on the part of parents. The child is the clear beneficiary. Parents who make the effort to understand their children’s point of view are likely to treat children fairly and in an age-appropriate manner.”
—David Elkind (20th century)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)