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:
“I am less disposed to think of a West Point education as requisite for this business than I was at first. Good sense and energy are the qualities required.”
—Rutherford Birchard Hayes (1822–1893)
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)