Fundamental Domain
The fundamental domain of a point group is a conic solid. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.
For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the disdyakis triacontahedron one full face is a fundamental domain. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.
Also the surface in the fundamental domain may be composed of multiple faces.
Read more about this topic: Point Groups In Three Dimensions
Famous quotes containing the words fundamental and/or domain:
“Wisdom is not just knowing fundamental truths, if these are unconnected with the guidance of life or with a perspective on its meaning. If the deep truths physicists describe about the origin and functioning of the universe have little practical import and do not change our picture of the meaning of the universe and our place within it, then knowing them would not count as wisdom.”
—Robert Nozick (b. 1938)
“You are the harvest and not the reaper
And of your domain another is the keeper.”
—John Ashbery (b. 1927)