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:
“What is wanted—whether this is admitted or not—is nothing less than a fundamental remolding, indeed weakening and abolition of the individual: one never tires of enumerating and indicting all that is evil and inimical, prodigal, costly, extravagant in the form individual existence has assumed hitherto, one hopes to manage more cheaply, more safely, more equitably, more uniformly if there exist only large bodies and their members.”
—Friedrich Nietzsche (1844–1900)
“While you are divided from us by geographical lines, which are imaginary, and by a language which is not the same, you have not come to an alien people or land. In the realm of the heart, in the domain of the mind, there are no geographical lines dividing the nations.”
—Anna Howard Shaw (1847–1919)