    Symmetry

Most of the surfaces studied by the Scientific Graphics Project and all those illustrated in these pages have one or more symmetries. Symmetries are attributed to an object based on its ability to remain invarient under certain rigid transformations. Such transformations are called symmetry operations for that object. Symmetry operations in R3 belong to one of the following types, where the italicized geometric objects specify the symmetry.

• inversion: point
• reflection (mirror): plane
• rotation: n-fold axis
• translation: vector
• rototranslation (screw): axis, twist, distance

In addition, the operation rotoinversion, or improper rotation, is sometimes included in this list. However, it can be constituted as a rotation followed by an inversion.

Applied to continuous surfaces, there are two types of rotation axes: two-fold axes lying on the surface, and n-fold axes which intersect the surface perpendicularly if they intersect the surface. This is reflected in the notation described below.

Notation used here

Where the symmetries are described for surfaces in the SGP web, the following notation is used:

• P - an inversion point
• M - a mirror plane
• I - a 2-fold rotation axis on the surface
• R.n - an n-fold rotation axis not on the surface
• T - a translation period
• S - an screw axis
An integer valued expression and asterix preceeding a symmetry designator indicates the number of such symmetries present. A boolean valued expression and question mark preceeding the symmetry designator (and integer expression if used) indicates whether the symmetry is present. In periodic surfaces (those having one or more translation groups), the symetry operations are counted in the quotient (single translation unit).

Symmetry Groups

As an alternative to describing all of the symmetry operations for each surface, one could also assign most of them to previously described symmetry groups. This would be particularly useful in the case of triply periodic surfaces since the symmetry operations tend to be numerous but each must belong to one of the possible 230 space (symmetry) groups.