Improper rotation
In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.
| Group | S4 | S6 | S8 | S10 | S12 |
|---|---|---|---|---|---|
| Subgroups | C2 | C3, S2 = Ci | C4, C2 | C5, S2 = Ci | C6, S4, C3, C2 |
| Example | beveled digonal antiprism |
triangular antiprism |
square antiprism |
pentagonal antiprism |
hexagonal antiprism |
| Antiprisms with directed edges have rotoreflection symmetry. p-antiprisms for odd p contain inversion symmetry, Ci. | |||||
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