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Helical symmetry

In 3D geometry and higher, a screw axis (or rotary translation) is a combination of a rotation and a translation along the rotation axis.

Helical symmetry is the kind of symmetry seen in everyday objects such as springs, Slinky toys, drill bits, and augers. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant angular speed, while simultaneously translating at a constant linear speed along its axis of rotation. At any point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the object rotates slowly and translates quickly, the coiling angle will approach 90°.

Three main classes of helical symmetry can be distinguished, based on the interplay of the angle of coiling and translation symmetries along the axis: