# Cylindrical Symmetry

## Symmetry of an Infinite Cylinder

An infinite cylinder is a paper towel roll or wooden dowel that goes on forever in both directions. It has a special type of symmetry as well, though its properties can't be summed up quite as neatly as for a sphere:

The symmetries of an infinite cylinder, or at least a picture of same
• An infinite cylinder has translational symmetry if you move it along its Axis (that is, the infinite line running through the core of the cylinder)
• An infinite cylinder has continuous rotational symmetry around its axis
• An infinite cylinder has 180° rotational symmetry around any axis perpendicular to the cylinder's axis.
• An infinite cylinder has reflection symmetry across any mirror which its axis lies on
• An infinite cylinder has reflection symmetry across any mirror which is perpendicular to the cylinder's axis
These properties are true for solid cylinders and cylindrical shells, which we say have infinite cylindrical symmetry. An infinite line (like the axis of an infinite cylinder) also has infinite cylindrical symmetry.