The natural world is full of things that are symmetric: they look the same when flipped, rotated, inverted, or transformed in some way. It also contains many things that are almost symmetric and that may be approximated by an appropriate symmetry.
Planar Symmetry: flipping from one side of a plane to another (for example, going from left to right) does not result in a change to the system. For example, the Charge on a Wire Figure has both up-down and left-right planar symmetry about the center of the wire.
Cylindrical Symmetry: rotation about a central axis does not result in a change to the system. For example, the Ring Figure has cylindrical symmetry about the center of the ring.
Spherical Symmetry: any rotation in three-dimensional space does not result in a change to the system. The concentric spheres in the Surface Area of a Sphere Figure have spherical symmetry.
Antisymmetry, where flipping over an axis results in an exactly negative value, can be as interesting or important as symmetry. In antisymmetric planar geometry, for example, everything on the left side of the plane will be the negative of the corresponding point on the right side of the plane.
In each example above, the symmetry is characterized by the direction in which the system does not change. For each symmetry, explicitly identify the direction(s) in which the system can change without breaking the symmetry.
From the persepctive of color, a solved Rubik’s Cube is not symmetric because the colors on opposite sides are typically different. From the perspective of geometry, the Rubik’s Cube has three different planar symmetries corresponding to each of the three directions.
With a little approximating, you can draw a line (or plane) through the center of each arm, which means there are five different planar symmetries! Because rotating the starfish does change where each arm is located, it does not meet the criteria for cylindrical symmetry. (In biological contexts, this is typically called radial symmetry.)
