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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 22.1 Symmetries in Physics

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.

Exercises Common Geometric Symmetries

  1. 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.
  2. Cylindrical Symmetry: rotation about a central axis does not result in a change to the system.
  3. Spherical Symmetry: any rotation in three-dimensional space does not result in a change to the system.

Note 22.1.1. Antisymmetry.

In physics, antisymmetry 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.

Exercises Activities

1. What Can Change?

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.
Answer.
  1. For left-right planar symmetry, the system can change along either of the other two perpendicular directions: up-down or in-out.
  2. For cylindrical symmetry, the system can change up and down along the axis or radially in and out from the axis.
  3. For spherical symmetry, the system can only change radially in and out from the center.

2. What Kind of Symmetry?

For each system below, identify the kind of symmetry, if any.
  1. The set of weights shown at the bottom of the Dog Weights Figure
  2. A solved Rubik’s Cube
  3. The electric field created by a single negative point charge
  4. A five-armed starfish
Answer.
  1. Cylindrical symmetry
  2. 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.
  3. Spherical symmetry
  4. 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.)
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