Dimensions and Degrees of Freedom

Scalar Dimensions Coordinate Geometry Net Motion
Energy Right Top Back
Unity
Speed Left Bottom Front
A B C Progression 0 displacements
20 = 1 DOF

Each scalar dimension can be displaced a single unit towards speed (s/t) or energy (t/s), from the neutral point of unity (speed of light). Larson refers to these as the first and second units of motion (which are actually displacements), so it has the appearance that three, scalar dimensions have a total of six "units of motion."

When these units are applied dimensionally, you have a maximum of 23 = 8 possibilities, not six. Students normally think of this as "2 directions, 3 dimensions," so it makes a 2x2x2 "cube" of motion. But it is not a cube, it is only the possible degrees of freedom. Additionally, the neutral point of unity must also be considered, so there are technically 33 = 27 possible degrees of freedom, which include zero, 1D and 2D displacements.

Click the mouse above, below or on switches to toggle the various possibilities. The "coordinate geometry" image is just to assist with visualization by mapping the three, scalar dimensions to coordinate axes. In "real life," it does not do that and you end up with the "Net Motion" image, where the dimensions are overlapped (a line is a line, no matter which way it points). I have arbitrarily selected the X axis to represent all linear motion and the X-Y plane to represent all planar (2D) motion.