Dimensions and Degrees of Freedom

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bperet
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Dimensions and Degrees of Freedom

Post by bperet » Thu Jul 11, 2013 10:13 am

In dealing with the concept of the Inter-Regional Ratio, one must become familiar with how the scalar dimensions produce various degrees of freedom in 1, 2 and 3 dimensions. Larson attempts to explain this by using 2 units of motion, across three, orthogonal axes to produce what some students mistakenly call "Larson's 2x2x2 cube." Larson was just using the cubic approach to show how these three dimensions with two directions produced 8 degrees of freedom, not 6 (2+2+2).

When dealing with the "unit of motion" concept, each scalar dimension can contain a maximum of two units of motion, the first being from 1 to "speed", s/t, and the second from 1 to "energy", t/s. It's an either-or choice, like flipping a switch, as motion cannot be speed and energy, concurrently, from the same perspective.

With that concept in mind, I did a quick JavaScript page to demonstrate the possible combinations of scalar motion, and the resulting degrees of freedom. And it demonstrates the DOF in 0, 1, 2 or 3 dimensions (27 total possibilities, 33). I guess you could say it shows the "Larson's point," "Larson's line," "Larson's square" and "Larson's cube."

The page is here: Dimensions and Degrees of Freedom

Use the mouse to click above, below or on the center of the switch to throw it, and the page is dynamically updated to show the results. The "Coordinate Geometry" image is a device I used to map the three, scalar dimension to coordinate axes, as Larson was doing with his "directions," for an aid to visualization, only.

I also include a 'Net Motion" image, which shows the practical effect of scalar motion--a line is a line, regardless of what direction it points, so no matter which scalar dimension is acting in a 1D fashion, it produces the same line.
Every dogma has its day...

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