Coordinate systems are artificial, make-believe things. Real physical quantities can't depend on one's choice of X axis (or Y or Z, or even T, the time dimension). That's the starting point for relativity in physics.
Go deeper, down to the simplest of shapes: paths in space, lines that twist and bend. The slope of a line is no good as an intrinsic quantity. It's not self-contained; it changes depending on how one rotates coordinate axes. What counts? Curvature — the rate of bending for the path. A flat arc of constant curvature closes on itself and makes a perfect circle. In three dimensions, constant curvature escapes into a helix, a barber-pole spiral along the surface of a cylinder, a strand of DNA, the path of an electron in a magnetic field. Let the tightness of the curl vary, and generate the lovely lines of Nature, from vine to flower to seashell to human body. Take the curvature into higher dimensional spacetime, link it to the distribution of matter, and find the laws of gravitation and motion. Awesome curves....
Friday, December 17, 1999 at 05:52:37 (EST) = 1999-12-17
(correlates: TheChronoliths, AppropriateUnits, Comments on ZimmermannSplinters, ...)