Fourier Closed Loops
Greg Volk
Every closed loop path, however complex, can be characterized in terms of a parameter running from 0 to 2 Pi. Thus, the three coordinates (x,y,z) describing a path are themselves periodic signals of the same parameter, and may therefore be broken into Fourier components. We can then recombine the coordinates (x,y,z) of each harmonic component, and obtain a series of closed loops which together reconstruct the original loop. This paper will show that each of these loops is in fact an ellipse, meaning that any closed loop path can ultimately be broken into a series of ellipses, completing their circuits in multiples of the original parameter.
Interestingly, this concept could be applied to knot theory, since the invariant information about a given knot is necessarily contained in the relative orientations of the ellipses comprising the Fourier series. It could be used as a tool for modeling electromagnetic structures that may exhibit the properties of elementary particles, and for designing a desired force characteristic, created by the actual geometry of a coil winding. One particular force this method of analysis could generate is the ?chiral force?, wherein repulsion or attraction between adjacent coils depends only on their relative handedness or helicity.
