# Interunion-Euclidean

- Author:
- Will Le

**is the act of**

*Interunion a ⋇ b**spinning fibers into threads*. In the simplest case, two fibers are spun into an epicycloid thread (check "Quadratic" to restrict all cases into epicycloid) or a hypocycloid (check both "Qradratic" & "Spatial"). In the general case, from the initial two fibers

*a*&

*b*, they are refined into many "remainder" fibers

*r*via Euclidean algorithm, and then all of them are spun together into a complex thread via partial Fourier series of those fibers. In that partial Fourier series, all harmonics are synchronized with a constant velocity (

_{i}*c*= 1), i.e. wavelength = period = amplitude × 2π = 2π·

*r*, so that those harmonics are touching circles together spinning a

_{i}**high-order cycloid**thread (epicycloid & hypocycloid are 2-cycloids, cycloid & circle are 1-cycloids). See the following images for comparison between epicycloid/hypocycloid (left) and complex (right) threads:

## Epicycloid versus complex thread for 12⋇5 = {12, 5, 2, 1}

## Hypocycloid versus complex thread for 12⋇-5 = {12, -5, 2, -1}

Let

*R*:= set of remainders, including a & b, and*N*= |_{r}*R*|, e.g. R = {12, -7, 5, 2, -1, 1},*N*= 6. The (extensional) product thread is an_{r}*N*-cycloid projected from the trace of_{r}*N*-D phase vector which is a diagonal line wrapping around and filling the hyper-box_{r}*N*-orthotope, as shown in the "Linear box" (only 2-box) in the applet. From Chinese remainder theorem, we have the extent (period) of that line = LCM(_{r}*R*). When LCM(*R*) > LCM(*a, b*), e.g. 420 > 84, we have the*N*-cycloid longer than the corresponding quadratic 2-cycloid._{r}