Imagine a straight wire. Bend this wire until its ends meet. You get a ring. Next stage.
Take another straight wire, bend it as before, but this time, don’t let the ends meet, instead continue bending the wire more and more, until you get a spring. Now, Think of this spring as the first wire, and bend it until the spring’s ends meet. We get a spring-ring. (See pictures below)
At this point it should be obvious – instead of making the ends of the spring meet, we can continue bending the spring into a meta-spring, and again make this meta-spring’s ends meet. Rinse and repeat ad-infinitum.
At first I had a hard time picturing it, and I thought it would be tough to write about it without drawing it. So I set with pen and paper, and tried to draw it. That was even harder than picturing it. So I thought I could write a script to draw it, and that turned out to be pretty easy.
To implement this, I wrote a function that draws any track. A track is defined to be a function takes a float argument between 0 and 1, and returns the appropriate 3d point in between, where 0 is one end of the track and 1 is the other. (Actually, I also decided that tracks should return a “right hand” vector as well). Then I wrote a ring:
def ring(t): pi = math.pi cos = math.cos sin = math.sin x = numpy.array((2*cos(2*pi*t),2*sin(2*pi*t),0)) return (x,-x/3)
Lastly I wrote a wrapper function, that takes any track, and springifies it. It was called springify_track. At this point I could do this:
(The constants are the radius of the spring loops, the sampling delta, and the number loops.)
t1 = ring t2 = springify_track(t1, 0.25, 0.1, 50) t3 = springify_track(t2, 0.06, 0.0011, 5000) t4 = springify_track(t3, 0.011, 0.000012, 500000)