The Jacobs Ladder

     The Jacobs ladder is the toy where the top block, when rotated in the right direction, initiates a cascade that will travel all the way down the ladder to the bottom. I have had one kicking around our house for forever, I think I may have stolen it from one of my friends. I apologize to them, but they may forgive me when they realize the use to which I have put their toy.
     No, I have not made a jacobs ladder simulation, maybe some day, what I have done, with the help of another friend, is figured out exactly what's going on. We even made one, out of duck tape.
     So the basic jacobs ladder has six blocks. Each block has ribbons on it's sides. Unless it's a block at the end, then it only has ribbons on one side. This was what we knew going in, along with the guess that the ribbons must go through the center of the blocks in some sort of configuration.
     We wanted to be able to express this whole thing as a diagram, and then maybe a mathematical formulation, so that we could describe all of the possible states of a jacobs ladder, and all of the dynamics that turned one state into another.
     The first thing we discovered, however, is that there is only one possible state for the Jacob's Ladder, just one. You can rotate and reflect (flip) this state, but you still get the same situation of each block and each ribbon, just upside down and backwards.
     So we drew our diagram:

     This is a side-on view of the jacobs ladder, where you can see the splits in the blocks, the purple is the two-strand line, and the green is the one-strand line. With this diagram, it is easy to see that there is a sort of back loop thing, where either the purple or the green will loop back on itself. This is what gives the ladder is's miraculous cascade ability. We then created a mathematical representation, where each side that did not have a ribbon is called a 0, each side with one ribbon is a I, and each side with two ribbons is a II:

     From this diagram, you can easily see that in all non-0 blocks, the color on one side is the same as the color on the other. But what about during the cascade, when everything is weird? Well let's see...

     Just through observation, we can see that this situation breaks the rules of the numerical representation, but, luckily, the graphical representation gives us an easy way to draw this:

     So now we know everything we need to know, and all that is left for us to do is to actually make one. I might make a video about  this, we'll see...