Yesterday I showed how I made a helical, or twisted, butterfly with Blender. As soon as I had something that I could make with simple rules with Blender, I knew I could figure out the equations behind those rules. And sure enough, it didn’t take too long to at least have the rough rules sorted out. Here are the initial curves:
The shape is still preliminary. I haven’t yet done anything to make sure that the spacing is correct or checked whether there are any problems in the structure. Also, the teardrop shapes are cut in one quarter segments, because if I showed anything more, there would only be one curve visible. This is due to the x-z and y-z projections overlapping.
The curves follow the same principles as I had already found for the doubly twisted structure. You basically have two different helices interposed on themselves. One which repeats slowly, and the other which repeats fast. Rather curiously the actual coming up with the equations was pretty easy. However, after figuring out the structure with Blender I had to wait almost a day just to want to take out a pen and some paper and write them down. I guess it had something to do with anticipating it to be more difficult than it was.
Next, I need to check that the equations are fully correct and not just something that produces a pretty, but meaningless, picture. But because I think the picture is cool, I decided to color it, just for the fun of it. Pretty cool for a simple equation.
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