getoutofmysunlight said: Hey, I was wondering if you can explain this post on your tumblr post/93506839901/underthesymmetree-fibonacci-you-crazy with the use of cycloids?
This took me a while to wrap my head around. Basically, the shapes are generated every time the faster one “passes” the slower one. In the first example, we had 13 and 8, so in one full cycle the faster one passes the slower one 5 times. Each time, it generates a cardioid. Cardioid are epicycloids, as seen here.
Since each pass will occur in an identical amount of time, we end up with rotational symmetry. Forgive my poor drawing skills:
It’s easy to miss the fact that they are cardioids because we are instead just seeing emergent patterns from all the overlap. More cardioids = more pedals and more layers of pedals. The outermost pedals are easy: 5 cardioids = five overlaps = five outermost pedals. Each cardioid overlaps with its nearest neighbor. As we move inward, we see overlap of non-nearest neighbors (ie, the first is overlapping with the third). Note: because of the cyclic nature here, “third” in one direction can also be considered “fourth” in the other direction, so two colors will have two different overlaps.
- Inner pedals = Red and Green
- Next Layer = Red and Yellow
- Outer Layer = Red and Blue
Hope that helps!
"The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought… The role that remains for the infinite to play is solely that of an idea."
"I think mathematicians do mathematics for reasons that are very similar to those of musicians playing music or any artist doing their art. It’s all about trying to contribute to a certain understanding of ourselves and of the world around us."
Princeton mathematician Manjul Bhargava, who has been awarded the 2014 Fields Medal, one of the most prestigious awards in mathematics. Read more about Bhargava and the award here and watch a video about him here. (via mathematica)