![]() ![]() This is a self-similar tree, a tree that can be regarded as a substitution system where a branching rule is applied recursively. For example, here is one of the first self-contacting golden trees that I discovered when I created my own version of “Tree Bender” in order to explore ternary trees (trees with three branches per node): After gathering some intuition and a rudimentary knowledge of the Wolfram Language, I encountered my first insights. Though I had to wait quite a while, I finally found the right tools: Mathematica, combined with Theo Gray‘s “ Tree Bender” Demonstration. After seeing Hans Walser‘s drawings of golden fractal trees in 2007, I was convinced that there was still space for exploration and new discoveries. The following findings aren’t a mere accident I’ve been working hard to grasp a glimpse of new knowledge since high school. Though it might sound strange, I will unveil new geometric objects associated with the golden ratio, which are the objects that illuminated my way when I attempted to map an unknown region of the Mathematical Forest. As we will see in this post, this number still has many interesting properties that can be investigated, some even dating back to the works of the ancient Greeks Pythagoras and Euclid, the Italian mathematician Leonardo of Pisa, and the Renaissance astronomer Johannes Kepler. ![]() Without doubt, the golden ratio is nowadays considered the most mysterious, magical, and fascinating number that exists: ![]()
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