If that’s the case, then the self-similarity of the tree implies that there is no tip disconnected, that is, the tip set is connected. See expressions at the bottom-left of the preceding diagram:įinally, to prove that this tree is self-contacting, we just need to show that two different leaves (or tips) connect in one of the two main tip-to-tip self-contacting points labeled in the diagram below: UR 2 U ∞ = RLU ∞ and its mirror-symmetric self-contacting point (left side). It’s also interesting to notice that the length of the successive branches can be represented using the Fibonacci numbers. = U ∞ describes the tip at the top of the tree:Īnd its width is the distance between tip LU ∞ and RU ∞: When the address is infinite, we call the limiting branch a tip or an attracting fixed point. So a string of letters is an address path that maps to a unique branch. The letter L is for branches turning left, R for branches turning right, and U for branches going up. As it has neither branch-crossing nor disconnected leaves, it is said to be “self-contacting.” Let’s look at some of its properties more closely:įollowing the notation introduced by Benoit Mandelbrot and Michael Frame for binary trees, I added a third letter U to map all the branches of a symmetric ternary tree. The angle of the symmetrical pairs is 72º from the middle branches. For this particular tree, the scaling factor is ϕ -1 for the middle branches and ϕ -2 for the symmetrical pairs. I call “golden trees” trees with branches scaling according to a multiple of GoldenRatio = ϕ. 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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