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### Author Topic: H-Tree Fractal: Hausdorff Tree Fractal Generator  (Read 320 times)

#### Boleeman

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##### H-Tree Fractal: Hausdorff Tree Fractal Generator
« on: August 03, 2024, 05:02:19 am »
Made a H-Tree Fractal (Hausdorff Tree Fractal)

I believe it is used to make high end Antennas.

At higher levels, the fractal has an intricate cross pattern. (See the 2nd attached png picture to see what I mean)
What I found interesting was that if you try to fill in one of those crosses in the higher level fractal curve, the whole background fills (implying that the whole curve is open).

Some info. on the H-curve:
This fractal is also known as an H-curve as it's entirely made out of many connected copies of the initial letter of Felix Hausdorff's surname. Modern literature often calls this fractal simply an H-tree and even a T-branching fractal, as the letter H can be created from two rotated T letters branching to the left and right. At the first iteration stage, the fractal is just a single letter H (or two T letters). At the second iteration step, four more H letters (each smaller by a factor of √2) are added to every vertex of the original H letter. At the third step, another sixteen smaller H letters are added, and so on. It is easy to see that the number of letters H (denoted by Hn) at the n-th iteration stage is calculated by the formula Hn = ∑4m (where m ranges from 0 to n-1). For example, at the 3rd iteration stage there are Hn = 4⁰ + 4¹ + 4² = 1 + 4 + 16 = 21 letters H. The number of letters H can also be determined by the recursive formula Hn = 4×Hn-1 + 1. For example, on the 4th iteration stage there are H4 = 4×H3 + 1 = 21 * 4 + 1 = 85 letters H. As the fractal's depth increases, the points of the H-curve come arbitrarily close to every point in the space.
« Last Edit: August 10, 2024, 12:03:44 am by Boleeman »