This simple equation helps you understand a fractal:

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … = 2 = Σ(1/2^k,k=0..∞)

Now here’s the construction of the H-tree fractal, in which the lines are divided in length by sqrt(2) = 1.41421356237… at each stage. Or multiplied by 0.70710678118… = &sqrt;0.5. This means that, after two divisions, the lines are 1/2 the size. So in the end they create a 1 x &sqrt;2 rectangle:

H-Tree fractal #1

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H-Tree fractal #2

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H-Tree fractal #3

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H-Tree fractal #4

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H-Tree fractal #5

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H-Tree fractal #6

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H-Tree fractal #7

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H-Tree fractal #8

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H-Tree fractal #9

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H-Tree fractal #10

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[…]
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H-Tree fractal #13

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H-Tree fractal #14

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H-Tree fractal #15

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Here’s an animation:

H-Tree fractal (animated at EZgif)

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And here’s the H-tree in black-and-white:

H-Tree fractal #3

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H-Tree fractal #6

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H-Tree fractal #6

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H-Tree fractal #12

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H-Tree fractal #15

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H-Tree fractal (animated at EZgif)

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Because the construction of the H-tree is governed by a string of directions — for example, left-right-right-left-left-left… or 211222… — you can perform tests on that string to create sub-fractals from the super-fractal. Like this:

count(1) = count(2) in string to step 12

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count(1) = count(2) in string (omitting lines)

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sum(string) = mul(string)

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sum(string) > mul(string)

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count(1) = 2 or count(2) = 2 after step 2

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count(1) < count(2)

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count(1) < 3 or count(2) < 3 after step 6

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value of string after step 8 > value of string at step 1

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value after step 8 > value at step 4

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value after step 8 < value step 1

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ispalindrome(string) to step 11

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ispalindrome(string) to step 18

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ispalindrome(string) to step 20

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alternating 121… or 212… in string after step 9

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ispolygonal(sum(string[i]-1),pol=10)

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isprime(sum(string))

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sum(string[i]-1) mod 13 = 0

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sum(string[i]-1) mod 13 = 1

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sum(string[i]-1) mod 16 = 0

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sum(string[i]-1) mod 18 = 0

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