# Hausdorff tree

## World's simplest fractal tool

This utility lets you draw colorful and unique Hausdorff fractals. You can control the drawing process through a number of easy to adjust parameters. You can set the order of the H-fractal, which changes the number of letters H, adjust the dimensions of the space, and set the drawing orientation for the initial H character (either vertical or horizontal). You can also select any color for the H letter and background. Fun fact – the H-fractal is a fractal canopy that fills the entire space with an exception of a couple of points that it can never reach. Fractabulous!

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Hausdorff tree tool
What is a hausdorff tree?

This online browser-based tool allows you to create a personalized Hausdorff fractal tree. 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 H

_{n}) at the n-th iteration stage is calculated by the formula H_{n}= ∑4^{m}(where m ranges from 0 to n-1). For example, at the 3rd iteration stage there are H_{n}= 4⁰ + 4¹ + 4² = 1 + 4 + 16 = 21 letters H. The number of letters H can also be determined by the recursive formula H_{n}= 4×H_{n-1}+ 1. For example, on the 4th iteration stage there are H_{4}= 4×H_{3}+ 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. Mind boggling and awesome at the same time, or as we love to say – fractabulous!
Hausdorff tree examples
Click to use

Hausdorff-85

This example illustrates the H-fractal at the fourth iteration stage. As discussed above, there are 4⁰ + 4¹ + 4² + 4³ = 85 H characters. We've adjusted the options so that the H chars are drawn horizontally in a 500x500 space with lagoon-blue line color.

**Required options**

Hausdorff fractal's order.
(Recursive depth or iterations).

Fractal width.

Fractal height.

H color.

Color for background.

Line segment's width.
(Width of drawing pen.)

Indent around H-curve.

H

_{3}FractalIn this example, we set the recursion depth to only 3 iteration steps, which means that there are 21 letters H (or 42 letters T, if you're creating it from Ts). We use a green-yellow color for the line and deep-sapphire color for the background. We also use 600x600px square canvas with padding of 25px and a line segment width of 10px.

**Required options**

Hausdorff fractal's order.
(Recursive depth or iterations).

Fractal width.

Fractal height.

H color.

Color for background.

Line segment's width.
(Width of drawing pen.)

Indent around H-curve.

Vertical Hausdorff Tree

In this example, we generate 6 iterations of a vertical Hausdorff tree. We've set the size to 600 by 1000 pixels and chosen the H orientation to be vertical. Let's try to calculate the total number of H letters in this fractal. H₆ = 4⁰ + 4¹ + 4² + 4³ + 4⁴ + 4⁵ = 1 + 4 + 16 + 64 + 256 + 1024 = 1365. Good luck shifting to the 6th gear!

**Required options**

Hausdorff fractal's order.
(Recursive depth or iterations).

Fractal width.

Fractal height.

H color.

Color for background.

Line segment's width.
(Width of drawing pen.)

Indent around H-curve.

Pro tips
Master online fractal tools

You can pass options to this tool using their codes as query arguments and it will automatically compute output. To get the code of an option, just hover over its icon. Here's how to type it in your browser's address bar. Click to try!

https://onlinefractaltools.com/draw-hausdorff-fractal?&width=500&height=500&iterations=4&background-color=white&line-segment-color=%23142f6e&line-width=6&padding=15&direction=horizontal

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