# 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. Created by fractal fans from team Browserling. Fractabulous!
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Order, Size and Orientation
Hausdorff fractal's order. (Recursive depth or iterations).
Fractal width.
Fractal height.
Horizontal or vertical H?
H-Fractal Colors
H color.
Color for background.
Curve Width and Indents
Line segment's width. (Width of drawing pen.)
Indent around H-curve.

## 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 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. 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
These options will be used automatically if you select this example.
Hausdorff fractal's order. (Recursive depth or iterations).
Fractal width.
Fractal height.
Horizontal or vertical H?
H color.
Color for background.
Line segment's width. (Width of drawing pen.)
Indent around H-curve.
H3 Fractal
In 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
These options will be used automatically if you select this example.
Hausdorff fractal's order. (Recursive depth or iterations).
Fractal width.
Fractal height.
Horizontal or vertical H?
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
These options will be used automatically if you select this example.
Hausdorff fractal's order. (Recursive depth or iterations).
Fractal width.
Fractal height.
Horizontal or vertical H?
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!
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