# 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!

Look what we made!
Browserling

Check out our project Browserling – your personal cloud browser.

A link to this tool, including input, options and all chained tools.

Can't convert.

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

All fractal tools

Quickly draw a custom McWorter dendrite fractal.

Quickly draw a custom canopy tree fractal.

Quickly draw a custom Gosper fractal.

Quickly draw a custom Z-order fractal.

Quickly draw a custom Hilbert fractal.

Quickly draw a custom binary v-fractal.

Quickly draw a custom Peano fractal.

Quickly draw a custom Heighway dragon fractal.

Quickly draw a custom twin dragon Heighway fractal.

Quickly draw a custom Heighway nonadragon fractal.

Quickly draw a custom Koch fractal.

Quickly draw a custom triflake fractal.

Quickly draw a custom Sierpinski triangle fractal.

Quickly draw a custom Sierpinski pentagon fractal.

Quickly draw a custom Sierpinski hexagon fractal.

Quickly draw a custom Sierpinski polygon fractal.

Quickly draw a custom Moore fractal.

Quickly draw a custom Cantor comb fractal.

Quickly draw a custom Cantor dust fractal.

Quickly draw a custom Levy fractal curve.

Quickly draw a custom ice fractal.

Quickly draw a custom Pythagoras tree fractal.

Quickly draw a custom t-square fractal.

Quickly draw a custom Hausdorff tree fractal.

Coming soon
These fractal tools are on the way

Generate a Hilbert Sequence

Walk the Hilbert fractal and enumerate its coordinates.

Generate a Peano Sequence

Walk the Peano fractal and enumerate its coordinates.

Generate a Moore Sequence

Walk the Moore fractal and enumerate its coordinates.

Generate a Hilbert String

Encode the Hilbert fractal as a string.

Generate a Peano String

Encode the Peano fractal as a string.

Generate a Moore String

Encode the Moore fractal as a string.

Generate a Cantor String

Encode the Cantor set as a string.

Generate a Dragon String

Encode the Heighway Dragon as a string.

Generate a Sierpinski String

Encode the Sierpinski fractal as a string.

Sierpinski Pyramid

Generate a Sierpinski tetrahedron (tetrix) fractal.

Cantor's Cube

Generate a Cantor's cube fractal.

Menger Sponge

Generate a Sierpinski-Menger fractal.

Jerusalem Cube

Generate a Jerusalem cube fractal.

Mosely Snowflake

Generate a Jeaninne Mosely fractal.

Mandelbrot Tree

Generate a Mandelbrot tree fractal.

Barnsey's Tree

Generate a Barnsley's tree fractal.

Barnsey's Fern

Generate a Barnsley's fern fractal.

Binary Fractal Tree

Generate a binary tree fractal.

Ternary Fractal Tree

Generate a ternary tree fractal.

Dragon Fractal Tree

Generate a dragon tree fractal.

De Rham Fractal

Generate a de Rham curve.

Takagi Fractal

Generate a Takagi-Landsberg fractal curve.

Peano Pentagon

Generate a Peano pentagon fractal curve.

Tridendrite Fractal

Generate a tridendrite fractal curve.

McWorter's Pentigree

Generate a Pentigree fractal curve.

McWorter's Lucky Seven

Generate a lucky seven fractal curve.

Eisenstein Fractions

Generate an Eisenstein fractions fractal curve.

Bagula Double V

Generate a Bagula double five fractal curve.

Julia Set

Generate a Julia fractal set.

Mandelbrot Set

Generate a Mandelbrot fractal set.

Mandelbulb Fractal

Generate a Mandelbulb fractal.

Mandelbox Fractal

Generate a Mandelbox fractal.

Buddhabrot Fractal

Generate a Buddhabrot fractal.

Burning Ship Fractal

Generate a Burning Ship fractal.

Toothpick Fractal

Generate a toothpick sequence fractal.

Ulam-Warburton Fractal

Generate an Ulam-Warburton fractal curve.

ASCII Fractal

Generate an ASCII fractal.

ANSI Fractal

Generate an ANSI fractal.

Unicode Fractal

Generate a Unicode fractal.

Emoji Fractal

Generate an emoji fractal.

Braille Fractal

Generate a braille code fractal.

Audio Fractal

Generate a fractal in audio form.

Draw a Pseudofractal

Create a fractal that looks like one but isn't a fractal.

Convert Text to a Fractal

Generate a fractal from any text.

Convert a String to a Fractal

Generate a fractal from a string.

Convert a Number to a Fractal

Generate a fractal from a number.

Merge Two Fractals

Join any two fractals together.

Draw a Random Fractal

Create a completely random fractal.

Iterate an IFS

Set up an arbitrary IFS system and iterate it.

Run IFS on an Image

Recursively transform an image using IFS rules.

Iterate an ICAF

Run infinite compositions of analytic functions.

Generate a Fractal Landscape

Create a surface that mimics a natural terrain.

Generate a Brownian Surface

Create a fractal surface via Brownian motion.

Generate a Self-similar Image

Apply fractal algorithms on your image and make it self-similar.

Find Fractal Patterns in Images

Find fractal patterns in any given image.

Find Fractal Patterns in Text

Find fractal patterns in any given text.

Find Fractal Patterns in Numbers

Find fractal patterns in any given number.

Fill a Plane with Fractals

Tessellate a plane with fractals.

Run a Cellular Automaton

Run a cellular automaton with custom rules.

Play Game of Life

Play Conway's Game of Life on an infinite grid.

Subscribe!
Never miss an update

Cool!

Notifications
We'll let you know when we add this tool

Cool!