onlinefractaltools logo

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!
i've redesigned my blog! catonmat
Super exciting news! I just redesigned my catonmat.net blog. Now it has a new minimalistic design and is entirely powered by alien technology. Check it out!
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.
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 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.
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.
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!
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!
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
Didn't find the tool you were looking for? Let us know what tool we are missing and we'll build it!
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 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 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 Cantor String
Encode the Cantor set as a string.
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.
Toothpick Fractal
Generate a toothpick sequence fractal.
Ulam-Warburton Fractal
Generate an Ulam-Warburton fractal curve.