# Pythagoras fractal

## World's simplest fractal tool

This utility lets you draw custom and colorful Pythagoras fractals. We offer you more than twenty options to create a truly personalized fractal. The options are divided into three categories. The first category is for adjusting the image size and canvas dimensions, iterations, tree angle, and the proportions of the base rectangle from which the whole tree evolves. The second category is for adjusting colors and gradient. You can paint the fractal with a five-color palette. The first two colors define the background gradient, the next two are responsible for the tree gradient, and the last one sets the line drawing color. The third category lets you quickly switch between predefined Pythagoras tree species. The species include coniferous, semi-coniferous, cypress, and randomized trees. Fun fact – there are infinitely many logarithmic spirals hidden in a Pythagoras tree and they can be found by starting with any square and always following the closest square that goes to the left or right. Fractabulous!

a new math project!
announcement

Love math as much as we do? Then you'll also love our new project – MATHURLS – a super neat math news aggregator. Check it out!

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

Can't convert.

Pythagoras fractal tool
What is a pythagoras fractal?

This online browser-based tool illustrates Pythagoras fractal trees. The Pythagoras tree is one of the most popular fractals in the fractal tree family that includes v-fractal and fractal canopy. The classic version of this fractal begins with a square but we've also added a modified version that begins with a rectangle. To construct the next iteration, take the square and build a right triangle on top of it so that the top edge of the square is the hypotenuse. Then, construct squares (or rectangles) on both sides of the triangle. This is the second level of the fractal. This construction is repeated recursively on each of the two new squares, and then recursively again on the new squares, until infinity. The limit of this construction is called the Pythagoras fractal (or the Pythagorean tree). If the sides of the interlevel triangle are equal, then the Pythagorean fractal will be symmetric. If the left corner is smaller than the right, the tree will tilt to the left side but if the right angle is less than the left, the tree will tilt to the right side. When the symmetric tree is drawn with a unit square as the base figure, the entire fractal (at infinite iteration) will fit exactly inside a 6x4 rectangle. Mind blowing and ingenious at the same time, or as we love to say – fractabulous!

Pythagoras fractal examples
Click to use

Regular Pythagoras Fractal

In this example, we set the left bending angle equal to 45 degrees, and the right angle is calculated automatically as follows: 90 - 45 = 45 degrees. With these angles, the interlevel triangles are isosceles, and as a result, the whole tree has mirror symmetry with respect to the vertical axis going through the middle of the trunk. We generate 10 levels of branches, using all five colors in the palette. We set a mountain-meadow-green to deep-sea-green color gradient for the background, a white to pink-flamingo color gradient for the tree, which is applied from the tree root to twigs, and a pompadour color line for the square border.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Bending angle same for all levels.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

Coniferous Pythagoras Fractal

In this example, the left and right angles alternate at each recursive level. As a result, the Pythagoras fractal has a shape of a coniferous tree. We set the first left bending angle to 35 degrees that makes our tree to be tilted to the left side. We also set the proportions of the base rectangle to 1:2 to make the tree thinner and higher. To make the tree fit nicely in the canvas, we slightly lengthen the height of the canvas to 900 pixels, with a width of 800 pixels, and draw 12 recursive levels of rectangles.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Alternate left and right angles
at each level.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

Semi-coniferous Pythagoras Fractal

In this example, we generate a semi-coniferous Pythagoras tree. That means that the left and right angles alternate every two levels. That is, on the first and second level, the left bend angle is 56°, on the third and fourth 90° - 56° = 34°, on the fifth and sixth 56° again, and so on. We set the width of the base rectangle to 20% less than its height and perform 13 iterative steps. We use an individual branch gradient from gorse-yellow to flamenco-orange color for the tree fill and curious-blue to Stratos-blue color gradient for the background.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Alternate left and right angles
every two levels.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

Cypress Pythagoras Fractal

In this example, the Pythagoras fractal grows to an extraordinary shape because the height of the base rectangle is set to be 3 times smaller than its width. With these base figure dimensions and the cypress tree type, it looks more like a bush than a tree. We iterate this bush for 11 stages and set the bending angle is 38 degrees. We also use a rectangular canvas of 800 by 600 pixels and padding of 10 pixels around the trunk and bush tips.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Alternate left and right angles
for each triangle.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

Partially Randomized Tree

In this example, the tool randomly selects the rotation angle for each iterative level and keeps it constant for all triangles on that level. However, as we have also activated the tree symmetrization option, the closer we get to the last iteration, the closer the angle gets to 45 degrees. At the last level, all the angles are 45 degrees. We set the rectangle ratio to 2:5 and generate 12 recursive stages. For the color palette, we use a deep-blue to congress-blue color gradient for the background and aqua to magenta color gradient for the branches.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Randomize angle for each level.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

Absolutely Randomized Tree

This example randomizes both key tree properties. The bending angle for each branch is chosen arbitrarily and the proportions of each rectangle are also selected randomly. Thus, we get an absolutely randomized Pythagorean tree with branches and twigs going in all possible directions. We color this tree with a paua to pompadour color gradient in the tree direction and calculate 15 iterations.

**Required options**

How many levels of branches
to generate?

Left bending angle of the tree.

Image width.

Image height.

Padding.

Base width.

Base height.

Rectangle edge thickness.

Canvas gradient starting color.

Canvas gradient ending color.

Trunk color (branch gradient
starting color.)

Twigs color (branch gradient
ending color.)

Palette for line around.

Randomize angle for each triangle.

Gradually change angle with each
level to reach 45° at the twigs.

Randomize base size every time.

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-pythagoras-fractal?&iterations=10&angle=45&width=900&height=900&base-width=100&base-height=100&line-width=1&padding=20&direction=up&gradient-from-trunk-to-twigs=true&background-from-color=%2317ada1&background-to-color=%23084a49&fill-from-color=white&fill-to-color=%23ff5cf1&line-segment-color=%23630059®ular-tree=true&symmetrize-tree=false&randomize-rectangle-sizes=false

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 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.

Subscribe!
Never miss an update

Cool!

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

Cool!