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# Pythagoras tree generator

World's simplest math tool
Free online Pythagoras tree fractal generator. Just press a button and you'll get a Pythagoras fractal. There are no ads, popups or nonsense, just an awesome Pythagorean tree generator. Press a button – get a tree. Created by math nerds from team Browserling.
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Tree Class
Angle of rotation at each level stays the same.
Left and right rectangles change rotation angle at each level.
Left and right rectangles change rotation angle every two levels.
Left and right rectangles change rotation angle every time.
Random rotation angle is selected for each level.
Random rotation angle is selected for all rectangles.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
Iterations, Size, and Tilt
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Pythagoras Tree Colors
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.

## What is a pythagoras tree generator?

This tool draws the Pythagoras tree fractal. The Pythagoras tree is created as follows – take a rectangle (or a square), then adjacent to its top side draw two more rectangles at an angle so that the sides of all three rectangles form a right-angled triangle. If this triangle is isosceles, then the tree will grow symmetrically in both directions. If the triangle's base angles are different, then the Pythagorean tree will be tilted to the side of the smallest angle. In this tool, you can set the tilt angle and see how the Pythagorean tree evolves with each step of the iteration. You can apply a gradient color the branches, add a contour line around each box, and set the background gradient. You can also set the distance that the tree will be drawn from the frame. Additionally, you can generate several different tree types – coniferous tree type, semi-coniferous tree type, alternating, and a random tree. Once you have generated the tree that you like, you can resize it to your desired dimensions by entering width and height in pixels. Mathabulous!

## Pythagoras tree generator examples

Click to use
Symmetrical Pythagorean Tree
This example generates a symmetric Pythagoras tree using twelve generations. It uses a square as the base figure. The outlines of recursively drawn squares are not visible here because it's set to zero. The fractal uses four colors to define two gradients. The first gradient is applied to the background and uses tolopea to black colors. The second is a smooth transition in the tree growth direction from a dodger-blue to spring-green color. Notice that as the branches twist, overlap and intersect, this fractal starts to look very similar to the Levy C curve.
Required options
These options will be used automatically if you select this example.
Angle of rotation at each level stays the same.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Regular Tree from Rectangles
In this example, we set the width of the initial rectangle to 100 pixels and height to 400 pixels. This base figure has a ratio of 4:1 and as a result, we grow a Pythagorean tree that's thin and tall. The rotation angle is 40 degrees, which makes the tree tilt to the right. We use an indigo to jaguar color gradient for the background and yellow to chartreuse color gradient from the Pythagoras tree, drawing it from the trunk to twigs.
Required options
These options will be used automatically if you select this example.
Angle of rotation at each level stays the same.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Coniferous Pythagorean Tree
In this example, we draw a coniferous Pythagoras tree, with individual segment gradient. In this type of tree, the angle of rotation alternates at every level. On the first level, the left square rotates 60 degrees, and on the second level, the right square rotates 60 degrees, and so on. We set the rectangular shape of the image (800x1000px) and generate 15 iterations, without using a border around squares.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle at each level.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Semi-Coniferous Pythagorean Tree
This example draws a semi-coniferous type of tree. This type alternates the rotation angle every two levels. On the first and second levels, the left square rotates by 50 degrees, and on the third and fourth by 90-50 = 40 degrees. The angles change up to the 11th iteration level this way. We fill the tree with a gradient from the roots to leaves and draw a limeade color line around branches.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle every two levels.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Realistic Pythagorean Tree
In this example, we generate a tree that looks very realistic, like you see in nature. This is achieved by selecting the semi-coniferous tree type option (where alpha and beta angles swap every two levels), setting the alpha angle to 34 degrees (beta is automatically set to 56 degrees), and using a non-square base rectangle. The height of each rectangle is 3 times greater than its width. This aspect ratio makes this Pythagoras tree look very realistic and all branches bend very smoothly. We also chose to use a gradient only for the background here and draw 14 iterative levels.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle every two levels.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Alternating Pythagorean Tree
This example draws an alternating type of tree. In this case, every pair of squares exchange base corner angles. Here we set the rotation angle to 58 degrees, which then turns to 32 degrees for the next pair of squares, then to 58 again, then 32, and so on. Also, in this example, we're using an interesting combination of colors for the tree – we don't use gradients and set the same color for the background and squares, and another for the contour. As a result, we get a transparent-looking tree.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle every time.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Extraordinary Pythagorean Tree
Believe or not, this is also a Pythagorean tree. You've got to agree that it has a very unusual appearance. The reason it's so weird is that the width of each rectangle is 5 times greater than its height. It's drawn using the alternating-angles method with the initial rotation angle of 50 degrees. It uses white fill color for all rectangles, Monza to flirt color gradient for the background and dark red color for the contour.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle every time.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Randomize Rotation Angles on Every Tree Level
In this example, we introduce some randomness in our Pythagoras trees. This example generates a Pythagoras tree that uses a random angle for every tree level. Note that one angle is randomly selected for every recursion level and it doesn't change on that level. For example, on the fifth level all triangles will have the same angle but fourth and sixth levels will have different angles. Every time you click on the example, you'll get something new because of randomization. Also, you can just press the Draw a Pythagoras Tree button to get a new, random Pythagoras tree.
Required options
These options will be used automatically if you select this example.
Random rotation angle is selected for each level.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Randomize all Rotation Angles
This example uses a different tree randomization method. Here the rotation angles are randomly selected for each pair of squares. The branches each time tilt in different directions, creating a chaotic tree shape. An interesting feature of this tree is its disappearance at the tips. As we've set only the lower color of the tree's gradient, its upper part becomes transparent, and only the white outline shows the shape of twigs and twiglets.
Required options
These options will be used automatically if you select this example.
Random rotation angle is selected for all rectangles.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Symmetrize Pythagoras Tree
This example uses the additional symmetrize function. This function gradually increases (or decreases) the angle to 45°. In our example, the tree starts at 10° and we generate 10 iterations of the tree. To achieve symmetry this angle has to be increased to 45°. To do it, at every iteration the angle increases by 3.5°. We can quickly calculate that by generating 10 levels we get 3.5° × 10 = 35° plus the initial 10° makes it 45°.
Required options
These options will be used automatically if you select this example.
Left and right rectangles change rotation angle at each level.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Tilted Pythagoras Tree
In this example, the angle of rotation of the left square is 30 degrees and the right is 60 degrees. As a result, left and right squares have different sizes and the whole tree is tilted to the left side. We also stretched the image horizontally by setting its size to 700x600px and set a black outline for squares with a thickness of 2px.
Required options
These options will be used automatically if you select this example.
Angle of rotation at each level stays the same.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
Random Rectangles
This example turns on the randomize rectangle sizes function. It does the obvious thing – the size of each rectangle gets randomized. Note that only the height can be randomized because the width is automatically computed so that every pair of rectangles formed a right triangle. In our case, the height is a random number from 1 to 200 (because we set the base rectangle's height to 200). This example is similar to the first example where we're drawing a regular Pythagorean tree with a base angle of 45 degrees but symmetry is lost here because of height randomization.
Required options
These options will be used automatically if you select this example.
Angle of rotation at each level stays the same.
At each level, the rotation angle increases or decreases so that it reaches 45 degrees in the last iteration.
Size of each rectangle is chosen arbitrarily.
How many times to recursively draw tree branches?
Tree width.
Tree height.
Base rectangle width.
Base rectangle height.
Angle of rotation of the left rectangle.
Tree branches outline thickness.
Space between the frame and the Pythagoras tree.
Starting direction of movement.
Tree background from color.
Tree background to color.
Rectangles fill from color.
Rectangles fill to color.
Color of the contour of the rectangles.
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Draw Minkowski Question-mark Curve
Generate a Minkowski Question-mark fractal.
Draw Thomae's Function
Generate a Thomae's function (also known as popcorn or raindrop function).
Draw Dirichlet's Function
Generate a Dirichlet's function.
Draw a Gabriel's Horn
Draw a geometric figure with infinite surface area and finite volume.
Convert Words to Numbers
Convert numbers as English text to actual digits.
Convert Numbers to Words
Convert numbers to written English text.
Convert Decimal Notation to Scientific Notation
Convert numbers written in decimal form to scientific form.
Convert Scientific Notation to Decimal Notation
Convert numbers written in scientific form to decimal form.
Round Numbers Up
Apply ceil operation to numbers.
Round Numbers Down
Apply floor operation to numbers.
Analyze Numbers
Count how many times each number appears.
Rewrite a Number as a Sum
Create a sum that adds up to the given number.
Rewrite a Number as a Product
Create a product that multiplies up to the given number.
Generate a Multiplication Table
Draw an n×m multiplication table.
Generate a Division Table
Draw an n×m division table.
Generate a Modular Arithmetic Table
Draw an n×m modular arithmetic table for any modulus.
Draw a Pie Chart
Draw a pie chart and show relative sizes of data.
Draw a Bar Chart
Draw a bar chart and visualize data.
Draw a Column Chart
Draw a column chart and visualize data.
Draw a Line Chart
Draw a line chart and visualize data.
Draw an Area Chart
Draw an area chart and visualize data.
Visualize Percentage
Draw a chart that shows percentage.
Tile a Plane
Fill a plane by convex regular polygons.
Flip a Coin
Toss a coin and get head or tails.
Roll a Dice
Throw a dice and get a number on its side.
Generate a Maze
Draw a maze of any size and complexity.