# L-system generator

## World's simplest math tool

Free online Lindenmayer system generator. Just enter the axiom and rewrite rules and you'll automatically get a visualization of the grammar. There are no ads, popups or nonsense, just an awesome L-system viewer. Press a button — visualize a Lindenmayer system. Created by math nerds from team Browserling.
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Grammar Options
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Rewrite rule 3.
Rewrite rule 4.
Rewrite rule 5.
Curve Options
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Colors
Background color.
Curve color.
L-system generator tool What is a l-system generator?
This tool draws and visualizes L-system rules. The L-system is also known as the Lindenmayer system. It consists of an alphabet of symbols, a starting axiom and a set of rules that define how symbols get rewritten. With this tool you can specify which symbols to visualize, which symbols to skip, and how many iterations of rewriting to perform. Because of the recursive nature of the rewrite system it often produces self-similar fractals. Symbols plus (+) and minus (-) have a special meaning and they change the curve direction by the specified angle. You can set the dimensions for your L-system and choose the drawing start direction. To make it more visually appealing, you can also set the color for the background and for the curve line, set the line thickness and padding. To share your creation you can click the link icon and get a URL to the L-system you created. Mathabulous!
L-system generator examples Click to use
Draw a Gosper Curve
This example uses L-system rules to generate a fractal Gosper curve. It uses four iterations and angle of 60 degrees. Background is set to green and line width to 4px.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Red Dragon Curve
This example generates a 10th order dragon curve using simple L-system rewrite rules. We set background to red as dragons love to spit fire. The L-system dimensions are set to 500x400px.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw an Icy Fractal
In this example a curve similar to freezing ice pattern is created by a single Lindenmayer system rule. We used cold colors to make it look like a real frost.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Fractal Tree
This example visualizes L-system rules that look like a tree. This curve has an angle of 36 degrees and is 3 iterations deep. We used special symbols [ and ] in the first rule that mean save the current curve position to stack, and restore the position from stack.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a William McWorter's Pentadendrite
This example generates a McWorter's Pentigree curve using just one L-system rewrite rule with a rotation angle of 72 degrees. It also sets iterations to 4, padding to 5px and width of curve's line to 2px.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Round Star
This example displays a non-standard L-system with an angle of 77 degrees. This curve consists of straight lines, which simultaneously form both a star and a circle.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Island Curve
This example generates a L-system, which consists of many small rectangles that look like islands. There are two special restrictions here. The F symbol is used for drawing but the b symbol is skipped (if you remember that an L-system is just a turtle that walks around a canvas leaving a trace for every alphabet symbol, then the b symbol makes turtle move ahead without drawing a line segment).
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Penrose Tiling
This example uses an L-system to generate a Penrose tiling. For this curve we use all 5 rules and use four characters for drawing. We also set the color of the background to black and yellow color for the line. Pretty!
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Rewrite rule 3.
Rewrite rule 4.
Rewrite rule 5.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Koch Curve
This example generates a variation of Koch star known as Cesero fractal with an angle of 80 degrees. It has a width of 600px and a height of 400px, it uses 6 iterations and the starting direction is set to left.
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Draw a Moore Fractal
In this example, we generate the Moore curve on a 200x200 pixel canvas. We tell the generator to run for 3 iterations and use dark-slate-gray and beige colors for the background and the curve. (Credit: Maurizio Butti.)
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Rounded Peano Curve
This example draws the rounded rhombus form of the Peano fractal. Here the rotation angle between the curve fragments is 45 degrees and the fractal is printed on a background with dimensions 300-by-300. (Credit: Maurizio Butti.)
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Peano Curve at 22.5 Degrees
This example also draws the rounded rhombus form of the Peano fractal, however, the angle of rotation in this curve is 22.5 degrees. This makes the curve much smoother. (Credit: Maurizio Butti.)
Required options
These options will be used automatically if you select this example.
Draw these symbols.
Skip these symbols.
Starting axiom.
Rewrite rule 1.
Rewrite rule 2.
Rewrite rule 3.
Curve width.
Curve height.
How many iterations to perform?
Angle of rotation.
Starting direction of movement.
Thickness of a curve.
Padding around the image.
Background color.
Curve color.
Pro tips Master online math 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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Quickly visualize rules of an Lindenmayer rewrite system.
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Quickly generate a list of Lucas numbers.
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Quickly find the inverse of a matrix.
Quickly calculate the determinant of a matrix.
Quickly find the sum of two or more matrices.
Find the difference of two or more matrices.
Quickly find the product of matrices.
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Quickly draw a Heighway triangle fractal.
Quickly draw a Harter-Heighway space-filling dragon fractal.
Quickly draw a Davis-Knuth dragon fractal.
Quickly draw a triple dragon fractal.
Quickly draw a quaddragon fractal.
Quickly draw a hexdragon curve.
Quickly draw a Peano space-filling fractal.
Quickly draw a Moore space-filling curve.
Quickly draw a Peano-Gosper space-filling curve.
Quickly draw a Sierpinski arrowhead fractal curve.
Quickly draw a Sierpinski sieve fractal.
Quickly draw a Sierpinski closed plane fractal curve.
Quickly draw a Sierpinski carpet fractal.
Quickly draw a Sierpinski maze fractal.
Quickly draw a Sierpinski square fractal.
Quickly draw a Sierpinski pentaflake fractal.
Quickly draw a Sierpinski hexaflake fractal.
Quickly draw a Sierpinski n-gon fractal.
Quickly draw a Koch island fractal curve.
Quickly draw a Koch antisnowflake fractal.
Quickly draw a Koch triangle fractal.
Quickly draw a Koch n-gon fractal.
Quickly draw a quadratic Koch flake fractal.
Quickly draw a Cesaro fractal.
Quickly draw a Cesaro n-gon fractal.
Quickly draw a triflake fractal.
Quickly draw a quadric cross fractal.
Quickly draw a self-similar Levy C fractal curve.
Quickly draw a Vicsek snowflake fractal.
Quickly draw a T-square fractal.
Quickly draw a Cantor set fractal.
Quickly draw an asymmetric Cantor set fractal.
Quickly draw a generalized Cantor set fractal.
Quickly draw an ε-Cantor set fractal.
Quickly draw a Cantor dust fractal.
Quickly draw a Pythagoras tree fractal.
Quickly draw a Fibonacci word fractal.
Quickly draw an H-tree fractal.
Quickly draw a V-tree fractal.
Quickly generate a Morton Z-order fractal curve.
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Calculate the n-th digit of Golden Ratio.
Find n-th Pi Digit
Calculate the n-th digit of pi constant.
Calculate e Digit Sum
Find the sum of e digits.
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Find the sum of Golden Ratio digits.
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Generate Champernowne Digits
Generate Champernowne constant's digits.
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Calculate the n-th digit of Champernowne constant.
Decode a Look-and-say Sequence
Perform the inverse operation on a look-and-say sequence.
Calculate p-adic expansions of arbitrary numbers.
Generate Pandigital Number Sequence
Create a list of pandigital numbers.
Generate Stanley Number Sequence
Create a list of Stanley numbers.
Generate Bell Number Sequence
Create a list of Bell numbers.
Generate Carmichael Number Sequence
Create a list of Charmichel numbers.
Generate Catalan Number Sequence
Create a list of Catalan numbers.
Generate Triangular Number Sequence
Create a list of triangle numbers.
Generate Composite Number Sequence
Create a list of composite numbers.
Generate Secant Number Sequence
Create a list of secant numbers.
Generate Golomb Number Sequence
Create a list of Golomb-Silverman numbers.
Generate Euler's Totient Number Sequence
Create a list of Euler's phi numbers.
Generate Juggler Number Sequence
Create a list of juggler numbers.
Generate Lucky Number Sequence
Create a list of lucky numbers.
Generate Motzkin Number Sequence
Create a list of motzkin numbers.
Generate Padovan Number Sequence
Create a list of Padovan numbers.
Generate Pseudoperfect Number Sequence
Create a list of semiperfect numbers.
Generate Ulam Number Sequence
Create a list of Ulam numbers.
Generate Weird Number Sequence
Create a list of weird numbers.
Generate Superperfect Number Sequence
Create a list of superperfect numbers.
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Generate Partition Number Sequence
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Generate Natural Number Sequence
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Generate Powers of Ten
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Generate a Dense Matrix
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Generate a Sparse Matrix
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Generate a Fibonacci flower head.
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Throw a dice and get a number on its side.