# Cesaro fractal generator

## World's simplest math tool

Free online Cesaro fractal generator. Just press a button and you'll get a Cesaro curve. There are no ads, popups or nonsense, just an awesome Cesaro curve generator. Press a button – get a Cesaro fractal. Created by mathematicians for mathematicians.

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Cesaro fractal generator tool
What is a cesaro fractal generator?

This tool draws custom Cesaro fractals. The Cesaro fractal is a generalization of the Koch fractal. Koch fractal is just one of many possible Cesaro fractals. Koch fractal is constructed by starting with an equilateral triangle and then replacing every side of this triangle with another equilateral triangle. Cesaro fractal generalizes this idea. Cesaro fracal can start with any regular polygon and bend all its sides at any angle, creating concave or convex triangles on the sides. With this tool you can specify the starting polygon, the angle (negative and positive) and how many iterations of this fractal to create. You can also specify 1 or 2 for number of sides of starting polygon which will make it iterate a single or two lines. For more fractal research, you can change the width and height of the fractal, which will let you create high resolution higher order fractals (use download to download full image). You can also colorize Cesaro fractal in three colors, or in two when the thickness of the line is equal to zero. You can also adjust the padding and select the starting direction for drawing the curve. Mathabulous!

Cesaro fractal generator examples
Click to use

Draw a Spiky Cesaro Fractal

This example generates a very spiky Cesaro fractal. It uses a negative -80 degree angle for bending triangle sides and with increasing number of iterations triangle gets spikes. It uses 4 iterations and sets right direction as starting direction.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

Generate a Cesaro Fractal with Eight Sides

The basis for this Cesaro curve example is an octagon. We painted the fractal in three colors and chose that each side bends at an angle of 60 degrees.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

Draw the Torn Square Fractal

This example generates the so-called torn square fractal, which is just a special case of Cesaro fractal. The incision angle is set to 85 degrees that makes a very sharp cut in the initial square at every iteration. Canvas size measures at 500 by 500 pixels and the thickness of the line is not set here, so the fractal is drawn using only two colors. Notice how another fractal appears inside of the square as the number of iterations increases.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

Generate a Koch Line

In this example, the number of sides for the initial figure is set to one. That means we're starting with a straight line. Iterating a straight line and bending it at 60 degrees at every step creates the Koch curve. In this example you can observe the expansion of the curve at the sixth iteration. As you can see Koch line is just a special case of Cesaro fractal. For a better look the width of the image is increased to 600px, and the height is decreased to 300px.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

Draw a Cesaro Fractal from a Triangle

This example generates a Cesaro fractal from an equilateral triangle. The bending angle is set to 60 degrees that makes incisions touch each other and separate the fractal into pieces.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

Generate a Cesaro Fractal from Two Lines

This example draws a Cesaro fractal starting with two touching horizontal lines. One of lines is iteratated upwards, the other downwards. As a result the fractal has horizontal mirror symmetry.

**Required options**

Cesaro fractal width

Cesaro fractal height

Cesaro fractal iterations.

Enter the number of sides for
the starting polygon.

Bend sides of the polygon
at this angle.
(Supports negative angles.)

Cesaro fractal background color.

Cesaro fractal line color.

Fill the inside of the fractal
with this color.

Thickness of Cesaro curve.

Padding around Cesaro fractal.

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!

https://onlinemathtools.com/generate-cesaro-fractal?&width=500&height=500&iterations=4&polygon=3&background-color=%23ad0255&line-segment-color=black&fill-color=%23fff700&angle=-80&line-width=4&padding=10&direction=right

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