# 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.

announcement
heads up!

We just launched a new site — Online IMAGE Tools — utilities for working with all image formats. Check it out!

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

Can't convert.

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

All math tools

Quickly visualize rules of an Lindenmayer rewrite system.

Quickly generate a list of primes.

Quickly generate a list of Fibonacci numbers.

Quickly generate a list of extended, negative Fibonacci numbers.

Quickly generate a list of Fibonacci primes.

Quickly generate a list of Fibonacci words.

Quickly generate a list of Tribonacci words.

Quickly generate a list of Lucas numbers.

Quickly generate a list of extended, negative Lucas numbers.

Quickly generate a list of Lucas numbers that are also prime numbers.

Quickly generate a list of perfect numbers.

Quickly generate random numbers.

Quickly generate a list of look-and-say sequence numbers.

Quickly create a list of Prouhet-Thue-Morse sequence values.

Quickly create a list of dragon curve sequence values.

Quickly create a list of Oldenburger-Kolakoski sequence values.

Quickly create a list of Gijswijt sequence values.

Quickly create a list of Golay-Rudin-Shapiro sequence values.

Quickly create a list of Baum-Sweet sequence values.

Quickly generate π constant's digits to arbitrary precision.

Quickly calculate Euler constant's digits to arbitrary precision.

Quickly calculate φ constant's digits to arbitrary precision.

Quickly calculate factorials.

Quickly create a triangluar array of binomial coefficients.

Quickly find the transpose of a matrix.

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.

Quickly draw a Hilbert space-filling curve.

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 triflake fractal.

Quickly draw a quadratic Koch flake fractal.

Quickly draw a quadric cross fractal.

Quickly draw a Cesaro 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 Fibonacci word fractal.

Quickly draw an H-tree fractal.

Quickly generate a Morton Z-order fractal curve.

Coming soon
These math tools are on the way

Find n-th e Digit

Calculate the n-th digit of Euler's number.

Find n-th Golden Ratio Digit

Calculate the n-th digit of Golden Ratio.

Find n-th Pi Digit

Calculate the n-th digit of pi constant.

Find Prime Factors

Given a number, factorize it and find its prime factors.

Test if a Number is a Prime

Find out if the given number is a prime number.

Test if a Number is a Fibonacci Number

Find out if the given number is a Fibonacci number.

Decode a Look-and-say Sequence

Perform the inverse operation on a look-and-say sequence.

Generate a Moser-de Bruijn Sequence

Create a list of Moser-de Bruijn sequence values.

Generate Abundant Number Sequence

Create a list of excessive 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 Composite Number Sequence

Create a list of composite numbers.

Generate Deficient Number Sequence

Create a list of deficient numbers.

Generate Secant Number Sequence

Create a list of secant numbers.

Generate Even Number Sequence

Create a list of even numbers.

Generate Odd Number Sequence

Create a list of odd numbers.

Multiply a Matrix by a Scalar

Multiply all matrix elements by a number.

Generate a Random Matrix

Create a matrix with random elements.

Generate an Identity Matrix

Create a matrix with ones along the diagonal.

Check if a Matrix is Singular

Determine if a matrix is degenerate.

Find the Co-factor Matrix

Given a matrix, find its cofactor matrix.

Find the Adjugate Matrix

Given a matrix, find its adjunct.

LU Factor a Matrix

Decompose a matrix into LU factors.

Find Matrix Eigenvalues

Find eigenvalues of a matrix.

Draw a Apollonian Gasket

Generate an Apollonian gasket fractal.

Draw a Mandelbrot Fractal

Generate a Mandelbrot fractal.

Draw a Julia Fractal

Generate a Julia fractal.

Draw a Rauzy Fractal

Generate a Rauzy fractal.

Draw Blancmange Fractal Curve

Generate a Blancmange fractal.

Draw Weierstrass Function

Generate a Weierstrass fractal.

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.

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.

Subscribe!
Never miss an update

Cool!

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

Cool!