# Cesaro polyflake generator

## World's simplest math tool

Free online Cesaro polygon fractal generator. Just press a button and you'll get a Cesaro polyflake. There are no ads, popups or nonsense, just an awesome Cesaro flake generator. Press a button – get a Cesaro polyflake. Created by math nerds from team Browserling.

A new project!
we made something new

We just created Online GIF Tools that offers utilities for quickly editing GIFs. Check it out!

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

Can't convert.

Cesaro polyflake generator tool
What is a cesaro polyflake generator?

This tool creates Cesaro polyflake fractals. A Cesaro polylake fractal is sometimes also called as Cesaro polygon, Cesaro n-gon, Cesaro n-flake, or simply Cesaro flake. This fractal is very similar to the regular Cesaro fractal. In both of these fractals, you specify an initial polygon shape that the fractal will evolve from. In the regular Cesaro fractal, each side of the polygon is bent only once at a certain angle, but in Cesaro polygon fractal you can set the transformation rule for the polygon sides yourself. To do this, you need to write down the rule for the new polygon side using L-system notation. You need to use letter F to denote a step forward, plus and minus signs followed by a positive or negative angle in parenthesis for rotating the line. By formulating this rule, you can explore Cesaro n-gon behavior for any number of iterations. You can also change color palette for the polygon sides, background, and polygon fill. You can also control polygon sides' thickness, padding and starting drawing direction. To get a high-quality picture, change fractal's width and height to higher pixel count. Mathabulous!

Cesaro polyflake generator examples
Click to use

A Five Segment Zigzag

In this example we create a transformation rule with five segments. What this means is each base polygon's side gets replaced with a five-segment zigzag. There are various angles between zigzag segments, some are positive, such as +(30) that makes zigzag rotate 30 degrees to the right and -(170) that makes it turn 170 degrees to the left.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

Burning Coal Fractal

In this example we use a short rule from four segments and apply it to a triangle. We use a black background color, a red outline, and a yellow fill. We get an unusual fractal shape, similar to burning coals. We also recurse this fractal for 6 iterations, change zigzag's width to 1 pixel and remove padding by setting it to zero pixels.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

Wavy Zigzag Rule

This example uses a transformation rule similar to a wavy shape. Each wave segment is rotated only by a small angle so that it's smoother. We create this fractal from a hexagon without coloring its sides and do 5 iteration steps.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

Star with Patterns

This example uses only positive and negative angles equal to 110 degrees in its substitution rule. Only after five iterations this Cesaro pentaflake acquires an unusual star-like shape with small patterns around its edges.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

Cesaro Phoenix

In this example we use acute, obtuse, and right angles for constructing the polygon transformation rule. By forming the fractal from a triangle and running it for 5 iterations, this transformation creates an unusual shape with tiny holes within the polygon and firey flames on the outside. We also changed some options such as set the starting movement direction to up and resized fractal to use a 500x500 pixels square frame.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

Preview Rule

This example demonstrates the Preview Rule option. This option helps you construct transformation rules. In this case, we've created a rule in the form of a heartbeat. Note that all angles of the transformation rule must add to zero so that the polygon stays closed. If you want to see what fractal will come out of this rule, just click on this example, and then remove the check mark from the Preview Rule option.

**Required options**

Cesaro flake's width.

Cesaro flake's height.

Cesaro flake's iterations.

Number of initial polygon's sides.

Transformation rule for
polygon's sides.

See the transformation rule.

Cesaro polygon's bg color.

Cesaro polygon's line color.

Cesaro polygon's fill color.

Thickness of Cesaro polygon's sides.

Padding around Cesaro polygon.

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-polyflake?&width=500&height=500&iterations=7&polygon=4&background-color=%2308027a&line-segment-color=&fill-color=%23a2ff45&transformation-rule=F%2B%2830%29F-%28170%29F%2B%2845%29F%2B%2895%29F&line-width=&padding=10&direction=left&preview-rule=false

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 create a list of excessive numbers.

Quickly create a list of deficient numbers.

Quickly generate a list of perfect numbers.

Quickly create a list of slightly defective 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 create a list of Moser-de Bruijn sequence values.

Quickly generate a list of even numbers.

Quickly generate a list of odd numbers.

Quickly find all factors of the given integer.

Quickly decompose integers into their prime factors.

Quickly find out if the given integer is a prime number.

Quickly check if the given number is a Fibonacci number

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 create a matrix with random elements.

Quickly create a unit matrix with ones along the diagonal.

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 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.

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.

Calculate Pi Digit Sum

Find the sum of pi digits.

Calculate e Digit Sum

Find the sum of e digits.

Calculate Golden Ratio Digit Sum

Find the sum of Golden Ratio digits.

Generate Champernowne Digits

Generate Champernowne constant's digits.

Find n-th Champernowne Digit

Calculate the n-th digit of Champernowne constant.

Decode a Look-and-say Sequence

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

Generate P-adic Expansions

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 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.

Partition a Number

Find all partitions of the given integer.

Generate Partition Number Sequence

Create a list of Partition function numbers.

Generate Arithmetic Progression

Create an arithmetic sequence of numbers.

Generate Geometric Progression

Create a geometric sequence of numbers.

Generate Polynomial Progression

Create a polynomial sequence of numbers.

Generate Natural Number Sequence

Create a list of natural numbers.

Generate Powers of Two

Create a list of numbers of powers of two.

Generate Powers of Ten

Create a list of numbers of powers of ten.

Generate a Sparse Matrix

Generate a matrix with very few non-zero elements.

Multiply a Matrix by a Scalar

Multiply all matrix elements by a number.

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.

Prettify a Matrix

Beautify a matrix by neatly aligning all its columns.

Reformat a Matrix

Convert matrix in one format to another format.

Draw a Fibonacci Spiral

Generate a Fibonacci spiral curve.

Draw Fibonacci Rectangles

Generate a drawing of Fibonacci rectangles.

Draw a Fibonacci Seed Head

Generate a Fibonacci flower head.

Draw a Padovan Fractal

Generate a Padovan equiliteral triangles fractal.

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.

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.

Generate a Multiplication Table

Draw an n×m multiplication table.

Flip a Coin

Toss a coin and get head or tails.

Roll a Dice

Throw a dice and get a number on its side.

Subscribe!
Never miss an update

Cool!

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

Cool!