# T-Square Fractal Generator

## World's Simplest Fractal Tool

This utility lets you draw custom and colorful t-square fractals. We have implemented three fractal types – the Regular T-square, Vertex T-square, and Diamond T-square. The difference between them is the number, location, and shape of new squares that get placed in further recursion levels (which can be set in the options). You can also customize the dimensions of the fractal, namely set its height, width, and padding. There are three colors you can change when painting a t-square – the background color, line color around the squares, and the square fill color. Fun fact – the t-square fractal has a boundary of infinite length bounding a finite area. Created by fractal fans from team Browserling. Fractabulous!

Tool Options

#### T-square's Type

Squares overlap the vertices
of the previous square.
Squares touch but don't
overlap the vertices of the
previous square.
Squares have a diamond shape,
they touch two vertices of the
previous square but don't
overlap them.

#### Depth, Size and Scale Factor

T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Scale factor must be greater
than or equal to the golden
ratio value (φ = 1.618…).

#### Colors, Line and Orientation

Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.
T-squares orientation.

## What Is a T-Square Fractal Generator?

This online browser-based tool illustrates t-square fractals. The t-square fractal is a closed finite two-dimensional fractal with an infinite length boundary. The fractal's name comes from the technical drawing instrument used for drawing horizontal lines on a drafting table. A t-square consists of a set of different size squares. Depending on their placement, size, and orientation, there are multiple types of t-squares. We currently support three types of fractal. The first type that we support is the classic textbook t-square version (we call it the Regular T-square). This classic t-square is iterated from a square by adding more squares centered at each vertex of all previous iteration's squares. The second type is the vertex t-square, which has the new squares placed around the vertices of all previous squares so that they touch at a single point. The third type is the diamond t-square that is a variation of the vertex t-square type but here new squares are rotated so they look like diamonds and they appear at two opposite vertices of the previous diamond. For all these types, you can customize the ratio (scale factor) of square side lengths between the current and next recursion. The scale factor must be greater than or equal to the golden ratio value (1.6180339…). At the nth iteration step, greater than 2nd, there are 4*3⁽ⁿ⁻²⁾ new squares added to the fractal (for diamond t-square it's 2⁽ⁿ⁻¹⁾) and in total, there are 2×3⁽ⁿ⁻¹⁾ - 1 squares (for diamond t-square it's 2ⁿ - 1). Mind blowing and ingenious at the same time, or as we love to say – fractabulous!

## T-Square Fractal Generator Examples

Click to try!
click me

### Classic Tsquare Fractal

In this example, we draw the original tsquare fractal after 6 iterations. At this stage, this fractal consists of 6 sets of overlapping squares of different sizes. The first set is a single unit-length square, the second set is four half-unit-length squares, the third is twelve quarter-unit-length squares, etc. We paint this fractal in three colors – white for the background, black for square edges, and Persian-blue for square fill.

Required options
These options will be used automatically if you select this example.
Squares overlap the vertices
of the previous square.
T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.
click me

### Vertex Tsquare Fractal

This example generates a 7th order vertex tsquare fractal. It consist of 1 + 4 + 4*3 + 4*9 + 4*27 + 4*81 + 4*243 = 1457 squares, in order from the largest to the smallest size. This matches the formula 2×3⁽ⁿ⁻¹⁾ - 1. It uses an 800 by 800 pixels canvas with 20-pixel padding and draws it with two bright colors.

Required options
These options will be used automatically if you select this example.
Squares touch but don't
overlap the vertices of the
previous square.
T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.
click me

### Golden Mean Tsquare

In this example, we illustrate a vertex tsquare fractal with the scale factor equal to the golden ratio φ. In this case, the squares grow so densely together that they fill the entire fractal space with the size of 1000x1000px. We draw the squares without the border and fill them with an orange peel color.

Required options
These options will be used automatically if you select this example.
Squares touch but don't
overlap the vertices of the
previous square.
T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.
click me

### Diamond Tsquare Fractal

In this example, we draw a diamond tsquare fractal on a rectangular canvas with a size of 1000 by 600 pixels. We perform 9 recursions that create 2⁹ - 1 = 512 - 1 = 511 diamonds. The ratio of diamond edge lengths between two recursion levels is equals to the golden mean value phi. We also set the horizontal direction for the diamond expansion, 3-pixel line width, and 12-pixel padding around all diamonds.

Required options
These options will be used automatically if you select this example.
Squares have a diamond shape,
they touch two vertices of the
previous square but don't
overlap them.
T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.
T-squares orientation.
click me

### Regular Scaled Tsquare Fractal

This example sets the scale factor to 2.5 and generates a regular tsquare fractal type. Unlike the classic fractal (where the scale factor is 2), the square sizes decrease significantly faster with each step, quickly diverging from each other. We generate 5 levels of squares, using a lavender color for canvas, ripe-plum color for the border and laser-lemon color for squares fill. The area of a square at 5th level is 1525x smaller than the center square, the perimeter is 156x smaller, and one side is 39x shorter.

Required options
These options will be used automatically if you select this example.
Squares overlap the vertices
of the previous square.
T-squares recursive depth.
Width.
Height.
Reduce the size of new squares
by this scale factor.
Background color for the fractal.
Fill color for the squares.
Border color for the squares.
Border width.

### Pro tipsMaster online fractal 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!

### All Fractal Tools

Quickly draw a custom McWorter dendrite fractal.
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Quickly draw a custom Hilbert fractal.
Quickly draw a custom binary v-fractal.
Quickly draw a custom Peano fractal.
Quickly draw a custom Heighway dragon fractal.
Quickly draw a custom twin dragon Heighway fractal.
Quickly draw a custom Heighway nonadragon fractal.
Quickly draw a custom Koch fractal.
Quickly draw a custom triflake fractal.
Quickly draw a custom Sierpinski triangle fractal.
Quickly draw a custom Sierpinski pentagon fractal.
Quickly draw a custom Sierpinski hexagon fractal.
Quickly draw a custom Sierpinski polygon fractal.
Quickly draw a custom Moore fractal.
Quickly draw a custom Cantor comb fractal.
Quickly draw a custom Cantor dust fractal.
Quickly draw a custom Levy fractal curve.
Quickly draw a custom ice fractal.
Quickly draw a custom Pythagoras tree fractal.
Quickly draw a custom t-square fractal.
Quickly draw a custom Hausdorff tree fractal.

### Coming Soon

These fractal tools are on the way!
Generate a Hilbert Sequence

Walk the Hilbert fractal and enumerate its coordinates.

Generate a Peano Sequence

Walk the Peano fractal and enumerate its coordinates.

Generate a Moore Sequence

Walk the Moore fractal and enumerate its coordinates.

Generate a Hilbert String

Encode the Hilbert fractal as a string.

Generate a Peano String

Encode the Peano fractal as a string.

Generate a Moore String

Encode the Moore fractal as a string.

Generate a Cantor String

Encode the Cantor set as a string.

Generate a Dragon String

Encode the Heighway Dragon as a string.

Generate a Sierpinski String

Encode the Sierpinski fractal as a string.

Sierpinski Pyramid

Generate a Sierpinski tetrahedron (tetrix) fractal.

Cantor's Cube

Generate a Cantor's cube fractal.

Menger Sponge

Generate a Sierpinski-Menger fractal.

Jerusalem Cube

Generate a Jerusalem cube fractal.

Mosely Snowflake

Generate a Jeaninne Mosely fractal.

Mandelbrot Tree

Generate a Mandelbrot tree fractal.

Barnsey's Tree

Generate a Barnsley's tree fractal.

Barnsey's Fern

Generate a Barnsley's fern fractal.

Binary Fractal Tree

Generate a binary tree fractal.

Ternary Fractal Tree

Generate a ternary tree fractal.

Dragon Fractal Tree

Generate a dragon tree fractal.

De Rham Fractal

Generate a de Rham curve.

Takagi Fractal

Generate a Takagi-Landsberg fractal curve.

Peano Pentagon

Generate a Peano pentagon fractal curve.

Tridendrite Fractal

Generate a tridendrite fractal curve.

McWorter's Pentigree

Generate a Pentigree fractal curve.

McWorter's Lucky Seven

Generate a lucky seven fractal curve.

Eisenstein Fractions

Generate an Eisenstein fractions fractal curve.

Bagula Double V

Generate a Bagula double five fractal curve.

Julia Set

Generate a Julia fractal set.

Mandelbrot Set

Generate a Mandelbrot fractal set.

Mandelbulb Fractal

Generate a Mandelbulb fractal.

Mandelbox Fractal

Generate a Mandelbox fractal.

Buddhabrot Fractal

Generate a Buddhabrot fractal.

Burning Ship Fractal

Generate a Burning Ship fractal.

Toothpick Fractal

Generate a toothpick sequence fractal.

Ulam-Warburton Fractal

Generate an Ulam-Warburton fractal curve.

ASCII Fractal

Generate an ASCII fractal.

ANSI Fractal

Generate an ANSI fractal.

Unicode Fractal

Generate a Unicode fractal.

Emoji Fractal

Generate an emoji fractal.

Braille Fractal

Generate a braille code fractal.

Audio Fractal

Generate a fractal in audio form.

Draw a Pseudofractal

Create a fractal that looks like one but isn't a fractal.

Convert Text to a Fractal

Generate a fractal from any text.

Convert a String to a Fractal

Generate a fractal from a string.

Convert a Number to a Fractal

Generate a fractal from a number.

Merge Two Fractals

Join any two fractals together.

Draw a Random Fractal

Create a completely random fractal.

Iterate an IFS

Set up an arbitrary IFS system and iterate it.

Run IFS on an Image

Recursively transform an image using IFS rules.

Iterate an ICAF

Run infinite compositions of analytic functions.

Generate a Fractal Landscape

Create a surface that mimics a natural terrain.

Generate a Brownian Surface

Create a fractal surface via Brownian motion.

Generate a Self-similar Image

Apply fractal algorithms on your image and make it self-similar.

Find Fractal Patterns in Images

Find fractal patterns in any given image.

Find Fractal Patterns in Text

Find fractal patterns in any given text.

Find Fractal Patterns in Numbers

Find fractal patterns in any given number.

Fill a Plane with Fractals

Tessellate a plane with fractals.

Run a Cellular Automaton

Run a cellular automaton with custom rules.

Play Game of Life

Play Conway's Game of Life on an infinite grid.

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