T-Square Fractal
Draw the T-square fractal: start with a square, then at every outer corner of the shape add a square centred on that corner with half the side of the squares added before, and keep going. Choose the level from 0 to 8 and the size of the start square. The tool places every square with whole-number arithmetic, so every square is exactly where the rule puts it, and it counts the new squares and works out the filled area as an exact fraction. Fill the squares in one colour or colour them by level, outline every square, or print just the outline of the shape to colour in. Download a PDF, a 300 DPI PNG or an SVG with every square as an exact polygon. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints the start square at the side you type when the whole shape fits on one sheet. The SVG holds every square as an exact polygon.
T-square
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every square, count and area is worked out by our own code with whole-number arithmetic. You may use what you make here, including commercially, with no credit needed.
How to draw and print the T-square fractal
- Pick a preset, or set the level.
- Choose filled squares, every square outlined or the outline of the shape, and the colours.
- Type the side of the start square and choose Fitted to one page or Real size.
- Check the square count and the filled area under the preview.
- Download the PDF and print at 100% (Actual size), or the PNG or SVG.
What you can use it for
- Maths lessons: powers of 3, halving and exact fractions
- Colouring sheets from the outline
- Posters, cards and wall art
- Plotter, laser and vinyl art from the SVG
- Recursion examples for coding classes
- Quilt and tile ideas
The 14 presets
| Preset | Level | Squares | Style | |
|---|---|---|---|---|
| Classic black, level 5 | 5 | 485 | filled | fitted |
| Fine detail: level 8 (13,121 squares) | 8 | 13,121 | filled | fitted |
| First steps: level 2, square outlines | 2 | 17 | square outlines | fitted |
| Colour by level: level 6, rainbow | 6 | 1,457 | filled | fitted |
| Blues by level: level 7 | 7 | 4,373 | filled | fitted |
| Purple to orange: level 6 | 6 | 1,457 | filled | fitted |
| Brown, red and gold: level 5 | 5 | 485 | filled | fitted |
| Greys by level: level 4 | 4 | 161 | filled | fitted |
| Outline of the shape: level 5 (to colour in) | 5 | 485 | outline of the shape | fitted |
| Outline of the shape: level 3 (easy colouring) | 3 | 53 | outline of the shape | fitted |
| Every square outlined: level 4 | 4 | 161 | square outlines | fitted |
| Real size: 80 mm start square, level 6 | 6 | 1,457 | filled | real size, start square 80 mm |
| Real size: 2 in start square, level 5, blues | 5 | 485 | filled | real size, start square 2 in |
| Level 1: the start square and 4 corner squares | 1 | 5 | square outlines | fitted |
Every count is worked out by the tool from the squares it draws.
Squares, area and outline at each level
| Level | New squares | Squares in all | Side of the new squares (start = 1) | Filled area (exact, start square = 1) | Filled area (decimal) | Width of the whole shape (start = 1) | Outline length (start side = 1) |
|---|---|---|---|---|---|---|---|
| 0 | 1 (start) | 1 | 1 | 1/1 | 1 | 1/1 | 4/1 |
| 1 | 4 | 5 | 1/2 | 7/4 | 1.75 | 3/2 | 8/1 |
| 2 | 12 | 17 | 1/4 | 37/16 | 2.3125 | 7/4 | 14/1 |
| 3 | 36 | 53 | 1/8 | 175/64 | 2.7344 | 15/8 | 23/1 |
| 4 | 108 | 161 | 1/16 | 781/256 | 3.0508 | 31/16 | 73/2 |
| 5 | 324 | 485 | 1/32 | 3367/1024 | 3.2881 | 63/32 | 227/4 |
| 6 | 972 | 1,457 | 1/64 | 14197/4096 | 3.4661 | 127/64 | 697/8 |
| 7 | 2,916 | 4,373 | 1/128 | 58975/16384 | 3.5995 | 255/128 | 2123/16 |
| 8 | 8,748 | 13,121 | 1/256 | 242461/65536 | 3.6997 | 511/256 | 6433/32 |
Areas, widths and outline lengths are exact: the tool works in whole units, with the start square 2 to the power n + 1 units wide.
What prints
| Download | What is in it |
|---|---|
| The shape fitted to one page or at real size, vector, with the square count, the filled area and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | Every square as an exact polygon at the printed size, or the outline as one path |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| Squares added at every outer corner, halving each time, with exact areas | T-Square Fractal (this page) |
| Squares on right triangles that branch like a tree | Pythagoras Tree |
| Mandelbrot and Julia sets, or line fractals | Fractal Generator |
| Repeating tile patterns | Truchet Tiles |
| Plain grid paper for your own drawings | Graph Paper |
Other tools: the Fractal Generator renders Mandelbrot, Julia and line fractals, Truchet Tiles makes tile patterns and Graph Paper prints grids.
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Frequently asked questions
What is the T-square fractal?
A flat fractal made of squares. Its name comes from the T-square, the drawing instrument, because the shapes in it remind people of one.
How is it built?
Start with a square. At every convex (outer) corner of the shape so far, add a square centred on that corner with half the side of the squares added in the step before, and repeat. Each new square has 4 corners: the one pointing back lies inside the square it grew from, and the other 3 are outer corners where the next squares go. Our tests check this rule on the generated squares: the new centres are exactly the corners that no other square covers.
How many squares are there?
Step 1 adds 4 squares and every later step adds 3 for each square of the step before, so step n adds 4 × 3 to the power n − 1 squares and level n has 2 × 3 to the power n − 1 squares in all, start square included. Level 8 has 13,121.
How much of it is filled?
Each new square overlaps the square it grew from by exactly a quarter of itself and nothing else, so it adds 3/4 of its area. Counting the start square as 1, the filled area after level n is (4 to the power n + 1 − 3 to the power n + 1) / 4 to the power n: 7/4 at level 1, 37/16 at level 2 and so on. The table gives every level as an exact fraction, checked against a separate count of the shape.
What are the three styles?
Filled squares paints every square, in one colour or in a colour per level. Every square outlined draws each square’s edges, including where squares overlap. Outline of the shape draws only the edge of the filled shape, a clean line drawing for colouring; its length is in the table.
Why does it stop at level 8?
Level 8 has 13,121 squares, and the newest are 1/256 of the start square’s side: on a 90 mm start square that is about 0.35 mm, close to what a printer can show.
Can it print at real size?
Yes, when the whole shape fits on one sheet: the start square is then printed at the side you type, and every square is within 0.2 mm of its true place in our tests on US Letter and A4, as long as you print at 100% (Actual size).
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every square and number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.