Rule 30
Draw rule 30 and every other elementary cellular automaton, and print it: a row of black and white cells where each new cell depends only on itself and its two neighbours in the row above, so the rows stacked from top to bottom show the whole history. Type any rule number from 0 to 255 (30, 90, 110, 184 …), start from one live cell, a random row with a seed or your own row of 0s and 1s, choose a row of 3 to 1,000 cells with ends that wrap round or stay 0, and draw up to 1,000 generations. The 8 neighbourhoods and the new cell each one makes are drawn above the diagram, worked out from the bits of the rule number, and the footer lists the middle column. Leave the last rows blank for a worksheet. Download a PDF, a 300 DPI PNG or an SVG with every cell as an exact rectangle. Nothing is uploaded. Not affiliated with Wolfram Research.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints on one sheet when the drawing fits; the check bar in the footer lets you confirm the scale. The SVG holds the drawing at the printed size.
Rule
Start row (generation 0)
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every generation is worked out by our own code from the rule number and the start row shown. You may use what you make here, including commercially, with no credit needed.
How to draw and print rule 30
- Pick a preset, or type a rule number from 0 to 255.
- Choose the start row: one cell, a random row with a seed, or your own 0s and 1s.
- Set the row width, the number of generations and the edges (wrap round or fixed 0).
- Tick the rule table, grid lines or blank rows for a worksheet; choose Fitted to one page or Real size and the paper.
- Download the PDF and print at 100% (Actual size), or the PNG or SVG.
What you can use it for
- Maths and computing lessons on cellular automata and binary numbers
- Work-out-the-next-row worksheets with blank generations
- Posters and wall art of rule 30, 90 and 110
- Plotter, laser and vinyl art from the SVG
- Knitting, weaving and cross-stitch charts from the cell grid
- Comparing all 256 rules side by side
The 15 presets
| Preset | Rule | Binary | Start | Width × generations | Edges | Live cells (last row drawn) |
|---|---|---|---|---|---|---|
| Rule 30 from one cell, 201 × 100 | 30 | 00011110 | one live cell | 201 × 100 | wrap round | 110 at generation 99 |
| Rule 30 from one cell, 999 × 500 (the largest width shown) | 30 | 00011110 | one live cell | 999 × 500 | wrap round | 506 at generation 499 |
| Rule 90 from one cell: the triangle pattern of Pascal’s triangle mod 2, 129 × 64 | 90 | 01011010 | one live cell | 129 × 64 | wrap round | 64 at generation 63 |
| Rule 110 from one cell, 201 × 200 (grows to the left) | 110 | 01101110 | one live cell | 201 × 200 | wrap round | 119 at generation 199 |
| Rule 110, random start (seed 1, 50%), 300 × 300 | 110 | 01101110 | random (seed 1, 50%) | 300 × 300 | wrap round | 175 at generation 299 |
| Rule 184, random start (seed 2, 40%), 150 × 150, wrap round | 184 | 10111000 | random (seed 2, 40%) | 150 × 150 | wrap round | 58 at generation 149 |
| Rule 30, random start (seed 1, 50%), 200 × 200 | 30 | 00011110 | random (seed 1, 50%) | 200 × 200 | wrap round | 98 at generation 199 |
| Rule 45 from one cell, 201 × 100 | 45 | 00101101 | one live cell | 201 × 100 | wrap round | 119 at generation 99 |
| Rule 73, random start (seed 3, 50%), 200 × 200 | 73 | 01001001 | random (seed 3, 50%) | 200 × 200 | wrap round | 93 at generation 199 |
| Rule 150 from one cell, 129 × 64 | 150 | 10010110 | one live cell | 129 × 64 | wrap round | 85 at generation 63 |
| Rule 105 from one cell, 129 × 64 | 105 | 01101001 | one live cell | 129 × 64 | wrap round | 44 at generation 63 |
| Rule 54 from one cell, 201 × 100 | 54 | 00110110 | one live cell | 201 × 100 | wrap round | 150 at generation 99 |
| Rule 126 from one cell, 129 × 64 | 126 | 01111110 | one live cell | 129 × 64 | wrap round | 127 at generation 63 |
| Rule 30 from one cell, 61 × 60 with fixed (always 0) edges | 30 | 00011110 | one live cell | 61 × 60 | fixed 0 | 31 at generation 59 |
| Worksheet: rule 30 from one cell, 15 × 8 at 8 mm, generations 0 to 3 drawn, 4 left blank (real size) | 30 | 00011110 | one live cell | 15 × 8 | wrap round | 6 at generation 3 |
Every count is worked out by the tool from the rows it draws.
The rule table for 30, 90, 110 and 184
| Neighbourhood (left, centre, right) | Value k | Rule 30 | Rule 90 | Rule 110 | Rule 184 |
|---|---|---|---|---|---|
| 111 | 7 | 0 | 0 | 0 | 1 |
| 110 | 6 | 0 | 1 | 1 | 0 |
| 101 | 5 | 0 | 0 | 1 | 1 |
| 100 | 4 | 1 | 1 | 0 | 1 |
| 011 | 3 | 1 | 1 | 1 | 1 |
| 010 | 2 | 1 | 0 | 1 | 0 |
| 001 | 1 | 1 | 1 | 1 | 0 |
| 000 | 0 | 0 | 0 | 0 | 0 |
Each column is bit k of the rule number, worked out by the tool. The rule 30 column matches the table in the Wikipedia article on rule 30 (read 12 November 2026).
Rule 90 and Pascal’s triangle
| Generation n | Live cells (tool) | 1s in n written in binary | 2 to that power |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
| 1 | 2 | 1 | 2 |
| 2 | 2 | 1 | 2 |
| 3 | 4 | 2 | 4 |
| 4 | 2 | 1 | 2 |
| 7 | 8 | 3 | 8 |
| 8 | 2 | 1 | 2 |
| 15 | 16 | 4 | 16 |
| 16 | 2 | 1 | 2 |
| 31 | 32 | 5 | 32 |
| 32 | 2 | 1 | 2 |
| 63 | 64 | 6 | 64 |
From one cell, rule 90 row n has a live cell wherever the matching number in Pascal’s triangle is odd, so its count is a power of 2.
Edges and conservation
| Run | What happens (worked out by the tool) |
|---|---|
| Rule 30, 61 cells, wrap round vs fixed 0 | the pattern reaches the ends at generation 30; the two rows first differ at generation 31 |
| Rule 30, 61 cells, wrap round, generation 59 | 31 live cells |
| Rule 30, 61 cells, fixed 0, generation 59 | 31 live cells |
| Rule 184, 150 cells, wrap round, random start (seed 2, 40%) | the number of live cells stays 58 in all 150 generations |
What prints
| Download | What is in it |
|---|---|
| The diagram fitted to one page or at real size, vector, with the rule table, the live-cell counts, the middle column and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The diagram at the printed size: live cells as exact rectangles (one row of touching cells per rectangle), the rule table, grid lines and the frame |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| One row of cells, any rule 0–255, its history drawn top to bottom | Rule 30 (this page) |
| A 2-D grid where cells live and die by their 8 neighbours | Conway’s Game of Life |
| An ant that turns on black and white cells | Langton’s Ant |
| Pixels you place by hand | Pixel Art Maker |
| Square grids to draw on | Graph Paper |
Other tools: the Pixel Art Maker lets you draw cells by hand, Graph Paper prints blank grids and the Cross Stitch Pattern Maker turns pictures into charts.
More free tools
Grids, patterns and printables?
Drawing and print tools that run privately in your browser, with no sign-up and no watermark.
Frequently asked questions
What is rule 30?
One of the 256 elementary cellular automata: a row of cells, each 0 (white) or 1 (black), where every cell’s next state depends on itself and its left and right neighbours. Rule 30 makes a new cell 1 for the neighbourhoods 100, 011, 010 and 001 and 0 for the other four; the Wikipedia article on rule 30 (read 12 November 2026) says it was introduced by Stephen Wolfram in 1983. From one live cell it grows a triangle with an irregular, non-repeating inside.
How does the rule number work?
List the 8 possible neighbourhoods from 111 down to 000 and write the new cell each one makes: for rule 30 that is 0, 0, 0, 1, 1, 1, 1, 0. Read as a binary number, 00011110 is 30. Every rule from 0 to 255 is made the same way, which is why there are exactly 256 of them. The tool draws this table above the diagram from the bits of the number you type, and our tests check all 256 rules against a separate stepper of our own.
Is there a formula for rule 30?
Yes: the new cell is left XOR (centre OR right), as the Wikipedia article on rule 30 gives it. Our tests check that this formula and the tool agree on all 8 neighbourhoods.
What does rule 90 draw?
From one live cell, rule 90 (new cell = left XOR right) draws a triangle of triangles: the rows match Pascal’s triangle with every odd number black and every even one white. Our tests check every cell of the 64-generation preset against the binomial numbers, and the table on this page shows that row n has 2 to the power of (the number of 1s in n written in binary) live cells.
What is the middle column?
The column straight below the first live cell. For rule 30 its first 40 generations are 1101110011000101100100111010111001110101, which matches sequence A051023 in the OEIS (read 12 November 2026). The footer prints the first 32 for any rule started from one cell. The Wikipedia article notes that Wolfram proposed this column as a source of pseudorandom numbers; this tool makes no claim about how random it is.
What do the edge settings do?
A row on paper has two ends. Wrap round joins them, so the right end’s neighbour is the left end, like a ring. Fixed 0 treats every cell beyond the ends as always white. The two give the same picture until the pattern reaches the ends; the table on this page shows where they start to differ, worked out by the tool.
Are many of the 256 rules the same?
Some are mirror images or black-and-white swaps of each other. The Wikipedia article on elementary cellular automata says 88 of the 256 are different once those are counted as the same. The tool draws every one of the 256 separately.
Can I make a worksheet?
Yes. Set a narrow row, turn on grid lines and leave the last few generations blank, then print at real size: pupils fill in the blank rows by hand with the rule table above as their key, and check against the full drawing.
How big can it go?
Rows up to 1,000 cells, up to 1,000 generations and up to 500,000 cells in one drawing; every preset draws in under 2 seconds in our tests. Grid lines work on rows up to 150 cells.
Can it print at real size?
Yes, when the drawing fits on one sheet: choose Real size and set the cell size. Every printed cell is within 0.2 mm of its true place and size in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every generation and cell is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.