Rule 30 is Rule 60 plus kicks: each square is its upper-left neighbour XOR the square above it, flipped once more when the square above is a kick, a white square with a black square to its right. That one equation can be read two ways. Down the page it builds the familiar pyramid, row by row from one black square. Sideways it is solved for the upper-left square instead, and builds the left half outwards from the centre column, beside a wall held at white, black, white… Either way every black square is an odd overlap of Sierpinski triangles, one spreading from each kick. Point at a square to see the kicks that decide it; click a kick to see the triangle it spreads.
Down the page, the equation fixes each square from the row above it, so every triangle spreads down and to the right. Sideways, the same equation fixes each square from the column to its right, so the left half can be built outwards from the centre column, and every triangle spreads down and back in time. The kicks are the same squares in both. Only where the dot is drawn differs: down the page it sits on the kick's own white square, whose triangle starts just below it; sideways it sits on the square it decides, one place further from the wall. Rule 30 can be solved this way for its upper-left square but not for its upper-right one, which is why only the left half can be built sideways.
Rule 30 makes a square black when its upper-left neighbour differs from (the square above or its upper-right neighbour). Rewritten, that says: do what Rule 60 does, black when the upper-left square and the square above differ, and then flip the square under every kick. Rule 60 on its own grows one black square into a Sierpinski triangle; press Kicks removed to see it. Each kick starts a triangle of its own, and where triangles overlap they cancel in pairs. So a square is black exactly when an odd number of triangles cover it.
The striped left side and the wild right side are built the same way, from the same kind of kick, and about a quarter of all squares are kicks on both sides; the readout counts them. What differs is how the kicks are arranged. On the left they fall in regular diagonal stripes, so their triangles stack into stripes; on the right they are scattered, and the overlapping triangles look like noise. Point at a square on the wild side: its triangle of kicks reaches back across the centre into the striped side.
Every triangle spreads down and to the right, never to the left. Rule 30 uses its upper-left neighbour exactly: flip that square and the square below always flips. Its upper-right neighbour counts only through the kick. That one-sidedness is also why the left side can be built sideways from the centre column outwards, as the sideways view does, and the right side cannot.
Run sideways, Rule 30 writes each row of this picture from the row above it. A cell is the row above at the next moment, XOR the row above now, XOR an edge event. The first two parts are one step of the Gray code in time: the row records where the row above flips.
Click a dot and you see the triangle it spreads: down the picture and back in time, three half-size copies of itself, each made of three smaller copies, the odd entries of Pascal's triangle. The cell under the pointer is covered by the triangles of exactly the dots inside the red triangle drawn above it, which is the same shape turned round, up the picture and forward in time. Where triangles overlap they cancel in pairs, so the cell is black exactly when its red triangle holds an odd number of dots. The card counts them and checks the count against the cell.
Some places in that triangle can never hold a dot, whatever the right half (marked ×): depths 2 and 6 never do, nor depth 1 at odd times or depths 4 and 14 at even times. Its slanted and upright edges never hold two dots side by side either (the rules below), so at most every other place on them is filled.
Rule 30 makes its own events: only the first row's come from outside, from column 1, and every deeper one is born where the pattern leaves a white cell beside a black one. Random photons keep the first row and scatter every deeper event at random at the same overall rate. What changes is what self-feedback does.
Because the picture makes its own dots, three exact rules hold in Rule 30 and nowhere else. A dot never has another dot straight below it, nor one below and to the left. Along a row, a dot is followed by another exactly when the cell under it is black, so each line of dots runs over black cells and ends on one white. Below and to the right, one more dot costs two more white cells. The “dots continue” readout counts how often the next cell in each direction also holds a dot: about 54%, 24%, 0% and 0% for Rule 30, and the overall rate, about 26%, in every direction for random photons.
Suppose a finite start, nothing black beyond some depth, made the wall repeat for ever. A change spreads at most one cell per step, so at time t everything deeper than that depth plus t would still be white: a wedge whose upper side drops one depth per step. Click or drag on the green line, or use the slider, to choose where the white edge starts.
In a real counterexample the wedge would go on for ever. This picture holds 200 steps, and row k is known only up to time 200 − k, because each row is one cell shorter than the row above it. So the wedge ends where the picture does, at its slanted right side, and with every row shown it closes into a complete triangle. Shaded cells are ones a counterexample must leave white; the darker ones are where this right half is black, so it does not come from a counterexample.
Down the page: the page rebuilds every square from the triangles alone (Lucas's theorem: a triangle started r rows up reaches the square d places to its right exactly when C(r, d) is odd) and compares the result with Rule 30 itself. The rewriting is exact algebra. The quarter is a measurement: from one black square, over rows 400 to 799, the left third of the pyramid is 25.8% kicks, the middle third 24.9% and the right third 25.0% (checked on 10 October 2026). Sideways: the right half starts as 24 random cells beside the wall and is run for 200 steps; its column 1 drives everything. The page rebuilds every cell from depth 2 on as a sum of Pascal triangles and compares it with the rule run sideways. Silent rows: depth 2 is proved in G240, depth 6 in CL048, depth 14 at white times in GC589 (with an exhaustive check in L309), and depth 1 at black times and depth 4 at white times too. The three dot rules are filed under G240. The sideways view was the Edge-Event Sieve until 10 October 2026; its old address now opens this page at that view.