A triangle, a square and a circle drawn over the same cells as Rule 30 runs. After Leonardo's Vitruvian Man, where the circle is centred on the navel and the square lower down: here too, no two shapes share a centre. Beside the plate is the run that makes the circle: column 0 is a wall held at white, black, white, and so on, the right half starts random, and column 1 (outlined) settles onto the wheel, a fixed pattern of 56 steps, and is knocked off it now and then. The dotted lines carry the wheel's point to its two clocks and to the cell of column 1 it stands for.
Each cell is decided by the three above it, so the pattern spreads at most one cell per row each way. Its own coordinates are its two diagonals. Every right diagonal repeats exactly, with a period that is a power of two.
In diagonal coordinates the triangle is the top half of a square standing on its corner. The prize column, the one Wolfram's prizes ask about, is that square's vertical diagonal.
Changes travel right at one cell per row but left at only about a quarter of a cell. So the chaotic core runs from a quarter of the way left to nearly the right edge. Like Leonardo's navel, its centre is not the square's centre.
Beside the wall, column 1 rides a wheel of 56 points, turning 17 notches per step. That is two clocks at once: 1 notch of an 8-point clock and 3 notches of a 7-point one, since 17 leaves 1 over 8 and 3 over 7. The chords trace the star of 56 points taken 17 at a time.
The 7-clock seems to run backwards because two steps of 3 make 6, one short of 7: seen as a pair of hands, it slips back one notch every two steps. The star seems to turn backwards because 23 steps of 17 make 391, one point short of 7 full turns. That is 7/23, the best small fraction near 17/56. A kick of k notches moves the wheel 2k points, so the 8-clock jumps by 2k mod 8 and the 7-clock by 2k mod 7. A common kick, +4, leaves the 8-clock alone and moves the 7-clock exactly one notch.
The pyramid is drawn to row 160 and then holds; the wall run keeps going. The core's edges are the record's measured rates (left edge about 0.25 cells per row; right strip about 2.8 log2 t cells), drawn as guides, not computed from these cells. Column 1 counts as on the wheel inside any stretch of at least 56 steps that follows it exactly; the run is computed 56 steps ahead so this is known as it happens. A kick is counted when column 1 takes up the wheel at a new phase. Most kicks are instant: column 1 jumps straight from one phase of the wheel to another without leaving it. Over millions of steps forward and backward kicks cancel.