The Chaos Game

Three corners are marked and the walk starts on one of them. A corner is drawn, the dot moves half the way from where it was to that corner, and the new position is marked. The corner drawn next has nothing to do with the corner drawn last, and the dot never returns to any position it has already left except by coincidence.

Watching forty of these moves is watching nothing much: a scatter of marks with a segment showing where each one came from. Watching fifty thousand is watching a specific triangle full of triangular holes appear. The build view shows the first kind and the accumulation view the second, and the rule generating them is the same rule.

Take the whole triangle and apply one move to every point of it at once, always towards the same corner. Halving every distance to that corner turns the triangle into a triangle half the size, sitting in the corner it was aimed at. Doing this for each of the three corners gives three half-size triangles, one in each corner of the original.

Those three copies do not cover the original. The upside-down triangle in the middle is missed by all three, so no move can ever land a dot there. The same argument applies inside each copy, which is why the holes appear at every scale rather than only in the middle, and it is the whole explanation of the figure.

Two runs from two seeds ink different cells and produce the same figure. That sounds contradictory until the two things being described are separated: the sequence of positions is a random object and depends on every draw, while the set those positions are dense in is not random at all. It is fixed by the corners and the ratio before a single draw is made.

Reseed the laboratory and watch which of the two changes. The individual dots move, the address bar changes, and the shape does not. That is the difference between a simulation estimating something and a simulation drawing something that was already determined.

For a corner $v_k$ and a ratio $r$ with $0<r<1$, the move towards that corner is

Then $|f_k(p)-f_k(q)|=(1-r)|p-q|$, so each $f_k$ is a contraction with the same factor $1-r$. A finite family $\{f_1,\dots ,f_m\}$ of contractions is an iterated function system, and it has exactly one non-empty compact set $A$ with

At three corners with $r=1/2$ that set is the Sierpinski gasket. At four corners of a square with $r=1/2$ it is the square itself, which is why that case has nothing to show.

The square. Each of the four maps sends the unit square onto the half-size square at its own corner, and those four half-size squares cover the unit square exactly, with no gap. So the square is a non-empty compact set equal to the union of its own images. Only one such set exists, so the square is the attractor and no pattern of holes is being missed by a run that is too short.

The gasket and the binomial coefficients. Play on the corners $(0,0)$, $(1,0)$ and $(0,1)$ at $r=1/2$, starting at the origin. After $n$ moves the position is

where $S$ is the set of moves that chose $(1,0)$ and $T$ the moves that chose $(0,1)$. A move chooses one corner, so $S$ and $T$ are disjoint and no power of two appears in both coordinates. Reading the leading six bits gives grid indices $i$ and $j$ with $i\mathrm{AND}j=0$, and by Kummer's theorem that is exactly the condition for $\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)$ to be odd. The occupancy of the grid and the odd entries of Pascal's triangle are therefore the same set, which is what the unit test checks cell by cell. Any other placement of the corners is an affine image of this one, and an affine map carries the system to the corresponding system, so it is the same figure in a different frame.

A finite run inks finitely many cells, so paper still showing is not evidence of a hole. At fifty thousand dots the accumulation view leaves many cells of the attractor empty, and the honest reading is the large holes that stay empty as the count rises rather than the speckle that closes. The ledger separates these: the theorem is proved and cited, the grid agreement is checked at one stated resolution, and the cloud on screen is recorded as one run at one seed.

The restriction toggle is the easiest thing on the page to misread. Forbidding an immediate repeat does not filter the same walk, it defines a different one, and the structure that appears belongs to the new system rather than being a hidden feature of the old. The four-corner square is the case worth sitting with: the rule looks identical to the triangle rule, and there is a proof that nothing is there to be found.

What is not settled by anything here is how many dots a run needs before its picture is recognisable. The attractor is determined, and the rate at which a random orbit fills it depends on the ratio, the corner count and the measure the draws induce, none of which this page measures.