The second method is mindblowing:
- Pick three points on a plane. These are the vertices.
- Pick a starting point anywhere.
- Randomly pick one of the 3 vertices and draw a point halfway between that vertex and your previous point.
- Repeat ad nauseam, using the most recent point each time.
As you do this, you will gradually see a Sierpinski Gasket emerge.
The above images are from this video. You can watch it to get a better sense of how to build the Gasket in this way. The chaos game producing a Sierpinski triangle (YouTube)
The choas game produces a clear Sierpinski Gasket, but you arrived at it by an entirely different method than the geometer's.
This is the Infinite Jest reader's methodology for building the Gasket. On the first pass through the book, the points scattered by your attention look like noise. That's certainly how I felt in my first read-through. As I reread the novel (I'm currently on my third), I fill in more and more points, and the structure of the novel and the relationships between its characters fill in a more complete internal representation of the novel, closer and closer to the geometer's Gasket.
Three features of the chaos game support this metaphor:
The first iterations are noise. Chaos-game simulations conventionally discard the first ~20–30 points as "burn-in". They haven't yet represented the Sierpinski Gasket. This is what a first reading of Infinite Jest felt like to me.
The starting point doesn't matter. The attractor (the Sierpinski Gasket) is determined entirely by the three vertices and the halving rule. Wherever you start, you converge to the same Gasket. Burn-in is why the starting point doesn't matter. It doesn't matter whether you came to Infinite Jest for Hal, Gately, or the Quebecois separatists. As you play the chaos game through rereading, you converge on the same Sierpinski triangle shape.
Each step depends only on where you just were. You don't need to remember the whole history of your previous moves to add the next point, just the most recent one. In rereading terms, this means you can dip back into the book at almost any scene and still add to the Gasket's picture. Infinite Jest rewards non-sequential rereading.