Chapter 7.2, Problem 34MS

A tighter weave. Look back at the Tight Weave story in Chapter 1 (page 9). There you constructed a pattern by dividing a purple square into a $3\times 3$ grid of nine squares, making the central square gold, and repeating the process in each of the remaining eight squares. The purple remaining after repeating that process infinitely often is sometimes called a Sierpinski Carpet. Here let’s modify the process. Instead of dividing the original square into a $3\times 3$ grid and making the central one gold, suppose we divide the square into a $5\times 5$ square and make the central one gold. Then take each of the 24 remaining squares, divide it into a $5\times 5$ grid and make the middle one gold, and continue this process forever. Starting with a square as stage 1, draw the first three stages of this replacement process.