Chapter 12, Problem 10E

Exercises 9 through 12 refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows:

Koch Antisnowflake

• Start: Start with a solid seed equilateral triangle [Fig. 12-36(a) ].

• Replacement rule: In each step replace any boundary line segment ____ with a (where the point is always facing toward the interior of the snowflake). [Figures 12-36(b) and (c) show the figures obtained at Steps 1 and 2, respectively.

Assume that the seed triangle of the Koch antisnowflake has sides of length $18$ cm. Let $M$ denote the number of sides, $L$ the length of each side, and $P$ the perimeter of the shape obtained at the indicated step of the construction. Complete the missing entries in Table $12-10$.

Table $12-10$

	M	L	P
Start	$3$	$18$ cm	$54$ cm
Step $1$	$12$	$6$ cm	$72$ cm
Step $2$
Step $3$
Step $4$
Step $5$