Chapter 12, Problem 12E

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 area $$ $A=729$. Let $R$ denote the number of triangles subtracted at a particular step, $S$ the area of each subtracted triangle. $T$ the total area subtracted, and $Q$ the area of the shape obtained at a particular step of the construction. Complete the missing entries in $\text{Table }12-12$.

Table $12-12$

	R	S	T	Q
Start	$0$	$0$	$0$	$729$
Step $1$	$3$	$81$	$243$	$486$
Step $2$	$12$	$9$	$108$	$378$
Step $3$
Step $4$
Step $5$