The Koch snowflake (described in 1904 by Swedish mathematician Helge von Koch) is an infinitely jagged “fractal” curve obtained as a limit of polygonal curves (it is continuous but has no tangent line at any point). Begin with an equilateral triangle (Stage 0) and produce Stage 1 by replacing each edge with four edges of one-third the length, arranged as in Figure 9. Continue the process: At the nth stage, replace each edge with four edges of one-third the length of the edge from the (n − 1)st stage.

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= (a) Show that the perimeter P, of the polygon at the nth stage satisfies Pn The snowflake has infinite length. (b) Let Ao be the area of the original equilateral triangle. Show that (3)4"-1 new triangles are added at the nth stage, each with area Ao/9" (for n ≥ 1). Show that the total area of the Koch snowflake is Ao. Pn-1. Prove that lim P₁ = ∞o. 11-00 A*** Stage 1 Stage 2 Stage 3