Answer Happy

71. (Extra Credit) 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 (book Stage 0) and produce Stage 1 by replacing each edge with four edges of one-third the length, arranged as in book 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. (a) Show that the perimeter, Pn, of the polygon at the nth stage satisfies Pn= Pn-1. Prove that lim P₁ = ∞o, that is, the snowflake has infinite length.. 818 (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 8 with area Ao/9", for n 2 1. Show that the total area of the Koch snowflake is A.