Homework discussion Read pages 388 – 391 Page 400: 49 – 52, 72

FIGURE 12-16

SELF-SIMILARITY A shape has self-similarity (symmetry of scale) if parts of the shape appear at infinitely many different scales.

FIGURE 12-19a

FIGURE 12-19b

FIGURE 12-19c

FIGURE 12-8 START. Start with a solid equilateral triangle of arbitrary size. (For simplicity assume that the sides of the triangle are of length 1.) STEP 1. (Procedure KS): Attach in the middle of each side an equilateral triangle, with sides of length 1/3 of the previous side. When we are done the result is a “star of David” with 12 sides each of length 1/3.

FIGURE 12-2 STEP 2. For each of the 12 sides of the star of David in Step 1, repeat procedure KS: In the middle of each side attach an equilateral triangle (with dimensions 1/3 of the dimensions of the side). The resulting shape has 48 sides, each of length 1/9. STEP 3,4, etc. Continue repeating procedure KS to the “snowflake” obtained in the previous step.

FIGURE 12-3 A rendering of the Koch snowflake.

FIGURE 12-4 (a) The Koch curve. (b) A portion of the curve in detail (magnified by 3).

FIGURE 12-4

RECURSIVE REPLACEMENT RULES A recursive process is a process in which the same set of rules is applied over and over, with the end product at each step becoming the starting point for the next step RECURSIVE REPLACEMENT RULE FOR THE KOCH SNOWFLAKE Whenever you see a line segment replace it with. Start with a solid equilateral triangle.

FIGURE 12-8

FIGURE 12-2

FIGURE 12-6a

FIGURE 12-8 START. Start with a solid triangle of arbitrary size. STEP 1. Procedure SG: Remove the triangle whose vertices are the midpoints of the sides of the triangle (We’ll call this the middle triangle.) this leaves a white triangular hole in the original solid triangle, and three solid triangles, each of which is a half-scale version of the original.

FIGURE 12-8 START. Start with a solid triangle of arbitrary size. STEP 1. Procedure SG: Remove the triangle whose vertices are the midpoints of the sides of the triangle (We’ll call this the middle triangle.) this leaves a white triangular hole in the original solid triangle, and three solid triangles, each of which is a half-scale version of the original.

FIGURE 12-2 STEP 2. For each of the solid triangles in the previous step, repeat procedure SG. (that is, remove the middle triangle.) this leaves us with 9 solid triangles (all similar to the original triangle ABC) and 4 triangular white holes. STEP 3,4, etc. Continue repeating procedure SG on every solid triangle, ad infinitum.

FIGURE 12-2 STEP 2. For each of the solid triangles in the previous step, repeat procedure SG. (that is, remove the middle triangle.) this leaves us with 9 solid triangles (all similar to the original triangle ABC) and 4 triangular white holes. STEP 3,4, etc. Continue repeating procedure SG on every solid triangle, ad infinitum.

Sierpinski Gasket RECURSIVE REPLACEMENT RULE FOR THE Sierpinski Gasket Whenever you see a triangle replace it with. Start with a solid triangle. RECURSIVE REPLACEMENT RULES A recursive process is a process in which the same set of rules is applied over and over, with the end product at each step becoming the starting point for the next step

FIGURE 12-9

FIGURE 12-10

FIGURE 12-13

FIGURE 12-14

FIGURE 12-15

FIGURE 12-2

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