Find the area of the region remaining at the th step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.
Sierpinski Carpet: Given: Starting area = 1 sq. unit Step 1: At each step, 8/9 of the area remains (1 square removed out of 9). Step 2: After n steps: Area = 1 × (8/9)^n = (8/9)^n sq. units. Answer: Area remaining at step n = (8/9)^n sq. units
Sierpinski Triangle: Given: Starting area = 1 sq. unit Step 1: At each step, 3/4 of the area remains (1 triangle removed out of 4). Step 2: After n steps: Area = 1 × (3/4)^n = (3/4)^n sq. units. Answer: Area remaining at step n = (3/4)^n sq. units
Explanation
The chapter establishes that for Sierpinski Carpet, each step removes 1/9 of the remaining area (since 1 of 9 squares is removed), so area remaining = (8/9)^n. For Sierpinski Triangle, each step removes 1/4 of the remaining area (since 1 of 4 triangles is removed), so area remaining = (3/4)^n.
Solution Steps
Step 1: For Sierpinski Carpet - Starting area = 1 sq. unit.
Step 2: At each step, 8/9 of the area remains (1 square removed out of 9).
Step 3: After n steps: Area = 1 × (8/9)^n = (8/9)^n sq. units.
Step 4: For Sierpinski Triangle - Starting area = 1 sq. unit.
Step 5: At each step, 3/4 of the area remains (1 triangle removed out of 4).
Step 6: After n steps: Area = 1 × (3/4)^n = (3/4)^n sq. units.