Question 2 of 13intermediate🔍 AnalyzeNumerical4 marks

Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.

Correct Answer

At Step 0: Holes = 0 Triangles remaining = 1

At Step 1: Holes = 1 Triangles remaining = 3

At Step 2: Holes = 1 + 3 = 4 Triangles remaining = 9

At Step n: Holes = 1 + 3 + 3² + ... + 3^(n-1) Triangles remaining = 3n3^{n}

Exercise: Figure it Out (Section 1) | Q: 2 | (Page Page 3)
For More Understanding

Explanation

The chapter shows that for the Sierpinski Triangle, each remaining triangle gives rise to 3 triangles in the next step (unlike 8 in the Sierpinski Carpet). So R_n = 3n3^{n}. For holes, each remaining triangle creates one new hole, so H_{n+1} = H_n + R_n, leading to H_n = 1 + 3 + 3² + ... + 3^(n-1).

Solution Steps

  1. Step 1: At Step 0, there is 1 triangle and 0 holes.

  2. Step 2: Each remaining triangle produces 3 new triangles, so R_n = 3n3^{n}.

  3. Step 3: Each remaining triangle creates 1 new hole, so H_{n+1} = H_n + R_n.

  4. Step 4: Calculate: R_0=1, R_1=3, R_2=9; H_0=0, H_1=1, H_2=1+3=4.

  5. Step 5: General formula: R_n = 3n3^{n}, H_n = (3n3^{n} - 1)/2 for n ≥\geq 1.

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