Question 19 of 19intermediate🔧 ApplyNumerical3 marks

To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '—' by a 'speed bump'. As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e. 3 times Powers of 4; this sequence is not shown in Table 1)

Correct Answer

The number of line segments in each shape of the Koch Snowflake follows the sequence: 3, 12, 48, 192, 768, ... which is 3 times Powers of 4.

Stage 1: 3 segments 3×43 \times 4⁰ = 3×13 \times 1 = 3

Stage 2: 12 segments 3×43 \times 4¹ = 3×43 \times 4 = 12

Stage 3: 48 segments 3×43 \times 4² = 3×163 \times 16 = 48

Stage 4: 192 segments 3×43 \times 4³ = 3×643 \times 64 = 192

Stage 5: 768 segments 3×43 \times 4⁴ = 3×2563 \times 256 = 768

Reason: Each line segment is replaced by 4 smaller segments (the 'speed bump' has 4 segments). ∴ Each stage multiplies the number of segments by 4.

Exercise: Figure it Out (Section 6) | Q: 5 | (Page Page 12)
For More Understanding

Explanation

The chapter explains that in the Koch Snowflake, each line segment is replaced by a 'speed bump' consisting of 4 smaller segments. Starting with 3 segments (a triangle), each iteration multiplies the total by 4. The sequence is 3, 12, 48, 192, 768, ... which equals 3×43 \times 4⁰, 3×43 \times 4¹, 3×43 \times 4², 3×43 \times 4³, 3×43 \times 4⁴, ... This is 3 times the powers of 4.

Solution Steps

  1. Step 1: Start with 3 line segments (initial triangle)

  2. Step 2: Each segment is replaced by 4 segments (speed bump)

  3. Step 3: Stage 1: 3 segments

  4. Step 4: Stage 2: 3×43 \times 4 = 12 segments

  5. Step 5: Stage 3: 12×412 \times 4 = 48 segments

  6. Step 6: Stage 4: 48×448 \times 4 = 192 segments

  7. Step 7: General formula: 3×43 \times 4^(n-1) segments at stage n

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