We previously explored the SierpiΕski Triangle. Now, let's look at another famous fractal: The SierpiΕski Carpet.
Instead of a triangle, we start with a solid square. We divide it into a $3 \times 3$ grid (9 smaller squares) and remove the central one. We then repeat this process infinitely for every remaining square. The mathematics of counting these squares and calculating their shrinking area relies perfectly on Geometric Progressions! Let's solve Exercise 8.3 Q7.
Instead of a triangle, we start with a solid square. We divide it into a $3 \times 3$ grid (9 smaller squares) and remove the central one. We then repeat this process infinitely for every remaining square. The mathematics of counting these squares and calculating their shrinking area relies perfectly on Geometric Progressions! Let's solve Exercise 8.3 Q7.
1 SierpiΕski Carpet: Square Count
2 SierpiΕski Carpet: Area Decay
Select the correct option:
In the SierpiΕski Carpet, each solid square is divided into 9 smaller squares, and the central one is removed. What is the common ratio (\(r\)) for the sequence of the number of solid squares?
Concept:
1
If 1 item produces \(X\) new items, the sequence of the total count forms a GP where the common ratio \(r = X\).
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Hint:
1
Stage 0 = 1 square. Stage 1 = 8 squares. \(\frac{8}{1} = ?\)
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Mistakes to Avoid:
!
Confusing counting with measuring: The number of squares is an integer count (1, 8, 64). The area of the carpet is a fractional measurement (1, 8/9, 64/81).
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Solution:
Because each square yields 8 new smaller squares, the number of squares multiplies by 8 at each stage.
Therefore, \(r = 8\).
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Select the correct option:
What is the common ratio (\(r\)) for the sequence of the remaining area in the SierpiΕski Carpet?
Concept:
1
If a fraction \(X\) of an object is removed, the remaining fraction is \(1 - X\).
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Hint:
1
1 out of 9 squares is removed.
2
\(1 - \frac{1}{9} = ?\)
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Mistakes to Avoid:
!
Choosing the removed fraction: The sequence tracks the area that is LEFT, not the area that was taken away! Don't select 1/9.
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Solution:
At every step, the area retains \(\frac{8}{9}\) of its previous size.
Therefore, \(r = \frac{8}{9}\).
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Select the correct option:
What happens to the total area of the SierpiΕski Carpet as the number of stages (\(n\)) approaches infinity?
Concept:
1
If the common ratio of a GP is a positive fraction less than 1 (\(0 < r < 1\)), the sequence exhibits exponential decay toward zero.
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Hint:
1
Calculate \((\frac{8}{9})^{1000}\). The result is incredibly small.
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Mistakes to Avoid:
!
Confusing limits: The sequence starts at 1 and drops to 8/9, then 64/81. It does not stop at 8/9. It keeps getting smaller forever.
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Solution:
Because the common ratio \(\frac{8}{9}\) is less than 1, multiplying the area by it repeatedly causes the value to shrink infinitely towards 0.
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Select the correct option:
If you want to find the number of solid squares in the SierpiΕski Carpet at Stage 3 using the explicit formula \(t_n = 8^n\), what is the correct calculation?
Concept:
1
Because Stage 0 defines the starting position, the exponent is simply \(n\). Substitute \(n=3\).
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Hint:
1
Calculate \(8^3\).
2
\(8 \times 8 \times 8 = ?\)
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Mistakes to Avoid:
!
Multiplying base by exponent: \(8^3\) is NOT \(8 \times 3 = 24\). An exponent means multiplying the base by itself!
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Solution:
\(t_3 = 8^3 = 8 \times 8 \times 8 = 512\).
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Select the correct option:
What is the defining difference between the SierpiΕski Triangle and the SierpiΕski Carpet?
Concept:
1
Both are fractals with an area that approaches 0. The difference is their geometric construction and their specific GP ratios.
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Hint:
1
Triangle Area Ratio: \(\frac{3}{4}\). Carpet Area Ratio: \(\frac{8}{9}\).
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Mistakes to Avoid:
!
Assuming they behave differently at infinity: Because both use a ratio less than 1, BOTH of their areas shrink to zero at infinity.
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Solution:
The Sierpinski Triangle uses a triangle base and an area ratio of 3/4.
The Sierpinski Carpet uses a square base and an area ratio of 8/9.
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