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Optional Quiz 4

# Optional Quiz 4 - S ∞ the Serpenski sponge Iterates 0 1...

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Name: Optional Quiz 4 Due Wednesday April 20, 2011 Staple your work 1. Consider the Serpenski sponge, constructed as follows. The zeroth iterate S 0 is a 1 × 1 square. The ﬁrst iterate S 1 is a 1 × 1 square with the middle 1 3 × 1 3 square removed, leaving eight 1 3 × 1 3 squares. The second iterate S 2 is S 1 with the middle 1 9 × 1 9 of each of the eight 1 3 × 1 3 squares removed. Continuing this process an inﬁnite number of times gives

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Unformatted text preview: S ∞ , the Serpenski sponge. Iterates 0 , 1 and 2 are shown below. S S 1 S 2 (a) Find the fractal dimension of S ∞ . fractal dimension 2 (b) Find the area of S n , the n th iterate in the construction of the Serpenski sponge. area of S n...
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Optional Quiz 4 - S ∞ the Serpenski sponge Iterates 0 1...

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