You cover a strange attractor with boxes of side ε and count N(ε). When you halve ε, you find N(ε/2) ≈ 4.3 × N(ε). What is the approximate box-counting dimension?
AD ≈ 2, because 2² = 4 and 4.3 is close to 4
BD ≈ log(4.3)/log(2) ≈ 2.10
CD ≈ 4.3, because the dimension equals the scaling ratio
DD ≈ 3, because the attractor lives in 3D space
Box-counting dimension satisfies N(ε/2) = 2^D × N(ε). So 2^D = 4.3, giving D = log(4.3)/log(2) ≈ 2.10. This means the attractor is slightly more complex than a surface (D = 2) — it has a layered, quasi-two-dimensional structure with fine fractal detail in the third direction. The dimension of the embedding space (3) is an upper bound but not the dimension of the attractor itself.
Question 2 Short Answer
The Kaplan-Yorke dimension of the Lorenz system with exponents (+0.9, 0, -14.6) is D_KY = 2 + 0.9/14.6 ≈ 2.06. Why does only the ratio of the positive to the negative exponent matter?
Think about your answer, then reveal below.
Model answer: The Kaplan-Yorke formula D_KY = j + (λ₁ + ... + λⱼ)/|λⱼ₊₁| measures how many dimensions the attractor 'fills' before the cumulative stretching is balanced by compression. The positive exponent stretches the attractor, trying to increase its dimension; the negative exponent compresses it, reducing dimension. The ratio λ₁/|λ₃| measures how much of the third dimension the stretching manages to fill before compression overwhelms it. A ratio of 0.06 means the attractor barely penetrates the third dimension — it's almost a surface with a thin fractal cross-section.
Think of it as a budget: the positive exponent 'spends' 0.9 units of stretching per unit time, and the negative exponent 'earns back' 14.6 units of compression. The attractor fills two full dimensions (the flow direction and the stretching direction), plus a fraction 0.9/14.6 of the third dimension before the compression budget is exhausted. The zero exponent (flow direction) contributes a full dimension but no net stretching or compression.
Question 3 True / False
A smooth curve has box-counting dimension 1, and a filled square has dimension 2. The Koch snowflake has dimension log(4)/log(3) ≈ 1.26. This means the Koch snowflake is 'more' than a curve but 'less' than a surface.
TTrue
FFalse
Answer: True
The Koch snowflake is constructed by repeatedly adding smaller triangles to each side, creating a curve of infinite length that encloses a finite area. Its dimension 1.26 reflects this: it fills more space than any smooth curve (dimension 1) but less than any surface patch (dimension 2). Box-counting captures this by measuring scaling: halving the box size more than doubles the box count (as it would for a curve) but less than quadruples it (as it would for a surface). The fractal dimension interpolates between integer dimensions to capture this intermediate scaling.
Question 4 Short Answer
Why are there multiple definitions of fractal dimension (box-counting, Hausdorff, correlation, information), and do they always agree?
Think about your answer, then reveal below.
Model answer: Different dimension definitions measure different aspects of a set's scaling. Box-counting is the easiest to compute but treats all parts of the set equally. Information dimension weights by the probability of visiting each box (more relevant for attractors where some regions are visited more often). Correlation dimension measures clustering of points. Hausdorff dimension is the most mathematically rigorous but hardest to compute. For self-similar fractals, they all agree. For strange attractors with non-uniform structure (multifractals), they generally differ, satisfying D_correlation ≤ D_information ≤ D_box-counting. The full spectrum of dimensions (the multifractal spectrum) characterizes the attractor's inhomogeneity.
The fact that these dimensions differ for typical strange attractors is physically meaningful: it reflects the fact that the attractor is not uniformly fractal — some regions are visited more densely than others, creating a hierarchy of scaling behaviors. Multifractal analysis, which computes the entire spectrum of dimensions, provides a much richer characterization than any single number.