Hausdorff and Moran: the similarity dimension

Solves Moran’s equation sum r_i^D = 1 for several self-similar sets and compares the answers with box-counting estimates computed from the sets themselves.

import numpy as np

from mathematicskit.fractals_chaos import (
    SierpinskiCarpet,
    SierpinskiTriangle,
    box_counting_dimension,
    koch_curve,
    similarity_dimension,
)

Moran’s equation against box counting

cantor_points = np.array([[sum(int(d) * 2 * 3.0 ** -(k + 1) for k, d in enumerate(f"{i:012b}")), 0.0] for i in range(4096)])
cases = {
    "Cantor set": ([1 / 3] * 2, cantor_points),
    "Koch curve": ([1 / 3] * 4, koch_curve(7)),
    "Sierpinski triangle": ([1 / 2] * 3, SierpinskiTriangle().generate(200000)),
    "Sierpinski carpet": ([1 / 3] * 8, SierpinskiCarpet().generate(300000)),
}
for name, (ratios, points) in cases.items():
    if name == "Cantor set":
        points = np.column_stack([points[:, 0], np.zeros(len(points))])
        sizes = np.logspace(-1, -3.3, 10)
        estimate = np.polyfit(np.log(1 / sizes), np.log([len(np.unique(np.floor(points[:, 0] / s))) for s in sizes]), 1)[0]
    else:
        estimate = box_counting_dimension(points).dimension
    print(f"{name:20s} similarity dimension {similarity_dimension(ratios):.4f}, box counting {estimate:.4f}")

Cantor set           similarity dimension 0.6309, box counting 0.6353
Koch curve           similarity dimension 1.2619, box counting 1.2771
Sierpinski triangle  similarity dimension 1.5850, box counting 1.5538
Sierpinski carpet    similarity dimension 1.8928, box counting 1.8194

Unequal ratios

print(f"\nratios (1/2, 1/4, 1/4): D = {similarity_dimension([0.5, 0.25, 0.25]):.4f}")
print(f"ratios (0.6, 0.3):      D = {similarity_dimension([0.6, 0.3]):.4f}")

ratios (1/2, 1/4, 1/4): D = 1.0000
ratios (0.6, 0.3):      D = 0.8594