Table 3.

Fractal dimensions from the present box-counting^72,73 compared to the corresponding Hausdorff dimensions of known geometrical patterns⁷³ shown in Fig. 11.

Known fractals	Hausdorff dimension	Present box-counting	Error ($\%$)
(a) von Kock snow flake	1.262	1.298	2.84
(b) Sierpinski triangle	1.585	1.572	$-$ 0.85
(c) 32 Segment $\frac{1}{8}$-scale quadratic	1.667	1.656	$-$ 0.62
(d) Ikeda map	1.680	1.648	$-$ 1.89
(e) Hexa-flake	1.771	1.705	$-$ 3.73
(f) $H-I$ de Rivera	1.771	1.752	$-$ 1.09
(g) Sierpinski carpet	1.893	1.876	$-$ 0.91
(h) Mandelbrot set	2	1.910	$-$ 4.5