A fanctal is a family of self-similar circular fractals, built by dividing a disk into equal angular sectors and inscribing, in some of them, a smaller tangent disk that repeats the same construction inside itself, indefinitely. Varying how many sectors there are and how many are recursed traces out an entire family of shapes — from sparse, spidery patterns to dense, almost space-filling ones — all sharing the same self-similar logic. Explore that family below, or read the summary and download a Python implementation further down the page.
Area calculation A(r, n, a)
About the fanctal
The original fanctal divides a disk of radius \(R\) into \(6\) congruent sectors and inscribes, in \(3\) alternating sectors, a tangent child disk; the construction repeats indefinitely inside each child.
Main result
The shaded area equals \(\pi R^2/4\) — proved by two independent methods. The perimeter added at each generation is invariant, \(\pi R + 6R\), but the sum over all generations diverges.
Generalization
The construction generalizes to a parametric family \(f(r,n,a)\): \(n\) angular sectors, of which \(a\) (preferably alternated) are selected to recurse, with scale ratio \(k(n) = \dfrac{\sin(\pi/n)}{1+\sin(\pi/n)}\). Its closed-form area is \(A(r,n,a) = \pi r^2 \cdot \dfrac{a\left(\frac{1}{n} - k^2\right)}{1 - ak^2}\), valid for every admissible \((n,a)\); setting \(n=6\), \(a=3\) recovers exactly \(\pi R^2/4\).
Fractal dimension
The family's self-similar attractor has Hausdorff dimension \(\dim_H(K) = \dfrac{\log a}{-\log k}\), which equals exactly \(1\) when \(ak=1\) — the same condition connected to the invariant perimeter of the original fanctal.
Yasser Díaz Arcila, Juan Manuel Díaz Gómez, and Elizabeth Solórzano Tovar.
A version restructured for peer review is in preparation for submission to Eco Matemático (Universidad Francisco de Paula Santander).
It will be published under a CC BY 4.0 license.
Reproduce it in Python
This script implements the same recursive construction described above — sector selection, scale ratio, and termination criteria — so you can regenerate every figure independently, in your own language if you prefer.