Geometric
n-flake (pentaflake)
For each n ≥ 3, the n-flake is a self-similar set built from a central regular n-gon with n smaller copies of itself attached around the perimeter. The pentaflake (n = 5) and hexaflake (n = 6) are the most common.
At a glance
| Family | n-flake for n ≥ 3 |
|---|---|
| Self-similar copies | n + 1 (central + n vertices) |
| Contraction ratio | 1 / (1 + 2 cos(π/n)) |
| Hausdorff dimension (pentaflake) | ≈ 1.8617 |
| Hausdorff dimension (hexaflake) | log 7 / log 3 ≈ 1.7712 |
Construction
- Pick a regular n-gon of side 1.
- Place a smaller copy of the n-gon at the centre and at each of the n vertices, scaled by
r = 1 / (1 + 2 cos(π/n)). - Recurse on each of the n + 1 sub-polygons.
Symmetry
The n-flake inherits the dihedral symmetry group Dn of the regular polygon. For n = 5 we get five-fold symmetry; for n = 6, six-fold (related to the Star of David / hexaflake).
References
- Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
- Edgar, G., Measure, Topology, and Fractal Geometry, Springer, 2008.
- Koch snowflake · Sierpinski triangle
Try it
Run an interactive playground at /tools/n-flake.