Understanding Sierpinski Polyflakes: The Mathematics of Regular $N$-Flakes
A Sierpinski Polyflake (or simply **$N$-flake**) is a beautiful generalization of regular polygon-based fractals. Created recursively by placing scaled-down copies of a regular polygon with $N$ vertices at its corners, it showcases the striking geometry of symmetric contraction mapping and self-similarity.
The N-Flake Scaling Factor Formula
To construct a regular $N$-flake where child copies touch perfectly at their vertices (or edges) without overlapping, we must solve a precise geometric constraint. The mathematical scaling factor $r$ for any regular $N$-sided polygon is given by the elegant formula:
$$r = \frac{1}{2\left(1 + \sum_{k=1}^{\lfloor N/4 \rfloor} \cos \frac{2\pi k}{N}\right)}$$
Where $\lfloor N/4 \rfloor$ represents the floor function. The distance to shift the child center is then calculated as $d_{\text{shift}} = R \cdot (1 - r)$.
Common Polyflake Scaling Factors
Depending on the number of sides $N$, the perfect non-overlapping scaling factor $r$ varies significantly:
| Polygon | Number of Sides ($N$) | Exact Scaling Factor ($r$) | Approximate Value |
|---|---|---|---|
| Triangle (Sierpinski Gasket) | 3 | $$1/2$$ | 0.500 |
| Square (Sierpinski Square) | 4 | $$1/2$$ | 0.500 |
| Pentagon (Pentaflake) | 5 | $$\frac{3 - \sqrt{5}}{2}$$ | 0.382 |
| Hexagon (Hexaflake) | 6 | $$1/3$$ | 0.333 |
| Heptagon (Heptaflake) | 7 | $$\approx 0.308$$ | 0.308 |
| Octagon (Octaflake) | 8 | $$2 - \sqrt{2}$$ | 0.293 |
Center Filled Variants
Just like the classic hexaflake, you can include a central child polygon inside the polyflake pattern. When the center copy is active, it forms a denser packing layout (e.g., $N+1$ copies per step instead of $N$), which dramatically changes the Hausdorff fractal dimension and visual density of the polyflake.
Frequently Asked Questions
Frequently Asked Questions
What does the "Vertices (Sides)" setting control?
It controls the base geometry of the regular polygon. Setting it to 3 generates a triangular polyflake, 5 generates a pentagonal pentaflake, 6 generates a hexagonal hexaflake, and so on.
Why does the max iteration decrease for larger polygons?
The number of drawn polygons grows exponentially as $N^d$ (or $(N+1)^d$ if centered). For an octagon ($N=8$), level 4 centered would try to draw $9^4 = 6,561$ shapes, which can slow down real-time browser rendering. The app dynamically balances this to keep interactions fluid and performant.
Can I download my custom designs as high-quality vector assets?
Absolutely! You can export your customized polyflakes as vector SVG files, which are infinitely scalable and perfect for print, laser engraving, or digital design. We also support high-resolution PNG downloads and raw JSON coordinates.
