Sierpinski, Koch, Menger - constructing infinite detail
The fractals on this page are constructed by simple geometric rules applied recursively. Each iteration adds more detail, and in the limit, they possess properties that seem paradoxical: infinite length in finite space, or infinite surface with zero volume.
These objects challenged mathematicians' understanding of dimension, measurement, and continuity - they were once called "mathematical monsters" before Mandelbrot showed they were everywhere in nature.
Start with a triangle. Remove the middle. Repeat for each remaining triangle, forever. The result has zero area but infinite boundary. Remarkably, you can also create it with the "Chaos Game":
Each triangle is divided into 3 smaller triangles. The fractal dimension is log(3)/log(2) ≈ 1.585.
Each side of a triangle is replaced with a bump, then each new segment gets a bump, forever. The perimeter grows without bound while the area converges to a finite value:
The Koch snowflake has infinite perimeter but finite area. Each iteration replaces every segment with 4 segments 1/3 the length - the perimeter grows by 4/3 each time!
The fractal dimension measures how a shape fills space. For self-similar fractals, we can calculate it:
Where N = number of copies and S = scale factor.
Fold a strip of paper in half repeatedly, then unfold at right angles. The resulting curve never crosses itself and eventually fills a region of the plane completely (dimension 2):
The Dragon Curve can be made by folding a strip of paper in half repeatedly, then unfolding at 90° angles. It tiles the plane and has fractal dimension 2!
The 3D analog of the Sierpiński carpet. Divide a cube into 27 smaller cubes, remove the center and face centers (7 cubes), repeat. The limiting object has infinite surface area but zero volume:
The Menger sponge has infinite surface area but zero volume (in the limit). Its fractal dimension is log(20)/log(3) ≈ 2.727. Drag to rotate, scroll to zoom.
| Fractal | Dimension | Area/Volume | Perimeter/Surface |
|---|---|---|---|
| Sierpiński | ~1.585 | 0 | ∞ |
| Koch | ~1.262 | Finite | ∞ |
| Dragon | 2 | Finite (fills) | ∞ |
| Menger | ~2.727 | 0 | ∞ |
These "monsters" turn out to be everywhere: