Sierpinski, Koch, Menger - constructing infinite detail
Take a shape, apply one rule, and repeat forever. You get objects with infinite length in a finite space, or infinite surface and no volume at all.
Raise the recursion depth to keep cutting out middle triangles. Then switch to "Chaos Game": random jumps halfway toward a corner build the same triangle.
Each triangle is divided into 3 smaller triangles. The fractal dimension is log(3)/log(2) ≈ 1.585.
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".
Slide the iterations up and watch every edge sprout a bump. Notice the perimeter keeps growing while the area barely changes.
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!
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.
Raise the iterations to fold the paper strip again and again. The curve never crosses itself, yet it fills in a solid region.
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!
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 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.
Drag to rotate and scroll to zoom, then raise the iterations to punch out more cubes. Each step removes 7 of every 27 cubes.
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.
| 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:
Spot a mistake or have an idea?