Sierpinski Fractal Generator 3D

Self-similar solids

The page builds the Sierpinski tetrahedron (4 copies at ratio 1/2), a Cantor dust (8 cube corners at 1/3) and the Menger sponge (20 of the 27 sub-cubes at 1/3). Each is the attractor of the maps x ← r x + (1 - r) vi, so a random orbit (the chaos game) draws it, and so does the exact recursion that replaces a solid by n copies scaled by r, k times.

Dimension, volume and surface

With n copies of ratio r the similarity dimension is D = ln n / ln(1/r): exactly 2 for the tetrahedron, ln 8/ln 3 ≈ 1.893 for the dust and ln 20/ln 3 ≈ 2.727 for the sponge. After k steps the volume is (n r3)k: (1/2)k for the tetrahedron and (20/27)k for the sponge, both tending to zero. The sponge's surface area (unit starting cube) is 2(20/9)k + 4(8/9)k, which grows without bound. The box-count dimension is the slope of ln N(ε) against ln(1/ε), measured here on a 128 x 128 x 128 grid of the random orbit.

What the 3D scene shows

In the chaos-game view up to 300 000 points are coloured by the vertex that sent them and sit on a turntable. In the exact view a Blender-modelled crystal (or cube) prototype is instanced up to 160 000 times with the same recursion; colours mark the top-level copy or the height. The scoreboard shows the live dimension, the box-count measurement and the counts.

Honest simplifications

The box-count slope is limited by the 64-cell grid (box sizes of 2 to 8 cells) and the finite orbit, so it is accurate to a few per cent only; for ratios that make copies overlap, the similarity dimension exceeds 3 and is no longer the dimension of the set. Rendering uses opaque instanced solids with flat shading, not a physical simulation. Educational model.

Model checks in this page

(1) the box-count dimension of a 400 000-point tetrahedron chaos game is 2; (2) exact tetrahedron recursion at depth 6 gives 4⁶ copies with summed volume (1/2)⁶; (3) the voxel model of the depth-2 Menger sponge has 400 cubes and an exposed-face area equal to 2(20/9)2 + 4(8/9)2.