3D Cantor Dust — Corner-Recursive Fractal

Recursion

Iteration depth n 3

Gap spacing 1.00×

Color mode

Live Readouts

Cube count 8ⁿ

512

Cube edge L₀/3ⁿ

0.0741

Fractal dim. D

1.8928

Volume fraction

2.60%

How it works

The classic 1D Cantor set removes the middle third of an interval, forever. This is the 3D analogue: split a cube into a 3×3×3 grid of 27 equal sub-cubes and keep only the 8 corner sub-cubes — discard the 6 face-centers, 12 edge-centers and the 1 center cube (19 pieces). Repeat inside each surviving corner cube, n times.

generation 0:  1 cube,  edge L0
generation n:  8ⁿ cubes, edge L0 / 3ⁿ
each axis independently: offset += (±1) · L0 / 3^level

Because each generation keeps 8 of 27 pieces at 1/3 the size, the self-similarity dimension is fixed by construction and never changes with n:

D = ln(8) / ln(3) ≈ 1.8928

That places 3D Cantor dust strictly between a 1D curve (D=1) and a 2D surface (D=2) — it is "dust": totally disconnected (no two kept cubes ever touch after the second generation) yet too dense to be a finite set of points. The total volume kept after n generations is (8/27)ⁿ of the original cube, shown live as it collapses toward zero.

Iteration depth n — rebuilds the cloud from scratch; 8ⁿ instanced cubes are drawn directly on the GPU.
Gap spacing — multiplies the per-level offset only (cube size is untouched), pulling the recursive clusters apart so the corner-keeping rule is visible at every scale.
By octant — colors every cube by which of the 8 first-generation corners it descends from, making the self-similar branching structure legible.
By radius — colors by distance from the center, highlighting the shells of the dust cloud.

Corner-kept cubes (8 of 27 per level)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.