Sierpinski tetrahedron (tetrix) 3D chaos game

Math essence

For a tetrahedron with vertices p_0..p_3, iterate x_{n+1} = (x_n + p_{c(n)}) / 2 where c(n) is uniform in {0,1,2,3}. After a burn-in, the orbit is a 3D point cloud on the tetrix. Project with a 3D rotation by pitch/yaw, splat each point into a small radius on the canvas with additive weight, and let a per-step decay blend frames so the orbit camera animation accumulates an organic soft glow.

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/fractals_3d_webgl/SierpinskiTetrahedron3D/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


def _tetra_chaos_iterate(
    n_pts: int, n_steps: int, seed: int | None
) -> tuple[np.ndarray, np.ndarray]:
    """Chaos-game iteration in 3D for the Sierpinski tetrahedron.

    Four regular-tetrahedron vertices A, B, C, D.  At each step pick
    one vertex uniformly at random and replace the current point by
    (current + vertex) / 2.  After a small burn-in the trajectory
    becomes dense in the tetrix.  Returns (xs, ys, zs) of shape
    (n_steps, n_pts).
    """
    rng = np.random.default_rng(seed)
    # Regular tetrahedron with centroid at origin
    s = 1.0
    verts = np.array(
        [
            [s, s, s],
            [s, -s, -s],
            [-s, s, -s],
            [-s, -s, s],
        ],
        dtype=np.float32,
    )
    x = rng.uniform(-0.1, 0.1, size=n_pts).astype(np.float32)
    y = rng.uniform(-0.1, 0.1, size=n_pts).astype(np.float32)
    z = rng.uniform(-0.1, 0.1, size=n_pts).astype(np.float32)
    xs = np.empty((n_steps, n_pts), dtype=np.float32)
    ys = np.empty((n_steps, n_pts), dtype=np.float32)
    zs = np.empty((n_steps, n_pts), dtype=np.float32)
    burn = max(8, n_steps // 50)
    for i in range(n_steps):
        which = rng.integers(0, 4, size=n_pts)
        v = verts[which]
        x = (x + v[:, 0]) * 0.5
        y = (y + v[:, 1]) * 0.5
        z = (z + v[:, 2]) * 0.5
        if i >= burn:
            xs[i - burn] = x
            ys[i - burn] = y
            zs[i - burn] = z
    return xs, ys, zs


class SierpinskiTetrahedron3D:
    """Sierpinski tetrahedron (tetrix) — Waclaw Sierpinski (1915).

    The 3D analogue of the Sierpinski triangle: start with a regular
    tetrahedron, replace it with four half-size copies at its corners,
    and recurse.  In the limit the volume goes to zero while the
    surface area stays constant; the Hausdorff dimension is
    log(4)/log(2) = 2.

    This implementation uses the *chaos game*: spawn many random