3D Menger sponge level-3 wireframe

Math essence

Per level, divide each cube into 3x3x3 = 27 equal sub-cubes; keep sub-cube (i, j, k) iff at most one of (i, j, k) is 1. This leaves 8 corner sub-cubes. After N levels there are 20^N kept cubes, each of side length 2/3^N. Here N = 3, so 8000 cubes. Project each cube centre by a 3D pitch/yaw rotation, sort by projected depth, rasterise each cube as 12 line segments using np.linspace on (x, y) endpoints, and blend the wire colour by (1 - depth) to dim the far side.

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/fractals_3d_webgl/MengerSponge3DLevel3/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


def _is_menger_kept(i: int, j: int, k: int) -> bool:
    """Return True if the (i, j, k) sub-cube (each in {0, 1, 2}) of a
    3x3x3 macro-cell is *kept* in the Menger sponge construction.

    Per Wikipedia (Menger sponge): keep the cube iff at most one of
    (i, j, k) is 1.  Equivalently, drop the centre, the six
    face-centres, and the twelve edge-centres; keep the eight corners.
    """
    ones = (i == 1) + (j == 1) + (k == 1)
    return ones <= 1


def _generate_menger_cubes(
    level: int, kept: bool = True
) -> tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
    """Recursively generate the kept (``kept=True``) or removed
    (``kept=False``) sub-cubes of a level-``level`` Menger sponge
    centred at the origin with side length 2.

    Returns (xs, ys, zs, sizes) for every kept cube, where each array
    has shape (N,).  Coordinates are cube centres; ``sizes`` is the
    side length of each cube.  N is 20^level (kept) or 7^level
    (removed).
    """
    cubes: list[tuple[float, float, float, float]] = [(0.0, 0.0, 0.0, 2.0)]
    for _ in range(level):
        new: list[tuple[float, float, float, float]] = []
        for cx, cy, cz, side in cubes:
            child = side / 3.0
            for i in range(3):
                for j in range(3):
                    for k in range(3):
                        is_kept = _is_menger_kept(i, j, k)
                        if (kept and is_kept) or (not kept and not is_kept):
                            ox = (i - 1) * child
                            oy = (j - 1) * child
                            oz = (k - 1) * child
                            new.append((cx + ox, cy + oy, cz + oz, child))
        cubes = new
    arr = np.array(cubes, dtype=np.float32)
    return arr[:, 0], arr[:, 1], arr[:, 2], arr[:, 3]


def _project_cube_wireframe(
    px: np.ndarray,
    py: np.ndarray,
    d: int,
    scale: float,
    ox: float,
    oy: float,
    color: np.ndarray,
    frame: np.ndarray,
    line_alpha: float = 0.85,