A Random Walk in ℤ⁵

from PIL import Image import numpy as np from typing import Annotated, TypeAlias, Tuple import numpy.typing as npt # %% # five dimensional integer vector Z5 = Annotated[npt.NDArray[np.int_], (5,)] RGB: TypeAlias = Annotated[npt.NDArray[np.uint8], (3,)] RGBTensor: TypeAlias = Annotated[npt.NDArray[np.uint8], (..., ..., 3)] # %% def random_walk_z5( width: int = 64, height: int = 64, strides: Tuple[int, int, int, int, int] = (1, 1, 3, 3, 3), n_iterations: int = 100_000, seed: int | None = None, ) -> Image.Image: """ Visualizes a random walk in ℤ⁵, the five dimensional space of integers, by using two coordinates for position and the other three for color. """ # validate arguments for n in (height, width, n_iterations) + strides: assert isinstance(n, int) assert n > 0 if seed is not None: assert isinstance(seed, int) # initialize the random number generator with the given seed. rng: np.random.Generator = np.random.default_rng(seed) # we'll stay inside this cube by wrapping bounds: Z5 = np.array([height, width, 256, 256, 256], dtype=int) # mutable vector state, starting at a random point vector: Z5 = np.zeros(5, int) vector[:] = rng.integers(0, bounds, dtype=int) # image is initially black image: RGBTensor = np.zeros((height, width, 3), dtype=np.uint8) for iteration in range(n_iterations): # take a step in a random direction along a random axis axis = rng.integers(0, 5) stride = strides[axis] step = rng.choice((-stride, stride)) vector[axis] += step vector %= bounds # interpret vector as (y, x, red, green, blue) and plot one pixel y: int = vector[0] x: int = vector[1] color: RGB = vector[2:].astype(np.uint8) image[y, x] = color return Image.fromarray(image) # %% # see a quick example scaled up x8 random_walk_z5().resize((512, 512), Image.Resampling.NEAREST) # %% # save to disk image = random_walk_z5() image.save("rwz5_64x64.png", format="PNG") # %% # a perfect cube of breathtaking symmetry random_walk_z5(256, 256, strides=(1, 1, 1, 1, 1), seed=0, n_iterations=2**24) # %% # 37 is the most random number: https://www.youtube.com/watch?v=d6iQrh2TK98 # 100 million iterations all but guarentees few black pixels remain, but # takes the better part of an hour to render. random_walk_z5(512, 512, n_iterations=int(1e8), seed=37)