"Movement in Squares"

It's May 7, 2026 afternoon just now. Earlier today, I saw a painting called "Movement in Squares". Here's a rough recreation.

$f\in\{0,1\}^{[0,1)\times [0,1)}$

$[0,1)=\bigcup_{n\in [0,k)_{\mathbb{N}}}[\frac{n}{k},\frac{n+1}{k})$

$\text{Odd}_k=\bigcup_{n\in [0,k)_{\mathbb{N}},n\text{ is odd}} [\frac{n}{k},\frac{n+1}{k})$

$\text{Even}_k=\bigcup_{n\in [0,k)_{\mathbb{N}},n\text{ is even}} [\frac{n}{k},\frac{n+1}{k})=[0,1)-\text{Odd}_k$

checkerboard¶

$f(x,y)=\chi_{\text{Odd}_{12}}(y)\chi_{\text{Even}_{12}}(x)+\chi_{\text{Even}_{12}}(y)\chi_{\text{Odd}_{12}}(x)$

In [17]:

import math
import numpy as np
import matplotlib.pyplot as plt

res = 1000
x = np.linspace(0, 1, res)
y = np.linspace(0, 1, res)
X, Y = np.meshgrid(x, y)

def chi_odd(k, x):
    assert 0 <= x <= 1
    n = math.floor(x * k + 1e-10)
    if n % 2 != 0:
        return 1
    else:
        return 0

def chi_even(k,x):
    return 1 - chi_odd(k,x)

def g(x):
    return 12 + math.floor(\
        12 * np.exp(-25 * np.abs(x - (2/3)))\
        )

def f(x,y):
    return chi_odd(12,y) * chi_even(g(x),x)\
          + chi_even(12,y) * chi_odd(g(x),x)

Z = np.vectorize(f)(X, Y)


plt.figure(figsize=(6, 6))
plt.axis('off')
plt.imshow(
    Z, extent=[0, 1, 0, 1],\
    origin='lower', cmap='Greys_r'\
    )
plt.show()