Checkerboard Periodic Patterns

📐 Definition

Checkerboard patterns arise from products of orthogonal sinusoids or sign-flipped lattices, yielding alternating quadrants across a grid.

A common construction is the signed product of two orthogonal sinusoids:

c(x,y) = \operatorname{sgn}\!\big(\sin(k_x x)\,\sin(k_y y)\big)

Domain and Codomain

Defined on two-dimensional spatial domains or grids. Outputs alternate between two levels (e.g. $\{-1,1\}$ ) across tiles.

⚙️ Key Properties

Fundamental periodicity follows the lattice spacings in each dimension.
Dominated by a small set of spectral peaks at the lattice frequencies.
Symmetry makes them useful for isolating cross-derivative behavior.

🎯 Special Cases and Limits

Periods: $2\pi/k_x$ in $x$ and $2\pi/k_y$ in $y$ .
$k_x=k_y$ yields isotropic tiling.
Smooth precursor: replacing $\operatorname{sgn}$ by the raw product $\sin(k_x x)\sin(k_y y)$ gives a smooth sinusoidal checkerboard.

Square waves are the one-dimensional analogue. Products of sinusoids provide the smooth precursor, and Fourier series represent the pattern as a structured sum of harmonics.

Usage in Oakfield

Oakfield provides checkerboard patterns as a built-in stimulus:

stimulus_checkerboard operator writes structured checkerboard-like forcing with configurable periods and phase parameters, for both real and complex targets.
Useful for stressing spatial operators and for generating strongly structured initial conditions without needing external assets.

Historical Foundations

📜 Periodic Test Patterns

Checkerboard-like patterns are common diagnostic inputs in numerical PDEs and signal processing because they isolate directional and mixed responses and have simple spectral structure.

🌍 Modern Perspective

They remain a standard stress test for discretizations, filtering, and anisotropic operators.

📚 References

Strikwerda, Finite Difference Schemes and Partial Differential Equations
Bracewell, The Fourier Transform and Its Applications