Calculating a symmetrical pattern efficiently starts with a fundamental domain. Pick a base shape defined by a function f(x, y) and constrain it to a minimal region, for example the first quadrant of the coordinate system. The full pattern is then obtained by reflecting this domain across the symmetry axes: P(x, y) = f(|x|, |y|). Because only half the absolute values are computed, you halve the workload. Use vectorized operations or GPU shaders to evaluate f for many points in parallel; this reduces time complexity to O(n) for n pixels. If you need higher‐order symmetry (rotations, reflections), apply the corresponding group operations: P(x, y) = min{ f(R_i(x, y)) | R_i ∈ G } where G is the symmetry group. By precomputing the transformation matrices R_i and reusing them, you keep the algorithm’s cost linear and avoid redundant calculations. This method yields a perfectly symmetrical pattern with minimal computational overhead.

You’re right; discontinuities in f(x, y) will produce artifacts after reflection. One fix is to define f on a closed interval with a continuous extension, for example by using a smoothing kernel or by enforcing boundary conditions that make the function equal at mirrored edges. If discontinuities are unavoidable, apply a post‑processing anti‑aliasing step: blur a small radius around the edges to blend the values. Regarding the min over a large symmetry group, a hash lookup of pre‑computed mask values can reduce the number of comparisons. Store the transformed coordinates in a lookup table and fetch the appropriate value in constant time. This trades a small amount of memory for the computational savings you noted.

You’re correct that preserving continuity in f eliminates the need for post‑processing. Design f as an analytic or piecewise‑smooth function; for instance, use polynomials or spline interpolation that match boundary values exactly. Then the reflected domain inherits that smoothness automatically. As for the hash, it’s only worthwhile if the group size G is small enough that the lookup fits in the L1 cache. For very large G, you’re better off pre‑computing the set of transformed coordinates once and reusing them in a tight loop, keeping the arithmetic cost minimal. The key is to keep the entire pipeline cache‑friendly and avoid unnecessary memory traffic.