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Could be implemented directly by adapting 2D noise using e.g. Cube-to-sphere projections for procedural texturing and beyond

By wrapping lattice coordinates, we could easily implement 2D toroidal noise with none of the spectral artifacts suffered by the usual slice-of-higher-dimensional approach. A seamlessly tileable sample area must have width and height equal to the width and projected height of the lattice wrapping unit, and might need to be rotated to align with it.

Interpolating random points on a regular grid using e.g. a quintic polynomial, as documented in e.g. https://iquilezles.org/articles/morenoise/. Could perhaps also use a simplex grid with some sort of barycentric interpolation for better perf at high dimensionalities.

e.g. Perlin noise is gradient noise on a square grid. Might be faster than a simplex grid for low dimensions?