Simplex noise

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Simplex noise is a type of gradient noise like Perlin noise, but designed to have fewer directional artifacts and to run faster in higher dimensions. It was created by Ken Perlin in 2001 to fix limitations of the classic noise in more dimensions.

The key idea is to divide space into simplexes (in 2D, triangles; in 3D, tetrahedra; in higher dimensions, shapes with n+1 corners) instead of using a hypercube’s corners. This means fewer corners to blend and smoother results.

Commonly used in 2D, 3D, or 4D (and sometimes higher), simplex noise follows four basic steps: skew the input coordinates so they lie on a simplex grid; determine which simplex the point is in; assign a gradient direction at each simplex vertex using a hash; and sum the contributions from the vertices with a smooth, distance-based kernel.

Each vertex has a pseudo-random gradient, and its contribution depends on how far the vertex is from the input point. A typical falloff uses an r^2 value; 0.5 gives smooth results, while 0.6 can improve visual quality in some cases. The final value is the sum of the contributions from the simplex’s n+1 vertices.

Note: earlier work on 3D+ texture synthesis tied to patents, which expired in January 2022.