Simplex noise

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Simplex noise is the result of an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with fewer directional artifacts, in higher dimensions, and a lower computational overhead. Ken Perlin designed the algorithm in 2001 to address the limitations of his classic noise function, especially in higher dimensions. The advantages of simplex noise over Perlin noise: Simplex noise has lower computational complexity and requires fewer multiplications. Simplex noise scales to higher dimensions (4D, 5D) with much less computational cost: the complexity is O ( n 2 ) {\displaystyle O(n^{2})} for n {\displaystyle n} dimensions instead of the O ( n 2 n ) {\displaystyle O(n\,2^{n})} of classic noise. Simplex noise has no noticeable directional artifacts (is visually isotropic), though noise generated for different dimensions is visually distinct (e.g. 2D noise has a different look than 2D slices of 3D noise, and it looks increasingly worse for higher dimensions). Simplex noise has a well-defined and continuous gradient (almost) everywhere that can be computed quite cheaply. Simplex noise is easy to implement in hardware. Whereas Perlin noise interpolates between the gradients at the surrounding hypergrid end points (i.e., northeast, northwest, southeast and southwest in 2D), simplex noise divides the space into simplices (i.e., n {\displaystyle n} -dimensional triangles). This reduces the number of data points. While a hypercube in n {\displaystyle n} dimensions has 2 n {\displaystyle 2^{n}} corners, a simplex in n {\displaystyle n} dimensions has only n + 1 {\displaystyle n+1} corners. The triangles are equilateral in 2D, but in higher dimensions the simplices are only approximately regular. For example, the tiling in the 3D case of the function is an orientation of the tetragonal disphenoid honeycomb. Simplex noise is useful for computer graphics applications, where noise is usually computed over 2, 3, 4, or possibly 5 dimensions. For...