Simplex noise

Simplex noise is the result of an n-dimensional <a href="/facts/Gradient_noise/1t96uV5b">noise</a> function comparable to <a href="/facts/Perlin_noise/Eg48246G">Perlin noise</a> ("classic" noise) but with fewer <a href="/facts/Isotropy/mc3QEY2x">directional artifacts</a>, in higher dimensions, and a lower computational overhead. <a href="/facts/Ken_Perlin/9a8zKPlR">Ken Perlin</a> 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:

<ul><li>Simplex noise has lower computational complexity and requires fewer multiplications.</li>
<li>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.</li>
<li>Simplex noise has no noticeable directional artifacts (is visually <a href="/facts/Isotropic/mc3QEY2x">isotropic</a>), 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).</li>
<li>Simplex noise has a well-defined and continuous gradient (almost) everywhere that can be computed quite cheaply.</li>
<li>Simplex noise is easy to implement in hardware.</li></ul>
Whereas Perlin noise interpolates between the <a href="/facts/Gradient/TsR7uukj">gradients</a> at the surrounding hypergrid end points (i.e., northeast, northwest, southeast and southwest in 2D), simplex noise divides the space into <a href="/facts/Simplex/0FCfNRN8">simplices</a> (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 <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral</a> 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 <a href="/facts/Tetragonal_disphenoid_honeycomb/Ae30mwnf">tetragonal disphenoid honeycomb</a>.
Simplex noise is useful for computer graphics applications, where noise is usually computed over 2, 3, 4, or possibly 5 dimensions. For higher dimensions, n-spheres around n-<a href="/facts/Simplex/0FCfNRN8">simplex</a> corners are not densely enough packed, reducing the support of the function and making it zero in large portions of space.

Simplex noise open-in-new

Simplex noise