Simplex noise

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Simplex noise is a method for constructing an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with a lower computational overhead, especially in larger dimensions. Ken Perlin designed the algorithm in 2001^[1] to address the limitations of his classic noise function, especially in higher dimensions.

The advantages of simplex noise over Perlin noise:

Simplex noise has a lower computational complexity and requires fewer multiplications.
Simplex noise scales to higher dimensions (4D, 5D and up) with much less computational cost, the complexity is $O (n)$ for $n$ dimensions instead of the $O (2 n)$ of classic Noise.
Simplex noise has no noticeable directional artifacts (is not anisotropic).
Simplex noise has a well-defined and continuous gradient everywhere that can be computed quite cheaply.
Simplex noise is easy to implement in hardware.

Whereas classical noise interpolates between the values from the surrounding hypergrid end points (ie: North South East West in 2D), Simplex noise divides the space into simplexes (ie: n dimensional equilateral triangles) to interpolate between. This reduces the number of data points. While a hypercube in $N$ dimensions has $2 N$ corners, a simplex in $N$ dimensions has only $N + 1$ corners.