Function Repository Resource:

KroneckerPoint

Get a quasirandom point based on Kronecker sequences

Contributed by: Ed Pegg Jr

ResourceFunction["KroneckerPoint"][n]

return , where ρ is the plastic constant.

ResourceFunction["KroneckerPoint"][dim,n]

return where x is a real solution of x^dim+1=x+1.

Details

The golden ratio is the real solution for ϕ²=ϕ+1, ϕ≈1.61803398874989….

The plastic ratio is the real solution for ρ³=ρ+1, ρ≈1.32471795724475….

The plastic sequence is a 2D quasirandom sequence (also called a low-discrepancy sequence).

Generate the first point:

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Numeric approximation of the first three points:

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The first three points with ρ (rho) as the plastic constant:

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$ResourceFunction["KroneckerPoint"][Range[3]] /. Root[-1 - # + #^3& , 1, 0] -> \[Rho]$

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Show the structure of the plastic quasirandom sequence:

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The first 500 points as Voronoi cells:

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Show the dimension 3 points:

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The plastic sequence can be folded into a triangle and still be a low-discrepancy sequence:

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