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 Pi

Plots of random walks based on the digits (168 graphics)

1D walks

1D random walks based on the digits of in various bases. If is even, a step to the right is taken, if is odd, a step to the left is taken.

1D random walks based on the digits of in various bases. If is even, a step to the right is taken, if is odd, a step to the left is taken. The left graphic shows the average distance of the walker from the origin, averaged over the first 100 even bases . The right graphic shows the binned scaled deviation of the individual walks (for a fixed basis ) from the average.

2D walks

2D random walks based on the digits of in various bases . For each , a step is taken in direction . For clarity, the th step is shown with a ‐value .

2D random walks based on the digits of in various bases . For each , a step is taken in direction . For clarity, the th step is shown with a ‐value .

2D random walks based on the digits of in various bases . For each , a step is taken in direction .

2D random walks based on the digits of in various bases . For each a step is taken in direction .

Platonic walks

3D random walks based on the digits of in various bases . For each , a step of size 1 is taken in direction of the th vertex of a Platonic solid.

2D Levy-type walks

2D Lévy‐type random walks based on the digits of in various bases and numbers of directions . For each , a step of length is taken in direction .

Higher-dimensional walks

2D projections of a D random walk based on the digits of in various bases . For each , a step is taken in direction (the is on position ). The resulting visited lattice points are projected into a 2D plane spanned by the vectors and .