Walk as we count integers, turning right when it's prime.

primes_walk.jl

using Primes
using StatsBase
using InteractiveViz
using GLMakie

function run(L, turns, zoom)
    function xy_to_grid(x, y)
        return round(Integer, L/2 + x*zoom), round(Integer, L/2 + y*zoom)
    end

    img = zeros(Int32, L, L)

    x = 0
    y = 0
    dx = 1
    dy = 0
    n = 1
    for p in turns
        while n < p
            x += dx
            y += dy
            i, j = xy_to_grid(x, y)
            if i >= 1 && j >= 1 && i <= L && j <= L
                img[i, j] += 1
            end
            n += 1
        end
        dx, dy = dy, -dx
    end

    return img
end

N = 1_000_000_000
T = primes(N)
#T = filter(x -> rand() < 1/log(x), 1:N)
@time img = run(2_000, T, 1//150)
@show entropy(img / sum(img)) / log(2)
iheatmap(log.(1 .+ img))