The Fractal Dimension of Sets Derived from Complex Bases

Source abstract

Abstract For each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λ n /log(n 2 + 1), where λ n is the positive root of λ 3 - (2 n - 1) λ 2 - ( n - 1) 2 λ - ( n 2 + 1).