Fractal surfaces from simple arithmetic operations

• A new and general method for constructing surfaces with fractal self-affine properties is presented.
• The method is based on generalized bitwise arithmetic on real numbers.
• All generalized bitwise operators working on a finite alphabet are constructed.
• Fractal surfaces are discovered already in most simple arithmetic operations.
• A roughness exponent is identified for these surfaces, larger values of the exponent leading to coarser surfaces.

Fractal surfaces (‘patchwork quilts’) are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent HH that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.