  Two discretizations, linear and nonlinear, of basic notions of the complex
analysis are considered. The underlying lattice is an arbitrary quasicrystallic
rhombic tiling of a plane. The linear theory is based on the discrete
Cauchy-Riemann equations, the nonlinear one is based on the notion of circle
patterns. We clarify the role of the rhombic condition in both theories: under
this condition the corresponding equations are integrable (in the sense of 3D
consistency, which yields also the existense of zero curvature representations,
B"acklund transformations etc.). We demonstrate that in some precise sense the
linear theory is a linearization of the nonlinear one: the tangent space to a
set of integrable circle patterns at an isoradial point consists of discrete
holomorphic functions which take real (imaginary) values on two sublattices. We
extend solutions of the basic equations of both theories to Z^d, where d is the
number of different edge slopes of the quasicrystallic tiling. In the linear
theory, we give an integral representation of an arbitrary discrete holomorphic
function, thus proving the density of discrete exponential functions. We
introduce the d-dimensional discrete logarithmic function which is a
generalization of Kenyon's discrete Green's function, and uncover several new
properties of this function. We prove that it is an isomonodromic solution of
the discrete Cauchy-Riemann equations, and that it is a tangent vector to the
space of integrable circle patterns along the family of isomonodromic discrete
power functions.
