Eigenfunctions of integrable planar billiards are studied — in particular,

the number of nodal domains, ν of the eigenfunctions

with Dirichlet boundary conditions are considered. The billiards

for which the time-independent Schrö dinger equation (Helmholtz

equation) is separable admit trivial expressions for the number

of domains. Here, we discover that for all separable and nonseparable

integrable billiards, ν satisfies certain difference equations.

This has been possible because the eigenfunctions can be

classified in families labelled by the same value ofm mod kn, given

a particular k, for a set of quantum numbers, m, n. Further, we observe

that the patterns in a family are similar and the algebraic representation

of the geometrical nodal patterns is found. Instances of

this representation are explained in detail to understand the beauty

of the patterns. This paper therefore presents a mathematical connection