  We characterise the eigenfunctions of an equilateral triangle billiard in
terms of its nodal domains. The number of nodal domains has a quadratic form in
terms of the quantum numbers, with a non-trivial number-theoretic factor. The
patterns of the eigenfunctions follow a group-theoretic connection in a way
that makes them predictable as one goes from one state to another. Extensive
numerical investigations bring out the distribution functions of the mode
number and signed areas. The statistics of the boundary intersections is also
treated analytically. Finally, the distribution functions of the nodal loop
count and the nodal counting function are shown to contain information about
the classical periodic orbits using the semiclassical trace formula. We believe
that the results belong generically to non-separable systems, thus extending
the previous works which are concentrated on separable and chaotic systems.
