  In this paper, we continue the study of eigenfunctions on triangles initiated
by the first author in \cite{Chr-tri} and \cite{Chr-simp}. The Neumann data of
Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being
equidistributed on each side. The proof of this result is remarkably simple,
using only the radial vector field and a Rellich type integrations by parts.
The equidistribution law, including on higher dimensional simplices, agrees
with what Quantum Ergodic Restriction would predict. However, distribution of
the Neumann data on subsets of a side is not well understood, and elementary
methods do not appear to give enough information to draw conclusions.
  In the present note, we first show that an ``obvious'' conjecture fails even
for the simplest right isosceles triangle using only Fourier series. We then
use a result of Marklof-Rudnick \cite{Marklof-Rudnick} in which the authors
show an interior {\it spatial} equidistribution law for a density-one
subsequence of eigenfunctions to give an estimate on energy distribution of
eigenfunctions on the interior. Finally we present some numerical computations
suggesting the behaviour of eigenfunctions on almost isosceles triangles is
quite complicated.

Analysis of PDEs;Spectral Theory