  "Period collapse" refers to any situation where the period of the Ehrhart
function of a polytope is less than the denominator of that polytope. We study
several interesting situations where this occurs, primarily involving
triangles. For example: 1) we determine exactly when the Ehrhart function of a
right triangle with legs on the axes and slant edge with irrational slope is a
polynomial; 2) we find triangles with periods given by any even-index
k-Fibonacci number, and larger denominators; 3) we construct several higher
dimensional examples. Several related issues are also discussed, including
connections with symplectic geometry.
