New examples of period collapse

"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.