Veech’s original examples of translation surfaces Vq enjoying what McMullen has dubbed “optimal dynamics ” arise from appropriately gluing sides of two copies of the regular q-gon, with q ≥ 3. We show that every Vq whose trace field is of degree greater than 2 has non-periodic directions of vanishing SAF-invariant. (Calta-Smillie have shown that under appropriate normalization, the set of slopes of directions where this invariant vanishes agrees with the trace field.) Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant. In an appendix, we prove various elementary results on the containment of trigonometric fields. 1.