It is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

We describe the period structure of the optimal continued fraction expansion of a quadratic surd, in terms of the period of its nearest square continued fraction expansion. The analysis results in a faster algorithm for determining the optimal continued fraction expansion of a quadratic surd.

We analyze the importance of the frequency of decision making for macroeconomic dynamics, in the context of a simple, well-known business cycle model with balanced budget rules. We explain how the frequency of decision making (period length) and the measurement unit of time (calibration frequency) differ and examine how local stability is affected by changes in the period length. We find that as the period grows longer, indeterminacy occurs less often. This may have significant quantitative implications: for the model at hand, there is a wide range of economically relevant labor tax rates (from 30% to 38%) for which the continuous-time model gives indeterminacy, whereas the discrete-time model has determinate dynamics.