Solution

{a.} In the figure, the motion along the trajectory is unfolded by mirror reflections into motion between two parallel straight lines. The corresponding points of the trajectories are marked with the same letters. From this figure it follows:

$$v' \approx \frac{x}{2A'B'} v \approx v \Delta \sqrt{2}; \quad \Delta \approx \frac{1}{2} \left[ \tan \left( \frac{\pi}{4} + \Delta \right) - 1 \right] = \frac{k}{2n},$$

where $k$ and $n$ are integers with no common divisor,

$$\tan(\pi/4 + \Delta) - 1 = k/n; \quad h_1 \approx 2a \Delta/k, \quad h_2 = 0.$$

f{b.} It is improbable that $\tan(\pi/4 + \Delta) - 1$ is exactly equal to a simple fraction, for example 0.03, since near this number there can be arbitrarily many other numbers, for instance numbers of the form $0.03 + \sqrt{2}/n$, where $n$ is an integer, which differ from 0.03 by an arbitrarily small amount. These numbers are called irrational, and in mathematics it is proven that the set of these numbers is more powerful than the set of simple fractions. If the number is irrational, then the trajectory is not closed.

{c.} $P=V/a^3$, since trajectory is not closed.