Rational surfaces vs. irrational surfaces

If the safety factor of a magnetic surface is a rational number, i.e., $q = n / m$, where $m$ and $n$ are integers, then this magnetic surface is called a rational surface, otherwise an irrational surface. It is obvious that a field line on a rational surface with $q = n / m$ closes itself after traveling $n$ toroidal turns and $m$ poloidal turns. An example of a magnetic field line on a rational surface is shown in Fig. 5.

Figure 5: Left: A magnetic field line (blue) on a rational surface with $q = 2.1 = 21 / 10$ (magnetic field is from EAST discharge #59954@3.1s). This field line closes itself after traveling 21 toroidal loops and 10 poloidal loops. Right: The intersecting points of the magnetic field line with the $\phi = 0$ plane when it is traveling toroidally. The sequence of the intersecting points is indicated by the number labels. The 22nd intersecting point coincides with the 1st point and then the intersecting points repeat themselves.