This is best illustrated using the Mercator projection , which is in fact why it is so good for marine navigation, since courses of constant compass bearing (loxodromes) appear as straight lines.

Another interesting property of the loxodrome is that for non-zero bearings measured clockwise from North, it circles a pole infinitely many times.

The y-coordinate of the Mercator projection tends to infinity as you get closer to the poles; the map above is clipped at latitudes ±85°.

The loxodrome appears as a straight line on the Mercator projection, so it’s not hard to see that it would have to cross the sides infinitely many times before reaching either pole.

Despite circling a pole infinitely many times, a loxodrome has finite length. The pole-to-pole length of a spherical loxodrome is the length of the meridian divided by the cosine of the loxodrome’s bearing measured clockwise from North.