I understand the geometry of double rainbows. They must be rare because I've only seen a third rainbow once.
Double rainbows are fairly common.
In the previous answer , we saw that the primary rainbow appears opposite the sun at an angular distance of about 42 degrees from the antisolar point. The primary bow is the result of a refraction as a ray of sunlight enters the drop, one internal reflection off the back of the drop, and a second refraction as the ray exits the drop. The geometry for the secondary bow is similar except that there are two internal reflections inside the drop. This geometry results in the appearance of the secondary rainbow at an angular distance of about 51 degrees, outside the primary bow. The secondary rainbow is much less brilliant than the primary because each time a ray crosses the surface of a drop from either side, part of it is reflected and part is refracted.