It's true rainbows appear to form perfectly rounded arches. But in reality, rainbows form full circles. Then why don't we see circles? When we're standing on the ground, we can only see light that's reflected by raindrops above the horizon. Thus, we can't see a rainbow's lower, hidden half. There is one way you may be able to see a full-circle rainbow, though. If you're a pilot or passenger in an airplane or helicopter – and thus can see below the horizon – you might see a rainbow as a full circle. Sometimes people climbing tall mountains can view circular rainbows as well [sources: Lewin, National Geographic].
Since a rainbow is a circle you'll never reach the end or the bottom. Rainbows seem to move when you do, because the light that forms the bow is always at a specific distance and angle from you [source: Howard]. Remember we said earlier that rainbows were optical illusions? That's why you'll never find your pot of gold, alas.