Why 42°
Every rainbow is a circle 42° around the shadow of your head. Inside one drop, two hundred rays per colour refract, bounce and leave, and the ones that leave crowd together at minimum deviation: 42.4° for red, 40.6° for violet. The sky beside it is built from that crowding, pixel by pixel — red outside, a fainter reversed bow at 51°, a dark band between them where no ray can land, and a bow that sinks as the sun rises and is gone once the sun is higher than 42°.
A rainbow is not a thing in the sky. It is a direction. Stand with the sun at your back and every raindrop that sits 42° from the shadow of your head sends some of its light straight at your eye, and the drops at that angle form a circle. The person beside you sees a different set of drops, so they see their own bow. You can never reach the end of one, because it moves with you.
Why 42°? René Descartes worked it out in 1637 by tracing thousands of rays through a glass globe of water, by hand. A ray that enters a drop refracts, reflects once off the back, and refracts out again, and how far it turns depends on where it hit. Most rays scatter widely, but near one particular impact point the turning angle stops changing: it reaches a minimum, and the rays on either side leave almost parallel. That crowding is the bow. On the left of the stage it happens in front of you: two hundred rays per colour go in evenly spaced, and come out bunched along one line.
Newton supplied the colours. Water bends violet light a little more than red, so violet's crowding angle is 40.6° and red's is 42.4°, which puts red on the outside. Two reflections inside the drop make a second, fainter bow at about 51° with the order reversed, and between the two no ray from either path can arrive at all. The sky there is darker than anywhere around it: Alexander's dark band, named for Alexander of Aphrodisias, who described it around 200 AD.
The same geometry explains the bow in a garden hose, which can appear below the horizon because the spray is below your eye, and the full circle seen from aircraft. Shrink the drops to fog and a different physics takes over: light is a wave, each tiny drop spreads every colour over several degrees, the colours overlap, and the bow turns broad and almost white. Sailors call it a sea-dog; hikers call it a fogbow.
Drop needles on floorboards, throw darts, pick random numbers, slam two blocks into a wall — π turns up every time. Only one of them is any good at it.