Rainbows are caused by light refracted and reflected by water droplets in the air. When light enters the water droplet, it splits into its component colors, the same way it does in a prism. Then, the light reflects off the back side of the water droplet. Because the surface of the water droplet is curved, the red hits it at a different angle than yellow, green, etc. This causes the light rays to cross over, producing the familiar rainbow with red on top.
Double rainbows are the light that reflects *twice* inside the water droplet. Because it reflected off a curved surface twice, the light crosses over twice, reversing the order of the colors again.
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