The rainbow shows us all the individual wavelengths of visible light in order, from the longer wavelengths (red) to the shortest that we can see (violet). These are the purest colours – each one has just a single type of light wave associated with it. But reality often isn’t so simple.
I’ve got a rose bush with bright pink roses outside my window, and I’m looking at it as I type this. The problem that my brain has is that the information from my eyes says that there is both red and violet light coming from the same rose.
A mixture of light from different regions of the rainbow is coming from the same place. What my brain needs to do is to interpret that information to distinguish it from objects that are just reflecting red light and those that are just reflecting violet. So it makes the interpretation that the rose is pink, a combination of red and violet.
We need to understand mixtures of the rainbow colours quickly, so our brain gives us a single colour for each mixture. This is what gives us the huge richness of colour in our world when there’s only the simple spectrum of the rainbow as the raw material. It’s all about combinations, and we see each different combination of lightwaves as a different single colour.
The reason that pink isn’t in the rainbow is that the rainbow shows colours in order. There’s nowhere in a rainbow where red and violet overlap, because they’re at opposite ends of the spectrum. So although pink is a real colour, it doesn’t have a place in the rainbow.