24 Feb Math prove that the world is more wonderful than imagination…. (IAN STEWART)
And now the most important; you asked me whether you would have to give up your sense of beauty to study mathematics, whether everything would become just numbers and equations to you, laws and formulas. Rest assured, Meg, I don’t blame you for asking this, since it’s unfortunately a very common idea, but it couldn’t be more wrong. It’s exactly the opposite of the truth.
What math does for me is this: It makes me aware of the world I inhabit in an entirely new way. It opens my eyes to nature’s laws and patterns. It offers an entirely new experience of beauty.
When I see a rainbow, for instance, I don’t just see a bright, multicolored arc across the sky. I don’t just see the effect of raindrops on sunlight, splitting the white light from the sun into its constituent colors. I still find rainbows beautiful and inspiring, but I appreciate that there’s more to a rainbow than mere refraction of light. The colors are, so to speak, a red (and blue and green) herring. What require explanation are the shape and the brightness. Why is a rainbow a circular arc? Why is the light from the rainbow so bright?
You may not have thought about those questions. You know that a rainbow appears when sunlight is refracted by tiny droplets of water, with each color of light being diverted through a slightly different angle and bouncing back from the raindrops to meet the observing eye. But if that’s all there is to a rainbow, why don’t the billions of differently colored light rays from billions of raindrops just overlap and smear out?
The answer lies in the geometry of the rainbow. When the light bounces around inside a raindrop, the spherical shape of the drop causes the light to emerge with a very strong focus along a particular direction. Each drop in effect emits a bright cone of light, or, rather, each color of light forms its own cone, and the angle of the cone is slightly different for each color. When we look at a rainbow, our eyes detect only the cones that come from raindrops lying in particular directions, and for each color, those directions form a circle in the sky. So we see lots of concentric circles, one for each color.
The rainbow that you see and the rainbow that I see are created by different raindrops.
Our eyes are in different places, so we detect different cones, produced by different drops.
Rainbows are personal.
Some people think that this kind of understanding “spoils” the emotional experience. I think this is rubbish. It demonstrates a depressing sort of aesthetic complacency. People who make such statements often like to pretend they are poetic types, wide open to the world’s wonders, but in fact they suffer from a serious lack of curiosity: they refuse to believe the world is more wonderful than their own limited imaginations. Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this. The ability to understand is one of the most important differences between human beings and other animals, and we should value it. Lots of animals emote, but as far as we know, only humans think rationally. I’d say that my understanding of the geometry of the rainbow adds a new dimension to its beauty. It doesn’t take anything away from the emotional experience.
Letters to a Young Mathematician