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30.0 Images, quantitatively by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

Breakfast Table, by Willem Clasz. de Heda, 17th century. The painting shows a variety of images, some of them distorted, resulting both from reflection and from refraction (ch. 31).

It sounds a bit odd when a scientist refers to a theory as “beautiful,” but to those in the know it makes perfect sense. One mark of a beautiful theory is that it surprises us by being simple. The mathematical theory of lenses and curved mirrors gives us just such a surprise. We expect the subject to be complex because there are so many cases: a converging mirror forming a real image, a diverging lens that makes a virtual image, and so on for a total of six possibilities. If we want to predict the location of the images in all these situations, we might expect to need six different equations, and six more for predicting magnifications. Instead, it turns out that we can use just one equation for the location of the image and one equation for its magnification, and these two equations work in all the different cases with no changes except for plus and minus signs. This is the kind of thing the physicist Eugene Wigner referred to as “the unreasonable effectiveness of mathematics.” Sometimes we can find a deeper reason for this kind of unexpected simplicity, but sometimes it almost seems as if God went out of Her way to make the secrets of universe susceptible to attack by the human thought-tool called math.

30.0 Images, quantitatively by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.