In the last column, we considered how symmetries can be used to reduce the difficulty of solving problems. In general, for each symmetry, one parameter can be eliminated. This holds from organizing subatomic particles to the structure of the universe. Even when symmetries are not exact, close approximations can help organize data. A common example is the bilateral symmetry of human being. We routinely assume people are the same if reflected left to right, but we know that is only an approximation. Our hearts tend to be bigger on the left side, and most people have slight asymmetries such as one ear being slightly lower than the other. But overall, most of us are the same left to right.

The expectation of bilateral symmetry in humans is so strong that serious research indicates that the desirability of another person as a possible mate is partially determined by how symmetrical that person is. One possible explanation is that a symmetrical person has all limbs in good working order and therefore can be a better producer and worker. While that seems to be a stretch, control of symmetry and the deliberate introduction of slight variations is important in human grooming. Most people part their hair on one side and introduce a bias to one side. This is a deliberate breaking of symmetry for effect.

Which brings us to mirrors. In the last column I challenged readers to consider the operation of mirrors from the point of view of symmetry: “…why are left and right reversed but not front to back or up to down? Is there one answer for both front to back and top to down or are these different types of functions? Can you configure a mirror or set of mirrors to reflect a non-reversed image? Why doesn’t a simple periscope reverse images?”

There are many ways to answer these questions, but one way is to consider that normal space has three dimensions and that a mirror preserves two dimensions and reverses one of them. When a mirror is placed in the normal fashion, it preserves the vertical dimension. But if a mirror is placed on the ceiling or floor, the images are upside down because in that orientation it reverses the up and down axis. In the normal orientation, a mirror cannot “see” behind you, so the image cannot reverse fore and aft.

To see yourself as others do, simply place two mirrors at right angles so that the image you see has bounced off two surfaces. Each bounce reverses one axis; the second reversal restores the original orientation. With such an arrangement, you can extend your right hand in greeting and the image will extend its right hand to shake with you. The effect can be disconcerting. What happens if you rotate this configuration so that the intersection of the mirrors is horizontal instead of vertical?

Once we understand the transformations that a mirror does from this point of view, the operation of a periscope is more obvious. Similarly the reason that we need two mirrors to see the back of our heads is more obvious.

What about other types of symmetry applied to mirrors? A concave reflecting mirror is radically symmetrical. Can that observation help predict what we will see in one? Three flat mirrors oriented in a corner mutually perpendicular reflect light back the same way it came – why? This configuration is often called a retro-reflector and is useful for reflecting radar waves. However, the return ray is spatially offset from the incoming ray. Assume you can build a similar mirror that exactly returns light in the direction it came from but with no spatial offset. That is, it is a perfect mirror. What would you see if you looked into such a thing?

The study of symmetries can be seductive. We all are familiar with simple mirrors, but by looking at their operation in detail and considering their function with the real-world symmetries in mind, we can learn new things. In the same way, various types of mirrors can be held up to the apparently random data of life and sometimes life can become simpler. While you might dismiss the discussion of mirrors as obvious, the mental set of analyzing mirror images using symmetry can be generalized to more difficult problems.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.