Exploiting The Symmetries Of A Problem

Often when faced with a complex puzzle or decision, one can simplify the situation by looking for and exploiting symmetries. Since a short column like this is insufficient to introduce the various types of symmetries and gush over the wonders of a symmetrical universe, I leave it to you to find a source. Here is a good place to start for non-specialists. For our purposes here, we will assume an intuitive sense of symmetry and not worry overly about rigorous definitions.

Recently I had a commission to restore and retouch a photograph that was both damaged and terribly composed, but a precious thing for the owner. The image itself was of four women, but it was an old Polaroid that had not been fixed correctly at the right side. In addition, the women were not centered, but biased toward the left. The result was that the end woman had her left arm unnecessarily amputated. For some reason, she had also closed one eye. Maybe she had been caught winking at the photographer. Another woman had both eyes closed. When I had completed the work, the four women were centered. They all had arms, and they all had both eyes open.

The owner was astounded. “How can you do that? It looks so natural.” She gushed. I just nodded and told her that it had been a difficult project, but I was glad she was happy. In fact, while the details were difficult, the bi-lateral symmetry of the human body helped greatly. I simply lifted the right arm, flipped it over, corrected a bit for lighting, feathered the edges, and fit it in where it belonged. That part was actually easier than constructing a continuation of the background that had also been lost.

A similar trick allowed me to copy the good eye, flip it, and insert it over the closed one. With care, the result is undetectable in normal viewing.

If we were not symmetrical (or nearly symmetrical), retouching photos would be much more difficult.

Sometimes symmetries can enter a problem in peculiar ways, and sometimes seemingly unrelated objects can be related by previously unnoted symmetries. The random distribution of events around a mean value often is symmetrical about the mean. Note that unlike the bi-lateral symmetry in mammals, this type of symmetry doesn’t exist in real space. It is inherent in the data. Plotting the data in a certain way makes the symmetry obvious even though a person simply looking at the individual events probably wouldn’t see it.

Sub-atomic particles are organized by realizing that they can be “changed” from one to another by reflections in non-obvious parameters. In a sense, neutrons and protons are different faces of the same type of particle.

All these examples can be valuable because they demonstrate that finding a symmetry is often equivalent to eliminating a variable. Eliminate enough variables, and your problem becomes easier to solve. Much of the tremendous advance in science, and physics in particular, is due to effective exploitation of natural symmetries. Once you become aware of the possibility of symmetry in data (and assuming you have generalized the concept enough to make it strong), you will see examples in places you never expected.

Since this series is supposed to have occasional puzzles, here is one involving symmetry of a common type - mirror images. We are all familiar with mirrors and the fact that our images have left and right reversed (remember that we are bi-laterally symmetrical), but 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?

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.

[tags]problem solving,puzzle,decision theory,symmetry,symmetries,symmetrical[/tags]