Some of the details, sadly including her name, have faded but others are brilliantly clear. On that day my eighth grade art teacher gave me something profound. It was fascinating at the time, but years later I began to realize how profoundly beautiful it was.
Our desks were bare save a straight edge, pencil and a sheet of paper. She told us to stand facing the window. "Close your eyes" We were to imagine being on the prairie looking a our feet. I opened an eye and saw most of us were looking at the floor. We were to imagine the line between our eyes and the point on the ground. Then we were to mentally draw a line to a point on the ground some distance away. "Don't worry about the numbers, but think about the angle the line makes with the ground..." Then we looked ten feet out, twenty, fifty, a hundred feet, the length of a football field. "What are the angles doing? How fast are they changing?" "a mile, ten miles, a hundred miles.. "how are they changing now?" Now off as far as we could imagine vanishing off into space. The angle was changing so little as we went further away. "Open your eyes" It may have been an eighth grade art class, but I felt myself gasp. We were looking straight ahead to the horizon and had touched infinity.
Next she showed us a photograph of railroad tracks receding into the distance. She asked us to close our eyes and scan along each of the rails and notice what happens. Of course they meet at infinity. But parallel lines can't do that. What if there were tracks a hundred feet to our left? She gave us our first lesson in projective geometry and none of us realized it.
We sat down the the straight edge and she showed us how to construct a grid of equally sized squares without resorting to numbers It was just drawing straight lines to the horizon and intersecting them with straight lines from other points. Very neat. That consumed the period. I couldn't wait for the next class.,
A few days later we met again. Now we were to imagine a semi-transparent piece of paper between us and perpendicular to the ground. We were told to think about where the lines to the "vanishing point" - the artist's infinity would intersect the paper. Then she took out a cheap print of a Renaissance painting, I remember it as the Giving of the Keys to Saint Peter by Perugino, and drew the perspective lines on it. She told us the great artists of the Renaissance had been struggling to create the most realistic painting possible and had developed perspective to bring some of the three dimensionality of the world onto a flat two dimensional sheet of paper.
My mind was blown. I started seeing the world differently. I knew that it had three dimensions and was Euclidian, but when you looked at anything a bit in the distance the two dimensional approximation of perspective was apparent. This was a powerful tool. It explained why movies and photography looked realistic. It started me thinking about what a lens does.
Years later I learned more about projective geometry - an area of math pioneered by the Renaissance artists. I can't appreciate it to the depth of an artist, but it is a good metaphor for other things.
Many concepts in physics seem strange or even bizarre to non-specialists. Many examples and popularizations aren't very good and can lead to even more confusion. The trick is to find metaphors and analogies that are good enough and easy to understand. I think projective geometry - perspective to an artist - is rich ground for tackling some of the basic ideas.
I'll mention four that are central to physics and, it turns out, to art. Relativity, invariance, symmetry and complementarity. Forget about any associations you might have heard about those words and physics (especially relativity). This is more of a fundamental level of how to look at Nature in general and learn to ask the right questions. It applies to art and probably many other fields. So taking them one at a time:
Relativity You can paint an object from many perspectives. All of them represent the same information about the object, it's just they're encoded differently. The same subject can be faithfully represented without any loss. (for physics think about simultaneity)
Invariance Bits of the subject can look different when you change your perspective, but there are some that never change. Consider straight lines on the subject that meet at a point. The fact that they intersect at a point doesn't change in different perspectives,. Those features that don't change in all possible representations are invariants.
Symmetry Now think of moving the subject rather than the artist - rotating a subject on a turntable for example. Arbitrary rotations may appear different, but the the sum of all views from all possible perspectives doesn't change. The sum of all of the perspectives is the projective description. Here rotation is a symmetry of the projective description.
Another way of looking at this is to consider a few special cases rather than the totality. Consider rotating a circle looking at it from one perspective - standing over it for example. No matter what rotation you make - 2°, 180°, 42°, 234234234234234° - it looks exactly the same. A circle is symmetric under rotation. Now think of an equilateral triangle. It only looks the same if you rotate it 120° - it has a three fold symmetry under rotation, a square has a four fold symmetry that pops up ever 90° of rotation Symmetry operations can be thought of as change without change.
Complementarity This one is related to the story telling from a few posts ago. There can be many views of the subject that are equally good, but when you describe the subject you usually have to pick one - hopefully one that makes sense in the context of how you describe it. This seems almost simple at first, but consider trying to take a picture of something very small - say an electron. The electron has a position and some momentum. When you make your picture you have to shine some light on it. The light its own energy and momentum and, in the process of interacting with the electron, manages to muck things up. If you're really clever you can set up your measurement to focus on one aspect of the electron - say it's position. Now the interaction will disturb the electron's momentum a great deal. You can also try to focus on it's momentum, but that will disturb the electron's position - ah the uncertainty principle. The point is you decide which description you want and the other descriptions are rendered invalid. Picasso would not approve.
The models and math can render physics incomprehensible to non-specialists, but these principals are often involved at a deep level. It is nice to think about them in terms of something much more tangible - perspective in art.
And if you wonder why I keep arguing for education in the arts...