I’m a little pressed for time this week so I thought I would try to provide some fun links.
Steven Strogatz, mathematician and writer, speaks on a radiolab broadcast about an early insight. It was in a high school math class where he says he was being taught how to use graph paper. The teacher gave them each a pendulum, whose length was adjustable. They were asked to choose a length, then swing the pendulum and plot the time it took for it to complete 10 swings. The students would then increase the length and plot the time again. As Strogatz plotted the times, he saw the parabola emerge and was wonderstruck. He tells the story on Radiolab’s The Wonder of Youth which also discusses the surprising order of the periodic table.
Radiolab produced a video of parabolic images inspired by Strogatz’s early experience that can be found here. The visual of the fingers, the clicks of the stop watch, the swinging pendulum, and penciled points on graph paper, help drive the point home. You see the parabola, the hidden image, emerging out of the order given to shapeless and distinct events.
Perhaps it was the direct experience of actually doing the distinct things – swinging, counting seconds, and plotting dots – that brought depth to Strogatz’s observation. He had already learned about parabolas in algebra, and now it seemed like this inanimate object produced one for him. It was then, he tells us, that he grasped the meaning of ‘a law of nature.’ And it was mathematics that could illuminate nature’s invisible structure. The simultaneous use of distinct ideas – ordered magnitudes, time, and defining the location of a point – all of them subjective, cast this light on the pendulum.
Another radiolab broadcast (Desperately Seeking Symmetry) centers around observations of mirror images (of which there are many in mathematics). It turns out that molecules have mirror images and, using our hands as a model, it is often said that there are left and right handed molecules. While not the same, these molecules are mirror images of each other, just like with our hands and so, not surprisingly, this property is often called the handedness of a molecule. All of the matter around us seems to contain a 50/50 split of right and left-handed molecules. Yet all living molecules are left-handed, an interesting mystery, and one perhaps anticipated by Alice when she wondered about the looking glass. She asked “is mirror milk any good to drink.” It likely wouldn’t be if it existed because changing the handedness of a molecule can lead to bad tastes and dangerous toxins (from another broadcast: Mirror Mirror) There are illustrations of the handedness of a molecule at this web site.
The symmetry of a reflected image is how light is vision. Yet, like the plotted times of the pendulum swings, a conceptual re-presentation of it gives us a way to perceive profound structure otherwise unavailable to the senses. Conceptual symmetry exists by virtue of experienced symmetries or (like an object and its mirror image) is one with its physical counterpart. Here’s a nice symmetry video, also from Radiolab.