If you've been following the new Lord of the Rings series, The Rings of Power, you may have noticed an interesting geometry highlight: a magic sequence with sand turning into different patterns on the rhythm of the opening song.
Some of the shapes have a special meaning in the saga, but believe it or not, the shapes are not restricted to the Lord of the Rings universe -- the ‘raw’ shapes can actually show up in the real world as well, and there's no magic here, just science. They're called Chladni figures.
Think about being on a swing for a moment. As you go back and forth and arc through the air, the swing moves smoothly. If you want to swing harder, you have to move your body with the rhythm of the swing. From a physical standpoint, the swing is an oscillator, and when you're swinging harder, you're increasing its amplitude.
Many physical systems have a frequency at which they oscillate easily -- this is called the natural frequency, and indicates a frequency at which a system tends to oscillate even in the absence of any driving force. Some systems (like guitar strings, for instance) have more than one natural frequency, which is why a single guitar string produces overtones (the same note at different octaves) when played.
If you force an object to vibrate, the vibration will happen at the frequency of the applied force. But if this frequency happens to be equal to the natural frequency, then the vibrations' amplitude increases manyfold, a phenomenon known as resonance. Going back to our swing example, if someone is trying to push you harder on the swing, they need to push at the right frequency -- if they're pushing on another frequency, sometimes they are pushing forward when the swing is still going backward, which would slow it down. Funny enough, the swing's natural frequency doesn't depend on the mass of the person on the swing, only on the gravity and the length of the chain holding the swing.
But how does this tie in with the Chladni figures?
How Chladni plates work
Natural frequency and resonance play an important role in the study of sound. Sound is essentially a vibration that propagates as an acoustic wave. It has a frequency, it can resonate, and it can also interact with other physical objects.
In the 1780s, Ernst Chladni, a German mathematician and physicist, found a way to produce visual patterns from sound waves. He used sand on a metallic plate which was vibrated by running a violin bow on its edge. As the plate vibrates at various frequencyes, the interference and resonance form complex but symmetrical patterns over its surface and the sand would reproduce these patterns.
Basically, when the plate resonates with the vibration from the bow, it becomes divided into regions that vibrate in opposite directions -- regions bounded by areas where no vibration occurs. So Chladni Patterns are formed when powder or sand settles at the nodal lines of a plate which is being driven at one of its natural frequencies.
Another way of looking at this is that Chladni patterns show the geometry of the different types of vibration of the metal plate, and the fine powder (be it sand or something else) is just a way of visualizing this physical process. When the plate resonates, the motion becomes large over most of the surface and this causes the sand to bounce around and only stay stable at or near the node.
If you put sand on the plates, the sand will move to these still, stable surface. In the video below, you can see how it was done back then.
Chladni traveled throughout Europe with his instruments giving demonstrations, and to this day, the phenomenon is striking.
However, modern Chladni plates are electronic, and we don’t need to know how to handle a violin bow. The plate is connected to something similar to a sort of speaker which itself is connected to a function generator, where you can pick the exact frequency you want.
Because the electronic system is placed in the center of the plate, there is always a small sand circle in the middle of each pattern. The patterns can form different shapes if the plates have different shapes. So, for example, inside a guitar, those shapes form according to its body, but we cannot see them. The higher the frequency, the more complex the figures on the plate.
In case you wonder if it is possible to play songs, it is. Songs are a bunch of sounds at specific frequencies. You can check this in a music video by Nigel John Stanford, with other cool demonstrations of resonance, using water, fire, and even more dangerous tools.
It's a striking experiment that shows how different things around us are bound by physics, often in unexpected ways.
But Chladni didn't pull this theory out of nowhere -- he was building on the work of Robert Hooke who, in 1680, was the first person to observe nodal patterns in vibrating glass plates. He was standing on the shoulders of giants -- as were the cast that produced Rings of Power.
So when you're enjoying the new series, take a moment to appreciate the patterns that show up. It's not just magic, it's science.