## Sunday, September 12, 2010

### Q: Could a flatlander utilize flatland technology to perceive something in the third dimension?

There are some big differences between living in a plane, versus living in Euclidean 3D and merely being able to perceive only a plane slice. Here's one: the wave equation in three dimensions is "nondispersive," meaning all frequencies travel at the same speed and the region of influence of an impulse is just an expanding circle. So if you say something 100 meters away, or transmit a signal from your radio, I will hear a delayed, quieter version of what you said.

In two dimensions (or any even number of dimensions), the wave equation is dispersive! You do not just hear a delayed, quieter version of stimuli at a distance; the sound itself is actually changed by traveling through the medium. The region of influence isn't a circle; it's a filled-in circle. Imagine throwing a rock into a pond -- the resulting ripples aren't just at the outside of the circle. The rock continues influencing the interior of the circle even after the first news of its plop has passed by. Similarly, the sound of thunder, constrained to a near-2D slice of the earth's atmosphere, is not just a bang when you hear it from far away. It gets transformed into a rolling sound because the different frequencies travel at different speeds.

So a flatlander who understood partial differential equations (this may be a small number of them...) could distinguish between these two possibilities (2d world, versus 2d slice of 3d world) by observing the behavior of waves as they propagate.

(NB: The proof that the wave equation is dispersive in even-dimensional Euclidean spaces and nondispersive in odd-dimensional spaces is really frickin' hard! I have seen it in a monograph and did not understand it at all.)

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