Flattered or Flattened

Last night I got an interesting question from my nephew Phineas: Do 2D objects exist in our world? How do they work? In my exhausted state, I managed to cobble something together about electrons in a conductor, but I felt I owed him a more complete answer once I thought a bit more.

First let's discuss a bit what it means for something to be 2D (or 3D, or even 0D). The number of dimensions refers to the extent of an object, or the space it takes up. However, we can talk about embedding an object of lower dimension in a higher-dimension space. For example, we can put a plane, which is 2D, in a 3D space:

The equation for this particular plane is z = 1, with x & y between -1 and 1. That means we can pick a point on the plane, like (0.5, -0.5, 1), but if we go even a tiny bit off of z, like (0.5, -0.5, 0.99999), we'll no longer be on the plane.

Even if it's mathematically possible, it's still hard to believe such a thing could exist in the real world. That's where my example from above comes in: Imagine a metal ball that we charge up with some free electrons. Metals conduct electricity, so the electrons can move around freely, but they also are trying to stay as far from each other as possible since like charges repel. If you imagine trying to find the largest separation between a set of points in a sphere, you'll see that the solution requires all the points to be on the surface. This surface is 2D, even if it's curved and embedded in 3D, like the plane above. You can tell it's 2D because you only need 2 coordinates to position yourself on the surface of a sphere, like latitude and longitude.

Now that answer may not be satisfying, since if we imagine ourselves like the electrons, moving around the surface of the Earth, we're not 2D, just the surface we're walking on. However, based on the measurements that have been made in particle colliders, electrons act as single points, or 0D objects. In response to this, Phineas asked, If electrons are 0D how do they exist? They must exist to become 1D, or 2D, or even 3D! I still don’t understand how if something is 2 or 1D how it exists. This is fair point – If something has fewer than 3 dimensions, it will have a volume of zero, so how can it have mass? If they do have mass, then they should have infinite density, which would make them a black hole. My feeling is that the answer lies in the fact that electrons, and similar sub-atomic particles, are quantum objects, and quantum mechanics works very differently from anything we're used to our own lives.

These questions are really at the limits of what we know right now – While looking up info to write this post, I found an article from 2020 about the discovery of 2D particles. If you're curious about this stuff, Phineas, you can become a physicist and maybe explain it to your UncOrion some day!