You can’t get there from here.
~Maine saying

My father once quoted a saying from Maine, where he spent some of his youth: “You can’t get there from here.” It refers to Maine’s winding road system, which often prevents a traveller from taking a direct route between two places. In physics and math terms, we might say that Maine’s road system is of fractional dimension: Less than two-dimensional, but more than one-dimensional.

Integer Dimensionality

Traditionally, we define the dimensionality of a space as the number of directions one can move in. For instance, a ski lift lives in a one-dimensional world. It can move forward or backwards along a pre-defined path, where the support cable lies. However, it can’t go off the cable, and the cable forms a simple circular loop.

A boat generally lives in a two-dimensional world. It can move forward and backward, but it can also turn left and right. In other words, it has two axes of motion. However, if a boat goes up or down (and it’s neither  a submarine nor that flying aircraft carrier from The Avengers), that’s probably a bad thing.

The flying aircraft carrier lives a three-dimensional world. It can go up and down, left and right, and forward and backward. It has three axes of motion.

Although we all live in a three-dimensional world, mathematicians and physicists often study systems that are more or fewer dimensions. String theory treats the universe as being 11-dimensional, although most of these dimensions are not accessible to us. General relativity combines space and time such that the universe is a four-dimensional object. (While we are pulled inexorably forward in time, without the ability to go backward or even stop, we do move in the fourth dimension.) Graphene, a one-atom-thick sheet of carbon, is so thin that electrons inside cannot move up or down at all; they behave as if they existed in a two-dimensional universe. A carbon nanotube–essentially a graphene sheet curled into a tiny cylinder–can be rolled so tightly that electrons can only move up or down along it, not around its circumference, making carbon nanotubes one-dimensional objects.

Although the dimensionality of a space or type of motion is often intuitively obvious (like for the ski lift, boat, and airship examples above), physicists and mathematicians working in very exotic spaces need a way to measure the dimensionality of a space. One method is to measure diffusion.

Diffusion and Spectral Dimension

If I release a ball of hot gas particles into an empty room, the gas particles will hit each other and push each other away, causing them to spread out in every direction. Eventually, they will be distributed more or less evenly around the room, filling it. This process is called diffusion, and it’s one of the major ways that chemicals move around. Professor Mark A. Peletier has a very nice video demonstration of this phenomenon.

Here’s the trick: The higher the dimensionality of the space, the more dimensions the particles can move in, and thus the faster they spread out. By simulating gas molecules diffusing through a space and measuring how fast they spread, we can figure out the dimensionality of that space. The dimension of  a space measured in this way is called the spectral dimension. Other ways to measure the dimension of a space include the Hausdorff dimension and the fractal dimension.

You Can’t Get There From Here

So far, we’ve talked about the dimensionality of a space as the number of directions in which you can travel through that space. However, that number is not always easy to define. For instance, consider the spiderweb of the header image. I’ll repeat it here again so you don’t have to scroll.

Imagine a tiny man–someone so small he can stand on a single thread–walking along that web. Most of the time, the man can only go in one direction. However, each time the man gets to a junction of threads, he can choose to go in several directions: left, right, forward, and backward. In other words, for most of his journey, the man lives in a one-dimensional world. But, sometimes, his world is suddenly two-dimensional. And as the Maine folks say, “You can’t get there from here.” If the little man wants to get from one strand of the web to another, he has to travel until he finds a junction, take the correct path, and walk to the strand he wants. How do we quantify the dimension of such a space?

The answer is that we measure the spectral dimension. We imagine millions of tiny little men, who start in a crowd somewhere on the web. The tiny men want to get as far away from each other as possible, so they will eventually be distributed evenly across the entire web. But “you can’t get there from here.” The tiny men are hindered as they look for web junctions to take. The time required for them to spread out will be less than for a one-dimensional system (like a string), but much more than for a two-dimensional system (like a dinner plate). The spider web has a fractional dimensionality.

A spider web is actually a pretty bad example of a fractionally dimensional system–it’s very close to one-dimensional. The canonical examples of fractional-dimensional surfaces are all fractals. You can find a list of the fractional Hausdorff dimension of fractals here.

The Quantum Foam

While fractional dimensionality sounds pretty esoteric, it actually may be very important. A major unsolved problem in physics right now is the question of how to merge the theories of quantum mechanics and general relativity. Quantum mechanics deals with the very small, while general relativity deals with the very massive. Usually these two domains are separate. However, there are objects that are both very small and very massive. Black holes are huge amounts of mass compressed to an infinitely small point. At the beginning of the universe, right after the Big Bang, all of the mass in the entire universe was compressed to a single point. We need a way to study these phenomena. Quantum gravity aims to solve this problem. Unfortunately, quantum mechanics and general relativity don’t play well together.

One thing quantum gravity has predicted, however, is that the universe on a very small scale looks very alien. The fabric of spacetime warps and bubbles in strange ways to form what we call the “quantum foam.” Although general relativity predicts a four-dimensional universe on human scales and larger, the spectral of the dimension of spacetime can be less than two when considered on the quantum scale. Causal Dynamical Triangulations (the theory I work on) predicts a spectral dimension of about 1.8, which is very weird and very cool.

Further Reading

Unfortunately, fractional dimensionality is a rather technical topic. I couldn’t find very much accessible stuff.

• If you haven’t read Flatland, you really should–it’s a classic, wonderful story about geometry written for laymen. A free version is available online here.
• This is a mostly-accessible introduction to fractional dimensions and fractals for the casual mathematician.
• You can find a fairly technical introduction to Hausdorff dimension here.
• This is an introduction to fractional dimensions. This is part of a master’s thesis, so it’s rather technical.
• For an introduction to causal dynamical triangulations (CDT), try this article by one of the inventors. The first few pages are pretty accessible.
• Scientific American also did a piece on CDT, but the piece is behind a paywall.
• You could also look at my paper on CDT. (Or I could do a post on CDT.)
• This blog post is an introduction to CDT. The sequel is here.

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