Lines, intersections and dimensions

Building on the interest in my post on knots and different dimensions, I thought I’d say a few words on some interesting issues on lines in different dimensions, as raised by a recent seminar. It’s an interesting mind game!

We explicitly consider only generic situations from here on, which means that we ignore special cases as physically unlikely. So parallel lines need not be considered — all lines will have some random angle between them.

Let’s start with two point particles. Put them on one dimension, that is, a straight line. Generically they will always collide at some time, either in the past or the future. But in two dimensions (on a flat surface), or indeed in more dimensions, two particles will generically never collide. See the diagrams below; the arrows are sample random velocities of the particles.

Now think about two infinitely long lines, moving in some direction perpendicular to the line. In two dimensions, they have a point of intersection at any given time. In three dimensions, they will generically have an intersection point at some time, either in the past or future, at some point along their infinite length.

But not in four (spacial) dimensions. Each line’s motion with time describes a plane, and in four dimensions these need only meet at a point. But that point corresponds (generically) to different times for the two lines, so they never actually intersect.

A physical implication

This abstract argument has some use. One of the (many) possible string theoretic starting points of the universe is a very small ten-dimensional surface of a “sphere”, or some similar closed (essentially, go far enough and you get back to where you started) space. This will have strings living on the ten-dimensional surface, carrying energy in three ways: momentum of strings within directions transverse to the string; oscillations in the string; and winding of the string around the entire space, like wool in a ball of yarn. As the universe evolves, various mechanisms can make some of those directions “expand” to form our universe. The energy in the momentum and oscillation modes behaves as a simple energy density, which acts like a pressure toward the universe expanding. But the wound strings have the opposite effect: as the dimensions around which they are wound expand, their length increases. Since they have tension, this increases their energy, so they act against expansion of the dimensions around which they are wrapped.

So unless there is a mechanism to get rid of winding modes, the universe can’t expand to much above the string scale, which is tiny. But if strings intersect they can interact and exchange their attachment, as in the diagrams below show. Two strings swap at the intersection point, and so (since the strings are closed into loops behind the sphere) one string results. Thus intersections reduce the number of wrapped strings.

But now we’ve seen that string intersections don’t generically happen in four dimensions, so this can be used to argue why our universe could have at most three “large” spacial dimensions — which is of course exactly what we observe. There are problems with the argument — not least of which is that particles in quantum mechanics interact at non-zero seperations (for example, electromagnetically), so we might expect the same to be true for strings in four or more spacial dimensions. Nevertheless, it’s always cool to go from a simple mind puzzle to an argument for why the universe has three dimensions!

Exercise for the reader: By thinking a little more about my arguments for why lines in four dimensions don’t intersect, find the number of dimensions in which planes will/will not intersect, generically. Use this to provide a simple argument for why there are exactly seven continents on Earth. If you succeed in the latter task, collect an Ig Noble Prize.