# The Higher Dimensions: Where Are They?

Physicist like to talk about some mind bending concepts. Frequently mentioned is the idea of multiple dimensions above the four spacetime dimensions that we encounter. This idea may seem far-fetched, but has become a more seriously discussed topic in modern physics. Recently string theory has used the idea of 10 and 11 dimensional spacetime in trying to unify gravity and quantum mechanics, but this idea has existed as far back as Einstein’s time with the Kaluza-Klein Theory. Another application for higher dimensions is the mind bending Holographic Principle proposed by Gerard t’Hooft in the mid 1990’s. In this theory, the spacetime curvatures that we experience in our universe are projections of information stored in higher dimensional space.

This all seems like a lot of fun, but these theories often leave people to wonder about the existence of higher dimensions. A common question is: Where are they? and What does a higher dimension even mean?

To demystify the topic a little, it should be understood that higher dimensional constructs are valid in mathematics. For instance, a cube can be given another dimension and evolve into a hypercube, more commonly known as a tesseract. When we add another dimension, all we are doing is adding another set of coördinates that will allow us to characterize a shape. Now a tesseract does show some strange properties, but it is not extremely difficult to solve for. In fact, using vectors and matrices, any college level calculus student can calculate the hypervolume of a tesseract rather quickly. It is not that difficult.

A depiction of a hypercube, also known as a tesseract. Note that this is just a 3 dimensional representation of a four-dimensional shape. We can not adequately draw this shape in our world, but we can describe it mathematically.

Like many ideas in mathematics, it is hard to transfer the concept of higher dimensional space to a real world situation.

Whether or not higher dimensions actually exist is an ongoing debate in astrophysics and particle physics. The math shows that they should exist, but does their existence in our mathematical models actually translate to something that occurs in reality, or is it just a mathematical construct?

The truth is, nobody really knows. But it is interesting to think about where these extra dimensions are located if they exist at all. I would like to point out two intriguing, but easy to grasp scenarios that show why we may not have experienced higher dimensions yet. (For this post I will only be referring to spacial dimensions unless I specify otherwise. We exist in four dimensions of spacetime. These can be labelled as x, y, z, and time but I will be ignoring time for the sake of simplicity.)

1. They are around, we just can not access them because of our geometry

In order to show this example we are going to need to take a simpler approach to the topic. Frequently, when an idea is too complex or too difficult to imagine, scientists and mathematicians will use a simpler model to experiment with properties that they would otherwise be unable to play with.

For this example, we will be imagining a 2-dimensional world (much like in the book Flatland).

Imagine that there is a person (named Claudius) who lived on a plane. A plane is a geometric object that we get when we restrict one of the variables in three-dimensional vector space. Our plane only has two dimensions of movement (x and y) and goes on forever in those two directions (in my picture I drew it restricted to the dimensions of the point {10,10,0} for convenience.) The movement in those directions is not restricted at all. Claudius can explore those directions as much as he wants, just as we are free to explore our 3 dimensions as much as we want (given adequate transportation technology).

Our 2 dimensional world. Notice that I have drawn it in reference to a 3 dimensional coördinate system. We can imagine that the plane we are talking about has the dimensions of (10,10,0) meaning that it has x and y dimensions but is restricted to the origin in respect to the z direction.

But what about the z direction (the up and down direction in our picture)? Well Claudius’s body does not even exist in that direction. He is as flat as can be. Not only is he flat, but he will be completely unaware that there is another dimension to travel in if he relies only on his own everyday experience. He might be able to do some math to prove that there is a higher dimension (maybe Claudius is a physicist and discovers the Holographic Principle in his world) but he would be otherwise completely unable to fathom or move in that direction.

This is similar to us. Any higher dimension would be inaccessible simply because we exist only in our three dimensions.

If we follow that line of reasoning, we are confronted with an interesting idea. Higher dimensions may exist right beside us, but we just can not access them. To prove, let’s add another plane to our previous picture.

We have now drawn another plane (in black) on our coordinate system. This plane has the same properties as our first plane except it has been moved to the coordinates (10,10,1) as defined by a point on the plane. This places it just above our first plane. Sorry that my drawing skills are not awesome.

Now this plane is very similar to our first except that it has been shifted up 1 in the z direction. We can imagine that this distance is really, really small. Maybe 1 yoctometer of distance (a yoctometer is about 1/100 of the diameter of an electron.) For all intents and purposes, these planes are nearly on top of each other. But would Claudius ever be able to travel to the plane above him?

The answer is no. Even though the second plane is so close, he would never be able to get to the next plane on his own accord. You may be screaming: “Just have him jump up!” Unfortunately for Claudius he can only move in two directions because he is a two-dimensional being. We can see the plight of his situation because we exist in 3 dimensions and can perceive all the planes that exist. We are omniscient in respect to Claudius’s restricted world.

Now what does this mean for us? If we apply these principles to our lives we get some interesting ideas. If higher dimensions do exist, there could be alternate realities all around us that we can not perceive. It is possible that floating 1 yoctometer in the fourth dimensional direction is a whole other world that we will never be able to reach. Our 3 dimensional existence is just limiting. However, any being living in the higher dimensions would be able to perceive all the 3 dimensional worlds separated by a higher dimensional distance. We would have no way of proving that these other worlds existed, but it is possible given these principles. (Maybe later I will discuss the religious implications that I find in that line of reasoning. But not today.)

2. Maybe the other dimensions are really small?

As string theory has progressed, the idea of super small extra dimensions has become more and more accepted. Whenever we are confronting the idea of other dimensions we have to deal with the fact that we somehow do not experience them. The theory mentioned above is one possibility, but the other is that these extra dimensions are so tiny that we never interact with them on a real level.

Here is an example to show this principle. Imagine a tight rope strung between two poles. If we are viewing this rope from a good distance away, how many dimensions does it seem to have? The rope looks like it is only one-dimensional, just a line. We know from experience that this tight rope is actually three-dimensional, but we only perceive it as one-dimensional from a distance.

Man… I am such a good artist.

As we get closer and closer to the rope, what do we see? We begin to see that the rope actually has thickness to it. It has another dimension. This will look rather rectangular. If you have a hard time imagining a rectangle, do not worry….

I’ve got this covered.

But we know from experience (and logic) that this rope has three dimensions. So we get closer and closer until we can eventually see that indeed, this rope is more like a cylinder than just a line. You can even see a bug on it!

In case you can not tell, this is the cowboy beetle (Chondropyga dorsalis)

As we keep zooming in we will eventually start to see even more dimensions. Current physics postulates that these will occupy a space really small, at a distance called Planck length. If we were to blow an atom up to the size of the universe, the Planck length would be an average tree.

This is so small that we can not actually see these dimensions until we get to that level. Our little beetle on the rope had no problem knowing that there were three dimensions on the rope, because he was small enough that he was forced to navigate them all the time. Any creature at Planck length (and there probably is not any.. but just for fun we will consider it) would feel completely natural in 9 to 10 spacial dimensions, even though this seems completely alien to us.

But if all these extra dimensions do exist on those scales, we actually do interact with them. They are just so small that we move through them too quickly in the macroscopic world to actually chart our motion in them.

This comes from string theory and is not proven. But it seems likely. When we start to consider space as higher dimensional, the math allows us to find middle ground between gravity and quantum mechanics. So hopefully this is how nature works.

Of course, there are more complex ideas about multidimensional space, but these are my two favorite theories that can be easily grasped without complex mathematics. For most of us, the existence of higher dimensions does not affect us, but it does offer some interesting thought experiments.