Dimensional leap: Visualizing and Understanding Dimensions

Jumping through dimensions

What picture comes to your mind if I say “multiple dimensions”? Chances are, you pictured some fictional realm, perhaps from “Doctor Who” or even “Phineas and Ferb.” The alternate dimensions in those stories can be extraordinary, with completely foreign inhabitants and even different laws of physics, to worlds different from ours only in a single choice made. In any case, access to the other dimension usually takes some portal or other futuristic device.

For physicists and mathematicians, the term ‘dimension’ means something different. Yes, dimensions take on standard measurements such as length, width, and height. The precise definition would be “a property of space” or “extension in a given direction.” Still too mathy? What if I give some example phrases: “You’re so one dimensional,” “2-D printer”, or “3-D glasses”? Let’s take a walk through each to get more familiar.

Dimensions

 

0^th Dimension:

Only existing as an abstract thought, an object in the zeroth dimension would be a dimensionless point. The picture to the right represents it but is not dimensionless. It is impossible to display without some dimensionality. The point would have to be infinitely small for it to be accurate.

 

1^st Dimension:

Still an abstraction, a first-dimensional object is now a line connecting two points. The only measurement that can be made on the line is length. Width and height are infinitely small.  This object casts a 0-dimensional shadow.

 

2^nd Dimension:

The object is now a flat plane. It has a length and width. But it still has no height, so it is infinitely flat. We are still in abstraction, so 2-D objects do not exist in the real world. A piece of paper is a good model, but it still has some height, so it is not truly 2-D. This object casts a 1-dimensional shadow.

 

3^rd Dimension:

Now, at last, objects that we see in the real world. Every object we come in contact with is three-dimensional, as we perceive things that way. The object now has three measurements, length, width, and height. These objects cast a two-dimensional shadow, the type of shadow we are most familiar with.

 

4^th Dimension:

This is where we start to go beyond what we can generally comprehend.  A 4th-dimensional object would have length, width, height, and a fourth measurement.  The closest way we can visualize that would be the hypercube.  This object casts a 3-dimensional shadow.

 

5^th Dimension and beyond:

Each higher dimension would have an additional measurement and cast a shadow of a dimension below it.  It is tough to wrap your head around those objects, let alone try and visualize them.  But the rules still apply.

When did people start visualizing other dimensions?  Why, in Flatland, of course!

 

Starting out in Flatland

Over a century ago, Edwin Abbot, a schoolteacher, wrote a short novel about a 2-dimensional square living in a world called Flatland. Since Flatland has only two dimensions, the inhabitants are squares, triangles, circles, and other shapes who live in open geometric buildings.

Since the flatlanders can only see the surface of things, their vision is restricted to lines. They are two-dimensional beings that see in one dimension. That may sound weird, but that is how we work too. We are three-dimensional but only really see in two dimensions. If we see another person, we only see 2-dimensional cross sections of their surface. To really see in 3-D, we’d have to be able to see all the way through them.

Back to the story.

Flatland starts with the protagonist, a square, explaining the nature of Flatland and how they live. The story continues with the appearance of a stranger from another land, Spaceland (the realm that we are from, the third dimension). This stranger appears to be an expanding and contracting circle but is a sphere. The sphere can show him the wonders of Spaceland and even look down from above on the square’s Flatland. But the sphere cannot take him to any higher dimensions, as the sphere can only interact with his own dimension and below; he has no concept of the ones above. In fact, the sphere doesn’t even believe there can be higher dimensions. Having experienced a dimension higher than his own already, the square is sure that there is.

It is an interesting story, especially since it was written around 1884. It gave readers a whimsical exploration of how to image other dimensions and the struggle to come to terms with even considering the existence of higher ones. Today, there is re-emerging interest in Flatland, with many spin-off stories and even an independent film.

But it brings up the question: How many dimensions are there?

 

Count the dimensions: 1, 2, and 3

In our everyday lives, we observe three spatial dimensions. We can talk about time being a dimension, too, though not a spatial one. So, all-in-all, we operate in a 4-D reality; three spatial and one time. But physics may tell us a larger story. A set of theories in physics called string theory sets the count at 10, 11, or even 26 dimensions, depending on what version of the theory you look at.

At its core, string theory postulates that all particles are vibrating strings; how those strings vibrate dictates how they appear. So a string vibrating one way will appear as one kind of particle, while a string vibrating another way will appear as a different kind of particle. Strings have never, and likely will never, are directly observed. Furthermore, the math behind getting string theory to work out is notoriously difficult.

To make the math work out in the equations, string theorists determined that there must be at least six additional dimensions than the four (length, width, height, time) we are familiar with.

The original string theory, now called bosonic string theory, predicted 26 dimensions. That’s quite a stretch. But there are more issues than that. The main one is that it predicts only the existence of bosons, a particular type of particle. Bosons include particles like photons (light) and the newsworthy Higgs boson. But it fails to predict the other primary type of particle, the fermion. Fermions include the particles we are familiar with, such as protons, neutrons, and electrons.

The next several string theory attempts fall under supersymmetric or superstring theory. There are five different versions of it, and they take into account fermions, as well as supersymmetry (hence the name). They predict ten dimensions, a nice break from the boggling 26 of the theory before, but still, six more than we are familiar with.

The most recent form of string theory is called M-theory. Its main feature is the postulation that the five supersymmetry theories are just the various limits of a more prominent, 11-dimensional theory. It was essential, as it sparked a new series of work and interest in string theory. Dr. Micho Kaku, a leading string theorist, goes into much more detail in his introduction to M-theory (http://mkaku.org/home/articles/m-theory-the-mother-of-all-superstrings/).

String theory has a lot of difficulties. It has extra dimensions that we don’t observe in everyday life. Math and equations are complicated. There is little to no observational evidence. But physicists keep working on string theory for a big reason: to find the fabled.

 

Theory of Everything.

The Theory of Everything would determine how the four fundamental forces in physics interrelate and explain all the physical phenomena in the universe. That’s no small task. Einstein, in his later years, unsuccessfully tried to find it. It is one of physics’s primary goals, so string theory proponents are not giving up anytime soon.

 

Can we visit?

Unlike the protagonist in Flatland, we are unlikely to visit any higher dimensions soon. We cannot get there ourselves, as we can only perceive and move in three spatial directions. If a 4th dimensional being visited us could pull us into the 4th dimension, as the sphere did for the square. But we have no way of knowing if 4th-dimensional life even exists. Unless the 4th spatial dimension comes to us, we won’t know.

As far as string theory’s extra dimensions go, the most straightforward explanation for not seeing the dimensions is that they are tiny and compact. Just as a three-dimensional object like a rope may appear one-dimensional (having just length) from a long distance off, the width and height dimensions would only be apparent upon closer inspection. Furthermore, if you were tiny and moving across the rope’s surface, you would think there were only two dimensions, forward-back, and left-right.

Another example is Planet Earth. The Earth looks like a single 0-dimensional dot from a great distance in space. But if you keep moving closer, you’ll eventually see that it is a 3-dimensional sphere. If you are already on the surface, you’d think it was a 2-dimensional plane. It is only when you get high enough into the atmosphere that you start to see the curvature since the Earth is so big. Many dimensions may be too small to see, but nonetheless present.

Dr. Brian Greene, another leading string theorist, does a phenomenal TED talk on explaining concepts and more. (https://www.ted.com/talks/brian_greene_on_string_theory?language=en#t-441037)

 

Conclusion

In our day-to-day lives, the universe operates like it has just three spatial dimensions and 1-time dimension.  Lower dimensions by themselves are mathematical abstractions, only able to be displayed as models.  Higher dimensions, if they exist, are hidden from our normal perception and do not normally impact our lives.  But if string theorists are right, the existence of higher dimensions may bring us closer to a Theory of Everything.  That would be a big impact on physics and the world.

 

References

http://www.geom.uiuc.edu/~banchoff/Flatland/

http://www.flatlandthemovie.com/

http://www.physics.org/article-questions.asp?id=47

https://en.wikipedia.org/wiki/M-theory

https://en.wikipedia.org/wiki/String_theory

https://en.wikipedia.org/wiki/Superstring_theory

https://en.wikipedia.org/wiki/Theory_of_everything