What is a Dimension?

When you read and watch a lot of bad science fiction, like myself, you get used to a lot of cliches and sci fi tropes.  The heroes will obviously have to travel through time, fight their doubles, and outsmart a supercomputer.  Despite their predictability, and no matter how far-fetched they are, I almost always love them.  The one thing I can not stand, though, is when writers use words incorrectly in order to make things sound “cool” and “sciency”.  Things like cosmic rays to explain the fantastic 4’s powers, any time quantum mechanics is brought up in fiction, and people saying infinite when they simply mean unimaginably large.  The worst one, though, is the misuse of the word dimensions.

quantum_mechanics

Quantum Mechanics image: Randall Munroe.

A dimension is not an alternate reality where everyone is evil and has facial hair.  Dimensions are properties of space that we can use to denote position.  A square and a circle have 2 dimensions, but a cube and a sphere have 3.  This is similar to how most movies are portrayed on a flat screen, but when watching 3-D movies you can put on special glasses and they suddenly have depth along with length and width.  This can go beyond 3 into 4, or 5, or n dimensions.

Movies and pop culture often shows its ignorance in many different dimensions.  You may have seen one of the many Marvel superhero movies featuring The Tesseract as a Macguffin.  A tesseract is the 4-dimensional version of a cube, just like how a cube is the 3-dimensional version of a square.  However, The Tesseract in the movies just appears as a blue glowing cube with magic powers.  It isn’t shown to have any properties unique to a real tesseract, or hypercube.  In the episode “Through the Looking Glass” of the TV show Farscape, the characters are separated into different colored dimensions.   Here dimensions are treated as if they are different universes each with a different color and a different effect on peoples senses.  Countless other tv shows, movies, games, and pulp novels provide just as poor and worse depictions of dimensions.

Glass_tesseract_animation

A rotating tesseract by Jason Hise.

 

In defense of these creators, thinking about 4-dimensional and higher objects can be difficult.  One way to help our imaginations is to think about going from 0 dimensions to 1 dimension to 2 dimensions and then up to 3 dimensions.  If we start with a point and stretch it some length we get a line, then if we stretch the line the same length we get a square, then if we stretch the square out we get a cube.  We can extend this to higher dimensions.  If we stretch the cube out into the fourth dimension we get a hypercube or tesseract. If we want to think of a 4-dimensional ball coming into our 3-dimensional space, we can similarly think of an analogy in lower dimensions.  You can imagine a ball dropping through the surface of a pool of water.  In this case, the ball is a 3-dimensional sphere going through the 2-dimensional plane of the water’s surface.  As the ball first touches the surface, a small solid circle would suddenly appear.  Then as the ball starts sinking down, the circle would grow and grow until it got half way in the water.  Once it passed the halfway mark, it would start shrinking until it was just a tiny circle and finally disappear.  Something similar would happen with a 4-dimensional sphere appearing in our 3-dimensional world.  It would appear as a small sphere, grow until it got halfway, and then start shrinking until it disappeared.  Basic geometric shapes are the easiest things to imagine in higher dimensions, because of all the symmetries they have.

385px-Dimension_levels.svg

Image by NerdBoy1392.

When you want to think about more complicated things, you need a way to represent the fourth dimension.  When we thought about the 4-dimensional sphere, we used time as the fourth dimension.  As time advanced the sphere passed through our space so that we could see its facets.  However, if we wanted to play a 4-dimensional version of chess time may not be the most useful method to think of the fourth dimension.  A regular chessboard is already 2-dimensional.  We can then stack the boards on top of each other to get a 3-dimensional box to play in.  If we want to extend the game further we need a way to represent the next dimension.  If we get 8 separate 3-dimensional game boxes each one can be treated as a different level in the the next dimension.  If we wanted to actually play chess we would also need a set of rules that work in higher dimensions, like this version.

Thinking of these higher dimensions may seem like it is only a fun exercise for your imagination, but they are meaningful and have some applications.  Einstein’s theory of special relativity combines the 3 spatial dimensions we live in with time to form Minkowski Space (Devlin 322).  This allows physicists to think geometrically about space and time.  Higher dimensions also comes up when looking at some higher mathematical ideas, like algebraic curves.  Graham’s number, one of the largest numbers ever used in a scientific paper, comes about when looking at cubes of n dimensions.  In 1868, Julius Plucker created a new type of geometry that doesn’t think of space as a bunch of points but rather as an infinite number of lines.  In this new line geometry, our normal 3-dimensional space made up of dots becomes an infinite amount of lines defined by 4 parameters (Boyer 498).  Thinking of higher dimensions can be a useful tool to help us analyze the world around us.

Not every work of fiction fumbles over depictions of different dimensions.  There are several examples of TV shows, games, and books accurately depicting higher and lower dimensions.  Futurama and The Simpsons are known for their many accurate references to mathematical ideas.  In “Treehouse of Horror IV”, Homer stumbles on the third dimension and becomes a computer generated 3-D version of himself.  In the Futurama episode “2-D Blacktop”, the Professor, Leela, Bender, and Fry end up smashed into 2-D versions of themselves with all the restrictions that brings with it.  When they start expanding back into 3-D, the characters are seen travelling amongst fractals, which have fractional dimensions.   The indie video game FEZ is about a creature who has experienced life in 2 dimensions being introduced to the third dimension by a 4-dimensional being.  Many of the game’s puzzles revolve around treating the 3-dimensional world as a 2-dimensional projection.  There is also the book Flatland by Edwin A. Abbott, which describes a square’s adventures to higher and lower dimensions.  All of these provide accurate representations of what dimensions are and what living in higher and lower dimensions would be like.

Higher dimensions are an abstruse but interesting concept.  This idea can be used as a tool to help with mathematics and physics, or as something fun to just spice up an old board game.  There is plenty of depth to the fourth and higher dimensions to write and create interesting sci fi stories.  These stories can be interesting, consistent, and can go beyond just using the word dimension as a way to sound more “sciency”. 

Sources

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print. 

Devlin, Keith. The Language of Mathematics: Making the Invisible Visible. New York, NY: W.H. Freeman and Company, 1998. Print.

Additional Materials

A video about dimensions

More about the TV trope of different dimensions

A little information on fractional dimensions

More on Graham’s number

More Numberphile with Ron Graham

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