How to Visualize Eleven Dimensions
One very offputting thing about trying to use “string theory” and “m-theory” to explain the behavior of the “physical universe” is that mathematicians and physicists talk about their formulas and equations in 10 or 11 dimensions (depending on who you listen to). Most people who talk about the “real world” pretty much limit themselves to four…width, length, height, and duration (a.k.a. time). If you get much further than four, most people can’t visualize it.
Let’s show an example of this by starting with the “geometry” version to see how most people were taught to visualize dimensions.
We start with the very simple ONE dimension. It’s just a line. It goes forever THAT way and forever the OTHER way. It’s a little unwieldy to visualize with, so for our purposes let’s take a short section of the line so we can easily think with it. It has a length, that’s about it.
Now, when you add a SECOND dimension, you can create concepts like “perpendicular”. Take two identical segments and arrange them so that the end of one connects with the middle of the other, and the angles on either side are equal (Figure 1). It’s commonly represented with a little “mini-square” in the corner where the lines meet. Many of us also recognize “Ninety degrees”. It is also referred to as “right-angles”.
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Fig 1. Perpendicular Line Segments.You see how complicated this is getting already? All this formality and rigor. I just want a square corner.
Alright, back to our dimensions. If we take our line segment and extend it an equal distance PERPENDICULAR in the SECOND dimension, we wind up with a square. It is a basic shape in two dimensions. It has square corners. It has a width and a height.
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Fig 2. A square.Now, if we add a THIRD dimension, where can it go? If we take our square and lay it flat, then smart money says “up”. So, if we extend our square PERPENDICULAR to its existing two dimensions, we wind up with a CUBE. Everyone knows what a cube is (snarky jokes about offices aside). You can visualize it.
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Fig 3. A two-dimensional representation of a cube.
I know, the perspective is weird. Go with me on this.So we have now created a cube. We want to visualize a FOURTH dimension. Hm…where can it go? In what direction can it be “perpendicular”? This is where the use of physical dimensions goes off the rails for most people. However, recognize from above that the picture in figure 3 is NOT really a cube. It’s a two-dimensional REPRESENTATION of a cube. We took our square, made a copy of it, and connected each VERTEX (place where lines meet) with its LIKE VERTEX. This gives us the illusion of three dimensions when we’re really only using two.
So, if you take two actual cubes, and INTERLOCK them, then connect all 8 vertices with their LIKE vertices, you can get a three-dimensional representation of a four-dimensional “cube”. Some people call this a “tesseract”. I don’t know why…google it.
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Fig 4. A tesseract.
Or, rather, a two-dimensional representation of
a three-dimensional representation of a tesseract.
Google for better pictures if you can't visualize this.However, this doesn’t really open the door to understanding how to extend dimensions even further. To get around this, some have tried to use “time” as the fourth dimension. I prefer to think of it as “duration at a position”. We measure this “fourth dimension” by noting where in THREE dimensions an object is, and how it changes over TIME. Anyone who has ever caught a ball or jumped into a pool or driven a car has seen this in action. You are able to follow the path of an object as it goes from HERE to THERE, tracing its path through TIME as it moves. Another two-dimensional representation analogy is a video game or a computer screen. What you see on the screen is in two dimensions. You get to see the objects there simulate three dimensions with tricks of perspective, overlapping, and such when the character (or car, or plane, or spaceship) has moved with respect to a background. As it changes simulated position over time, thus it simulates a fourth dimension.
However, these are all just simulations. How about in the real world? Which direction would you point to show the way to “perpendicular” to the other three dimensions. And this is where it all goes off into the realm of “Huh?”. How would you visualize a FIFTH dimension? A “spinning tesseract”? Where would it go?
So, as a public service and a way to oversimplify, I have come up with a method to help all us other regular joes and janes out there wrap our respective heads around what 10 or 11 dimensions means. I’ll put it in the context of something we’re all familiar with…letters printed on a page. It is a trick used by computer programmers and data scientists to group data into meaningful buckets, to look for patterns where none might be obvious.
Let us begin with the most simple dimension. It is the equivalent of a geometric line…a typed line of text on one sheet of paper. It doesn’t matter what letters…they might be words, or numbers, or a bunch of random characters that look like this: #$%^&*. It may look like cartoon swearing but it isn’t. Please trust me with this.
Next, we’ll go to two dimensions: imagine the same line of %^&*()@#$%^& characters typed on an entire page. You can see that if you drew a line vertically on the page, each letter or symbol the line passed through would be in the “same position” in the context of this second dimension. So, character number 3 in line 1 might be “x”, but character number 3 in line 2 might be “@”. And in line 3, it might be “8”, and so on. Each character is distinct, but it occupies the same position in a different line.
Three Dimensions: imagine the same page repeated as an entire bound book. How would you represent getting each letter “in the same position” in that dimension? Easy. Close the book, then get a drill press. Every page would be pierced at the same spot. So each letter the hole passes through is “in the same position” with respect to the page it’s on. Or, it would have been if we hadn’t destroyed them. Just think about the letter immediately to the right or left if destruction bothers you.
Note: In real life, this method wouldn’t really work. In a “real book” each page is printed on both sides. It’s cheaper that way. So, if each page was read left-to-right, then the drill press hole would show in a different orientation to how we’re trying to think.
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Front Back (flipped over)So let’s just pretend that each page is printed on only one side. Maybe we could put a pretty watercolor picture on the other side, like a Beatrix Potter book. But I digress.
Four Dimensions: imagine the exact same book 10 times on one shelf. How would you represent getting each letter “in the same position” in that dimension? Here’s how.
Think about the first book. Now think about page 53. Go think about the 10th line, the fourth letter in from the left. Got it? Anybody who’s every looked up a word in a dictionary has sort of done this. Please tell me you’ve used a dictionary instead of googling or wiki-ing everything. I think wiki-ing is a word. Maybe I’ll google it.
Anyway, once you’ve located your letter, get the NEXT book from the shelf. Again, go to page 53, line 10, letter 4 from the left. Now, get the next book, do it again, and then the next book…you get the picture. Each of the selected page/line/letters is IN A STRAIGHT LINE in your new fourth dimension. You couldn’t really use a drill press for this case.
Five Dimensions: imagine the exact same shelf from the 4-dimension example, stacked 5 high as a bookshelf. Maybe oak. Or mahogany. I’ve always liked mahogany. Remember, we’re trying to visualize here. So if you pick the fourth book from each shelf, then open each book to page 53, line 10, letter 4.
Six Dimensions: imagine the exact same bookshelf stood side by side as a library aisle. Mahogany would probably be too expensive in that case. Maybe just some pine or metal. Anyway, on each bookshelf, go to the second shelf, pick the fourth book, open each book to page 53, line 10, letter 4.
As we get further up in the dimensions, it gets harder to identify “straight lines”. So, I’ll give you some tricks for how to visualize the overall intent of the dimension, and leave it as an exercise for the bored or pedantic to figure out how to locate “letters in the same position” in each example.
Seven Dimensions: imagine the same aisle from dimension six, back to back with other aisles in a room. Think of a school library. Shhhh.
Eight Dimensions: imagine the same room side by side with other rooms on an entire floor. Maybe it would look like Powell’s in Portland. Look it up.
Nine Dimensions: imagine the same floor stacked 5 floors high as an entire public library. Again, SHHHHH.
Ten Dimensions: imagine the same entire library beside other libraries on the same street. This should be easy to do…just think of that street in every town that has all the fast-food restaurants. Now imagine them as libraries.
Congratulations! You are now competent to understand string theory. But wait, there’s more!
Eleven Dimensions: imagine the same street identically reproduced in another city. We already got to imagine that every city has a fast-food restaurant street…change them ALL to libraries. See? Easy! Good luck finding french fries, though.
Congratulations! You are now competent to understand m-theory. Please publish your findings for the rest of us.
And yet, we can even go higher:
12 dimensions: imagine the same city from 11 dimensions along with other cities in the same country
13 dimensions: imagine the same country with other countries on the same continent
14 dimensions: imagine the same continent with other continents on the same planet
15 dimensions: imagine the same planet with other planets in the same star system
16 dimensions: imagine the same star system with other star systems in a constellation
17 dimensions: imagine the same constellation with other constellations in a spiral arm
18 dimensions: imagine the same spiral arm with other spiral arms in the same galaxy
19 dimensions: imagine the same galaxy along with other galaxies in a local group
20 dimensions: imagine the same local group with other local groups in a cluster
21 dimensions: imagine the same cluster with other clusters in a supercluster
And so on, and so on. Dimensions without end, as far as your imagination can take you. Enjoy! And, if you conquer the cosmos with your new-found understanding of dimensions, please be kind to the citizens.
