Flatland

I’ve been thinking about how to finish up my post on the Fibonacci sequence, and what’s been holding me up is how to illustrate my point. I mean that literally. How am I going to draw a golden rectangle and the spiral that can be formed inside repeated rectangles that corresponds perfectly to certain shells in nature? I can’t find my compass, and I’ve lost my momentum.

So here’s a Fibonacci spiral from wikipedia:

The ratio of the larger side of the rectangle to the smaller is the same as the ratio of a larger Fibonacci number to one just preceding it, a number that approximates 1.61803.  It’s known as the golden ratio, and it’s been used since ancient times in art and architecture.

I’m stopping there because I have become completely absorbed in something else: the concept of dimensions.

This was a later lecture in my math series, and it touched on a subject that I’ve run across in various books by physicists like Brian Greene and Richard Feynman. Every time I read something about dimensions beyond the third, I get curious. The physicists think they’ve proved 17 dimensions now, but it’s all theoretical. Still, I like the idea.

One of the math professors in this lecture series began to illustrate the dimensions we have experience with. First we start with a point. A point is invisible, mathematical, and has no dimension. It is completely self-contained.

Next we have a line. A line stretches between two points. It moves only in a forward or backward direction. It is one-dimensional.

Next would be two perpendicular lines. Together they would form a plane. We could draw a square on the plane, and it would have four corners, or vertices. It would have four sides, all made of one-dimensional lines. The square is two-dimensional.

And finally we could draw a line perpendicular to the plane (to go up and down). We could turn the square into a cube. It would have eight corners, or vertices, and six sides made up of two-dimensional squares. The cube is three-dimensional.

If we kept up this pattern, a four-dimensional cube would have 16 vertices and eight sides made of three-dimensional cubes. To see the fourth dimension we’d need to be able to draw a line perpendicular to the third dimension. And of course we can’t do that, so this theoretical concept is one we can’t visualize.

You know how you can draw a cube, a three-dimensional object, in 2-D? Example:

You look at it, and the sides are not all equal and the angles aren’t all 90 degrees, but it doesn’t matter because we have experience with cubes and so are able to visualize that this drawing represents one.

Well, one of the professors constructed a 3-D model of a 4-D cube. It was sort of the same thing as the drawing above, only it was three-dimensional and had eight sides. He suggested that to someone in the fourth dimension it would make sense, like the drawing above makes sense to us. Click here for a movie of what it looks like.

Science fiction writers call the 4-D cube a tesseract. I first ran across that term as a kid reading A Wrinkle in Time. It’s still one of my favorite books.

The professors suggested that a better way to wrap our brains around the concept of dimensions is to go lower rather than higher. We live in 3-D. But what would be different about life if we lived in only two dimensions?

This is a concept Edwin A. Abbott explored in his groundbreaking 1884 novel, Flatland. The book is in the public domain now, and you can read it in its entirety here.

The main character is a square, and he describes his existence. Take a penny, for instance. You lay it on a table, and you can see that it’s round. But what happens if you put it on the edge of the table and lower your eye so that your view is in the same plane as the penny? It becomes a small, copper, horizontal line. In Flatland, the square explains, everything is on a plane, so everything to the creatures in Flatland appears to be a line. He goes into a bit of detail about how they learn to recognize people and places, and if you’re interested, you can check out the link.

This was a part that grabbed my interest: in Flatland, the surface of each figure is its outline. To a square then, the outline of a square is its skin. It is incapable of seeing anything contained within the outline without cutting itself open. So far as a square is concerned, the area within it is as contained as our organs are within our bodies. But to a three-dimensional creature, the insides are visible.

Below is an example I drew. On the left is a girl as we would expect to draw her in a 2-D space. On the right is a girl as Abbott might have drawn her. For her to see or hear, her eyes and ears would need to be on the perimeter (surface) of her body. Additionally, her insides, while invisible to her and her compatriots, would be visible to us.

So, to extrapolate this idea to the fourth dimension, fourth-dimensional creatures would be able to see our insides just as we can see those of two-dimensional creatures. Pretty freaky.

If you’re wondering why such a weird concept has grabbed my attention, the answer is that I’m reworking a time-travel novel I wrote, and I’ve been looking for a good way to explain how it works. So I’m kicking these thoughts around in my head. Perhaps time travel can be explained through other dimensions, blah, blah… I’m still figuring it out. But it’s fun for me.

Hope you enjoy it too!

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Categories: Brain Workouts

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1 reply

  1. amazing. my brain must be flabbier than yours. i got into the spiral and i couldn’t get much further. it sure is a pleasing spiral.

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