A Globe Can Help You Understand How Space Can Be Curved

There are theories out there that space itself can be curved. This is a confusing idea for some people. A quick exercise with a globe can make it a little more understandable.

If you hangout among physicists long enough, you'll hear about curved space, higher dimensions, and the possibly weird shape of the universe. What do these ideas mean? When we look around us, they mean nothing. Our universe looks the same, from the inside, whether it is shaped like a piece of paper, or a saddle, or a globe. There are very few ways to get any perspective on the idea of a curve to our universe, or a curve to our dimension.

One point often made to illustrate the consequences of living on a curve is the fact that the geometry of a curved world isn't anything like geometry of a flat one. Shapes that are impossible on a flat piece of paper materialize if given enough space on a curved world. A simple way to picture this is to grab a globe and start tracing lines with your finger. You're going to make an impossible triangle.

First draw a triangle, any triangle, on a flat piece of paper. No matter what triangle you drew, its angles add up to 180 degrees. On a flat surface, that's part of the definition of a triangle. To see how impossible it is to draw a triangle with angles that add up to more than 180 degrees, try drawing a right angle - a perfect 90 degree angle. At the end of one of the lines protruding from the right angle, draw another right angle. That's 180 degrees, and what do you have? A square that's missing a side. There's no way to make it a triangle.

Now get the globe. (You can try doing the same thing by tracing lines on a tennis ball, or an orange, or any spherical object, but a globe is helpful because it has lines already sketched on it.) Imagine you are an amoeba on that globe—right on the equator. At the nearest line of longitude, you take a 90 degree turn north toward the pole. Keep going up that line of longitude. When you get to the north pole, take another 90 degree turn, going down the appropriate line of longitude. That's 180 degrees. In a flat space, it would be impossible for you to make one more turn and describe a triangle. But eventually, you hit the equator. You make a 90 degree turn, and end up where you started. From your perspective, you have just made three 90 degree turns, and walked three straight lines, and you have made a giant triangle. That's not possible.

From the perspective of someone who can see the globe, on the other hand, it's perfectly possible to make this triangle. It works for the same reason you can walk in a straight line on a globe, and end up where you started. This is what people mean when they say that we could be a curled up dimension inside higher dimensions. In a small space the 270 degree triangle isn't visible. But if we were to send a spaceship crew around the universe, and the universe we move through is sufficiently curved, they should make the amoeba's voyage. They would think they were charting out squares on a grid, only to find themselves back where they started, because they made a triangular voyage accidentally.

[Via Surfing Through Hyperspace.]