Today we talked about why we know that triangles have three sides, a question that came up in an earlier discussion. Looking at them in three dimensions, it looks as if triangles had five sides: the three lines that form their circumference and the surfaces at the top and the bottom. The mistake, of course is that triangles exist in TWO dimensions.

Abbott’s “Flatland”, a short book from 1884 can help us imagine what two-dimensional space is like. (You can read it for free online, but it is written in old-style language.) All kinds of shapes (or Flat People) live in Flatland, – squares, circles, triangles, – but the inhabitants only see each other as lines when they look at each other. We checked this today by cutting out paper triangles, painting them to be Triangle People and looking at them sideways, so we couldn’t actually see what we have drawn on them. Flat People cannot “turn sideways”, because there is no sideways in Flatland. This is what people in flatland would look like to each other. As Space People, we are looking down at Flatland from a THIRD dimension that doesn’t exist for the Triangle People. When a three dimensional sphere visits Flatland in the book, this is what he says to the Flat Person he is talking to:

“The higher I mount, and the further I go from your Plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending; now I can see your neighbour the Hexagon and his family in their several apartments; now I see the inside of the Theatre, ten doors off, from which the audience is only just departing; and on the other side a Circle in his study, sitting at his books. Now I shall come back to you. And, as a crowning proof, what do you say to my giving you a touch, just the least touch, in your stomach? It will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive.”

There is a film based on the book. The two minute trailer that will help you see what Flatland would look like. The whole film seems to be on YouTube as well, but I haven’t watched it, so I have no idea if it’s any good. Please remember, you are watching this clip from the Third Dimension!

There is a video by TED, which is a bit harder, but only 5 minutes long, so worth watching.

Where Are We In Space?

If someone asked you where you lived, you would give the name of your city, town or village. All points on earth can be easily found on a map if you know their latitudes and longitudes, which are called their coordinates. Maps (just think of the map of the earth) are two dimensional representations of three dimensional space. (Paintings, films and photos of people are also two dimensional representations of people.)

But in reality, we do not live on a flat surface. Where we are is quite hilly. Woldingham’s coordinates are 51.2860° N, 0.0337° W, but would someone be able to find you if you were hiding in a hole in the ground or on top of a tree? We need to add a third dimension, which in this case is our distance from sea level. The sea is at the same height all over the world, so we can compare whether we are higher or lower than the sea. The elevation of Woldingham is 700 feet: we are 700 feet above sea level.

Time

But even if you know where in space someone is, that might not help you to find them. If you tell someone that you are in Woldingham, give them the coordinates and elevation and also, maybe, the train timetable, would that be enough to meet up? Not if you didn’t tell them when you will be home! We move not just through space, but also time. This is why many people think of time as the fourth dimension. Then you can say things like “four dimensional space-time” and “manifold”, which sound cool even if you have no idea what they mean.

Another simple exercise helps you to think about time as another dimension. Pictures of people are in two dimensions, but films are really three dimensional. Two dimensions are in space and the third dimension is time, as the two dimensional images move around.

This article explains the whole thing well, but it is (again) a bit more complicated than what you are used to right now.

Finally, here is a film of the astrophysicist and novelist Carl Sagan (he has written the book that the movie Contact is based on) talking about Flatland and then telling us what it would be like for us to move to a fourth dimension in space, rather than time, and what four-dimensional hypercubes or tesseracts would be like. (Remember the point about him being three dimensional in this film clip: flat, but moving in time.)