 The chalkboard, um, paper, um, computer screen is a two-dimensional surface. So how can we represent a higher dimensional objects on a two-dimensional surface? And there are three traditional approaches. First, we might retain information, even if the drawing doesn't look right. We actually see this in medieval art. So pictures like this may look a little strange to us, but what's important here is the objects on the table can be seen and identified very easily. The other possibility is we might try to make it look right, even if we do lose information. So here, because the objects on the table are shown as they would appear, it's sometimes a little difficult to see what they are. And the third possibility is we can look at this a two-dimensional slice at a time. And so we get something like this sequence of CAT scan images of a brain. Now one way to get a better understanding of how to translate two-dimensional views into multi-dimensional objects goes back to 1885 when Edwin Abbott wrote Flatland, a mathematics-themed satire on Victorian society. Flatland describes the inhabitants of a two-dimensional plane. And it's important to remember that all objects in Flatland are on the same plane, and in three dimensions we might view these objects from above, but these look very different from a Flatlander's perspective. To understand this, we might look at it in the following way. We live in three dimensions, but when we see the world, our images of the world are in two dimensions. Similarly, Flatlanders live in a two-dimensional plane, so they would perceive their world in one dimension. And we can mimic this by imagining looking at objects through an infinitely narrow slit. So suppose our Flatlander walks around an object. They can infer what it looks like in two dimensions by putting together the different images. So if they walk around this object, the fact that the width of the object remains the same no matter which direction they look at it from says that this figure must be a circle. On the other hand, a figure whose width changes as you walk around it, well that could be some sort of triangle or some other polygon potentially. And well, this one's odd because it disappears completely from certain perspectives. And the only thing that can do that, well that must be some sort of line segment because if you look at it along an endpoint, it has no width. Now suppose our Flatlander tries to perceive a three-dimensional object. So suppose we look at the different layers of a three-dimensional shape through our infinitely narrow slit. So again, this is what the Flatlander would be seeing. We can try to infer details of the shape by what we see. For example, if as we look up and down the object the shape always has the same size, well that suggests an object of constant width and because we live in three dimensions, we know that an object of constant width might look like a cylinder. Or maybe the shape grows larger at a constant rate. And again, thinking about what this must look like as a three-dimensional object, this might be some sort of cone. Or maybe the shape grows larger, then smaller, and from our three-dimensional perspective, it might be a sphere. The important thing here is this gives us a way of visualizing objects in three or more dimensions. So we see the world through a two-dimensional window, but given different views we can try to reconstruct what a three-dimensional object looks like. And since we actually live in three dimensions, this is possible. We can view the object from three different angles. And because we know what three-dimensional objects look like, we can use our imagination to infer what the three-dimensional object actually is. What about four, five, or seventy-dimensional objects? Well, the idea is that we can attempt the same thing. We can view a three-dimensional slice and try to combine these slices together to view our n-dimensional objects. But here's the problem. Because we don't live in four or more dimensions, we can't really see these objects. But that doesn't really matter as long as we can work with them.