###### The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

Glancing out the window we might see a crow sitting atop a cramped flagpole experiencing zero dimensions, a robin on a telephone wire constrained to one, a pigeon on the ground free to move in two and an eagle in the air enjoying three. But as we’ll see, finding an explicit definition for the concept of dimension and pushing its boundaries has proved exceptionally difficult for mathematicians. It’s taken hundreds of years of thought experiments and imaginative comparisons to arrive at our current rigorous understanding of the concept. The ancients knew that we live in three dimensions. Aristotle wrote, “Of magnitude that which (extends) one way is a line, that which (extends) two ways is a plane, and that which (extends) three ways a body. And there is no magnitude besides these, because the dimensions are all that there are.”

Yet mathematicians, among others, have enjoyed the mental exercise of imagining more dimensions. What would a fourth dimension — somehow perpendicular to our three — look like? One popular approach: Suppose our knowable universe is a two-dimensional plane in three-dimensional space. A solid ball hovering above the plane is invisible to us. But if it falls and contacts the plane, a dot appears. As it continues through the plane, a circular disk grows until it reaches its maximum size. It then shrinks and disappears. It is through these cross sections that we see three dimensional shapes.

Similarly, in our familiar three-dimensional universe, if a four-dimensional ball were to pass through it would appear as a point, grow into a solid ball, eventually reach its full radius, then shrink and disappear. This gives us a sense of the four-dimensional shape, but there are other ways of thinking about such figures. For example, let’s try visualizing the four-dimensional equivalent of a cube, known as a tesseract, by building up to it. If we begin with a point, we can sweep it in one direction to obtain a line segment. When we sweep the segment in a perpendicular direction, we obtain a square. Dragging this square in a third perpendicular direction yields a cube. Likewise, we obtain a tesseract by sweeping the cube in a fourth direction.