A Mathematician's Guided Tour Through Higher Dimensions

At first glance, the notion of dimension seems intuitive. We might be able to see the crow perched on a flagpole with zero dimensions. A robin restrained to one on a telephone wire, a bird on the ground free to move in two, and an eagle flying in the air enjoying three dimensions.
Mathematicians have had a difficult time defining the concept of dimension, and pushing its boundaries. To arrive at the current, rigorous understanding of this concept, it took hundreds of years of imaginative comparisons and thought experiments.

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Ancient people knew that there were three dimensions to life. Aristotle said, "A magnitude is one that extends in one direction is a line. A plane that extends in two directions is a plane. A body that extends in three ways is a body." There is no other magnitude than these because there are only dimensions.

Mathematicians have, like many others, enjoyed the mental challenge of thinking about more dimensions. How would a fourth dimension, perpendicular to the three we have seen, look like?

One popular method is to imagine that our universe is a 2-D plane in 3-D space. We are not able to see a solid ball hovering above the plane. If it touches the plane and falls, however, a dot will appear. It continues to travel through the plane until it reaches its maximum size. Then it shrinks and disappears. These cross sections are what allow us to see three-dimensional shapes.

A plane's inhabitant would only see the cross-sections of three-dimensional objects. Illustration: Samuel Velasco/Quanta Magazine

Similar to the three-dimensional world, a four-dimensional ball passing through would appear as a point. It would then grow into a ball and shrink until it reaches its full radius. However, this gives us an idea of the four-dimensional shape. There are other ways to think about these figures.

Let's take an example: By building up, we can visualize the four-dimensional equivalent to a cube. To create a line segment, we can start with a point. We get a square when we sweep the segment in a perpendicular manner. This square can be dragged in a third direction perpendicular to form a cube. We can also get a tesseract if we sweep the cube in another direction.