The notion of dimension at first seems intuitive. 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?

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