flat

What’s excellent about Sagan’s explanation is that he uses simple and relatable objects: an apple and a Tesseract, or a hypercube.

Sagan began by discussing how a two-dimensional being living in a flat world would perceive a three-dimensional object like an apple.

“Imagine we live in this ‘flatland’/2-D plane with no concept of ‘up’ or ‘down.’ Then along comes a 3-D object like an apple. We do not even notice it until it crosses our plane of existence — and even then, we have no idea what the apple is,” Sagan explains. “We see only a fragment as it passes through our plane. There is no way we can comprehend the 3-D quality/dimension of the apple, because it is more than we can understand. We only have the evidence of what has passed through our plane.”

Sagan then related this two-dimensional experience of the third dimension to how we might try to understand the fourth. To do so, he used the Tesseract, a four-dimensional cube, to demonstrate how difficult it is for us to perceive or visualize dimensions beyond our own three. At this point, Sagan is asking the viewer to expand their minds to understand the fourth dimension metaphorically.

Sagan’s demonstration of the fourth dimension isn’t just a wonderful explanation of a scientific idea that many of us find difficult to comprehend; it’s also a great example of how to teach complex ideas by combining clear explanations with thought-provoking visuals.