Guided tour of a mathematician through higher dimensions


The notion of dimension seems at first glance intuitive. Looking out the window we could see a crow sitting on top of a cramped flag pole in zero dimensions, a robin on a telephone wire forced into a, a ground pigeon free to roam in two and an eagle in the air taking advantage of three.

But as we will see, finding an explicit definition of the concept of dimension and pushing its limits has proven to be exceptionally difficult for mathematicians. It took 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: “In magnitude that which (extends) a way is a line, that which (extends) two ways is a plane, and that which (extends) three ways is a body. And there is no greatness outside of these, because dimensions are all there is.

Yet mathematicians, among others, have enjoyed the mental exercise of imagining more dimensions. What would a fourth dimension – sort of perpendicular to our three – look like?

A 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 makes contact with the plane, a point appears. As it crosses the plane, a circular disc grows until it reaches its maximum size. It then shrinks and disappears. It is through these cross sections that we see three-dimensional shapes.

An inhabitant of an airplane would only see cross sections of three-dimensional objects.Illustration: Samuel Velasco / Quanta Magazine

Likewise, in our familiar three-dimensional universe, if a four-dimensional ball passed through it, it would appear as a spike, turn into a solid ball, eventually reach its full radius, then shrink and disappear. This gives us an idea of ​​the four-dimensional shape, but there are other ways to think about such figures.

For example, let’s try to visualize the four-dimensional equivalent of a cube, known as a tesseract, by building it. If we start with a point, we can sweep it in one direction to get a line segment. When we sweep the segment in a perpendicular direction, we get a square. Dragging this square in a third perpendicular direction results in a cube. Likewise, a tesseract is obtained by sweeping the cube in a fourth direction.

By scanning blue shapes to violets, we can visualize cubes of different sizes, including a tesseract.

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