How to visualize the fourth dimension

Is it possible to visualize four dimensions in your head?

Because we exist in a universe that has only three dimensions, you might assume that the answer is no.

However, it turns out that you have a lot more experience visualizing four dimensions than you might think.

Every day, not only do you imagine where things are in three dimensions, but you also imagine when they’re happening.

Time is obviously different from the three dimensions of space, but you can treat time as an extra dimension and use it to visualize and solve four-dimensional geometry problems in your head.

It’s really true, and it’s perfectly legal.

The Problem

Consider the following four-dimensional geometry problem.

In three dimensions, it’s pretty easy to imagine what the intersection of two planes looks like.

In three dimensions, two planes intersect at a line.

So, the question is, in four dimensions, what do two intersecting planes look like?

Is it possible to visualize the solution to this problem in your head?

The Solution

Yes it is. To do it, treat time as a fourth dimension of space.

Consider each of the two planes separately.

Think of the first plane as a line, dragged through time over the span of a five seconds, creating a plane.

Now think of the second plane as a plane that pops into existence at time 3, just for an instant, in three dimensional space, and then disappears.

Now consider what’s happening with the first plane during this single instant.

This is the only instant in time in which both of the planes exist. What do they look like then?

The first plane, as the line dragged through time, looks like a line, and the second plane looks like a plane.

And when a line and a plane intersect in three dimensions, which is all that exists in this instant, the intersection is a point.

So that’s the answer. In four dimensions of space, two planes intersect at a point.

Tada.

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More 4D Visualization

Thanks for reading. While you’re here, please check out the app I made that lets you understand what rotating objects look like in four dimensions by playing with a tesseract.