Some people say it’s impossible to visualize 4-dimensional things. But lots of people I know can do it.
How do they do it?
Well, how do you visualize 3-dimensional things? A computer screen is just 2d, but we can draw a 3d object on it by picking some diagonal direction—southeast in the picture below—to stand for ‘forwards, towards our eyes’. Similarly, we can draw a 4d object by picking another diagonal direction—northeast in this picture—to stand for ‘forwards in the 4th dimension’.
Here we are using this trick to draw 0d, 1d, 2d, 3d and 4d cubes. The first dimension, often called the x direction, points along the red arrow. The second, called the y direction, points along the green arrow. The third, the z direction, points diagonally along the blue arrow. And the fourth, sometimes called the w direction, points diagonally along the yellow arrow.
There’s nothing sacred about these names or these particular directions; we can implement this idea in lots of different ways. It’s ‘cheating’, but that’s okay. A vague, misty image can be a lot better than no image at all.
Of course, we need to think about math to keep straight which lines in our picture point in the w direction. But that’s okay too. A mixture of equations and visualization lets mathematicians and physicists make faster progress in understanding the 4th dimension than if they used only equations.
Physicists need to understand the 4th dimension because we live in 4d spacetime. Some mathematicians study much higher-dimensional spaces, even infinite-dimensional ones, and here visualization becomes more subtle, and perhaps more limited. But many mathematicians working on 4d topology take visualization very seriously. If you’ve ever seen the elaborate, detailed gestures they make when describing 4-dimensional constructions, you’ll know what I mean.
Visualizing shapes in 4 dimensions takes practice, but it’s lots of fun! As this series of posts continues, I want to give you some practice while talking about nice 4-dimensional shapes: the 4d relatives of the Platonic and Archimedean solids. In the series so far, we’ve spent a lot of time warming up by studying the 3d Platonic and Archimedean solids and developing a technique for classifying them, called Coxeter diagrams. All that will pay off soon!
Here’s a great series of videos that explains higher dimensions:
• Jos Leys, Étienne Ghys and Aurélien Alvarez, Dimensions.
The only problem is that it’s tough to navigate them. Click on your favorite language and you’ll see part 1 of the series. After you start playing it you’ll see an arrow at the lower right of the video that lets you jump to the next one. This is good if, like me, you’re impatient for the 4th dimension! That starts in part 3.
There’s a guide to all nine parts here:
• Jos Leys, Étienne Ghys and Aurélien Alvarez, Tour.
but you can’t get to the videos from there! They need a bit of help from a good website designer.
The picture above is a shot of the glorious 120-cell… one of the six Platonic solids in 4 dimensions. But more on that later! We’ll start with a simpler one: the 4-cube.