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.
Azimuth 
I watched all nine of these excellent videos by Jos Leys, Étienne Ghys and Aurélien Alvarez. The classification of polyhedral and their generalizations as dimensional polytopes is a more recent and surprising interest to me. I greatly appreciate your efforts.
About the videos, can the publicity be avoided? I felt taken hostage: “Please bear with our 30-second nonsense, or else, you’ll get 90 seconds of the same.”
I don’t know what you’re talking about… it’s been a long time since I watched those videos.
John, I went to the video, ‘Dimensions’. I have been looking for something like this for 20 years. This is beyond fantastic. Thank you so much for the link to this beautiful presentation.
You’re welcome—glad you like it! It’s pretty cool.
“They need a bit of help from a good website designer.”
Heh.
I read this right after wishing, for the tenth time, that there were some easy way to get from one of these “Symmetry and the Fourth Dimension” posts to its successor.
Most of the posts (all but the first?) have back links, but I’m trying to read through the series, so after each one I have to Google for the next. Just recalling the URL and changing part-10 to part-11 won’t get me there because the URL embeds the post date.
This is a nuisance on my tablet.
I suspect you can’t predict when the next post in the series will be, since it’s a blog and you are writing about other things, too. One alternative to going back and putting in forward-links is to make the articles less interesting, so we don’t want to read the next ones.
Nice series. Thanks.
Indeed, I can’t predict when the next post in the series will be—but I don’t want to make the articles less interesting. In many of the articles I start by mentioning the last one, creating a back link to a previous post—and that back link appears in the comments on that previous post, so you can also use it as a forwards link! However, I see that part 11 has no back link to this one.
In the long run I plan to take this series of blog posts and make it into a well-organized bunch of webpages on my website. I’ve already done this with other series of blog posts, like the network theory series, the information geometry series, the rolling circles and balls series, the game theory series and the integral octonions series. But I’m a lazy bum, so I haven’t gotten around to doing it with this one yet.
I will probably write some new posts in this series when my fall classes end in mid-December. I’ll be on sabbatical from January to June, so I’ll have some more time for it then, though I also plan to finish up a bunch of papers and travel around Europe.
[…] (picture taken from: Azimuth) […]
Separate cells of the 5-cell (pentachoron).
Separate cells of the 8-cell (tesseract).
Separate cells of the 16-cell (hexadecachoron).
Separate cells of the 24-cell (icositetrachoron).
I’m new to dimensions, I thought anything times 0 is 0.