In 3d we’re used to rotations *around an axis*, but we need to break that habit when we start thinking about other dimensions. More fundamental is rotation *in a plane*.

What do I mean by ‘rotation in a plane’?

This is clearest in 2d space, which is nothing but a plane. Put your finger on a piece of paper and spin the paper around. That’s what I’m talking about!

Or look at this:

This picture lies in a plane. It’s rotating, but not *around an axis* in that plane.

Of course we can imagine an axis at right angles to that plane, and say the picture is rotating around that axis… but that requires introducing an extra dimension, so it’s artificial. It’s unnecessary. Worst of all, it’s misleading if you’re trying to think about how rotations work in different dimensions.

Here’s the general fact that works in any dimension of space. You can always take any rotation and break it into separate rotations in different planes that are at right angles to each other.

For example, consider 3d space. You can pick any 2d plane and get the points in that plane to rotate. But now there’s a 1d line at right angles to that plane that doesn’t move. Note the numbers here:

Next consider 4d space. Now you can pick any 2d plane and get the points in that plane to rotate. This leaves another 2d plane at right angles to the rotating one. Note the numbers:

This movie shows a 4-cube rotating in a single plane in 4d space:

We aren’t drawing the plane of rotation, just the cube. And the true 4d picture has been squashed down to a plane using perspective! So, it’s a bit hard to understand. But the rotation is taking place in the plane that contains the ‘left-right’ axis and the ‘in-out’ axis. The ‘up-down’ axis and the ‘front-back’ axis are left unmoved.

But we can also do something fancier! We can break 4d space into a rotating 2d plane and, at right angles to that, another rotating plane. This is what’s happening here:

The two rotations just happen to be going on at the same speed—otherwise it would look a lot more complicated.

The story continues in 5 dimensions. Now the equation that matters is this:

That means we can pick any 2d plane, get that to rotate, then pick another 2d plane at right angles to the first one, get that to rotate… and we’re left with a 1d line of points that don’t rotate.

So, rotations are very different depending on whether the dimension of space is odd or even! In odd-dimensional space, any rotation must leave some line unmoved. In even-dimensional space, that’s not true. This turns out to have huge consequences in math and physics. Even and odd dimensions work differently.

For more on rotations in 4d space, see:

• Rotations in 4-dimensional Euclidean space, Wikipedia.

Also check out this:

• Plane of rotation, Wikipedia.

I’ve shown you movies of rotating 4-cubes where the planes of rotation are neatly lined up with the axes of the 4-cube. For a detailed study of more tricky ways that 4-cube can rotate, read this:

• Greg Egan, Hypercube: mathematical details.

Then you’ll understand pictures like these:

### Image credits

The rotating 4-cubes were created by Jason Hise using Maya and Macromedia Fireworks. He put them into the public domain, and they reside on Wikicommons here and here.

The rotating yin-yang symbol was created by Nevit Dilmen.

The other pictures were created by Greg Egan.

Hi John, I think you have a typo! In the line below where you write

5 = 2+2+1

and then write “That means we can pick any 3d plane, get that to rotate …”. But I suspect that should say “That means we can pick any 2d plane, get that to rotate …”.

Yes—fixed!

Fascinating series, and great blog post, John! I was wondering whether you meant in the following sentence discussing rotation in 5 dimensions, instead of :

“That means we can pick any 3d plane, get that to rotate, then pick another 2d plane at right angles to the first one, get that to rotate… and we’re left with a 1d line of points that don’t rotate.”,

“”That means we can pick any [2d plane], get that to rotate, then pick another 2d plane at right angles to the first one, get that to rotate… and we’re left with a 1d line of points that don’t rotate.” ?

The correction is in brackets in the paragraph below your original version. So, I think you meant 2d plane instead of 3d plane. Otherwise, there’s a lot of stuff packed in this blog post–excellent! Thank you!

Yes, thanks for catching this: I meant to write “2d plane”, not “3d plane”. My finger must have slipped.

I’m resisting saying a lot

moreabout rotations in different dimensions, because this is a huge subject in its own right….Understood, John, and look forward to the continuation of this series–it is utterly fascinating and helpful, indeed! :)

Minor typo under 5 = 2+2+1. We are just picking 2d planes to rotate.

This is a fun series. :-)

You can download a math vis toy for Mac or Windows to play with this at http://www.PacificT.com/FreeStuff.html

The documents under Four Dimensions in the examples menu are functions of a complex variable graphed as (u+iv)=f(x+iy) and projected to 3D. The controls on the right rotate the surface through rotations in the xy-, xu-, yu-, xv-, yv- and uv-planes. The white grid shows the xy plane at u=v=0.

Going from simple multi-hyperplane 4D objects to 8D E8, we have

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