a2.4.0 (feature) - Clifford Torus

Added a new four-dimensional object, the Clifford torus (Wikipedia/Polytope Wiki)!

You can play around with it in the new scene FixedCliffordTorus! But note that especially in the WebGL build it is quite slow, which is also the reason why I have decided against adding it to the normal, non-fixed scenes.

I hear you ask, what is a clifford torus? Imagine taking a circle in the XY plane. Now, at every point of the circle, add a perpendicular line pointing in the Z axis, essentially extruding this circle out in the third dimension. You get what looks like a wireframe of a cylinder. In the FixedCliffordTorus scene, if you have the scene slider to the left, you can rotate this wireframe of a cylinder essentially in 3D space using the YW and ZW sliders.

This cylinder has an interesting property. If you moved along a normal square, you will notice that you kind of have “borders”; within that square you can only go left, right, or upwards and downwards until you reach the edge of the square. But for a cylinder it’s a little different, while you will still reach an edge if you move up/down, moving left and right will cause you to wrap back around infinitely!

This “wrapping back around” is what a lot of videogames do as well, like for example the retro game Asteroids, which, when you move around, you will notice that you wrap back around, meaning that you can move right/left/up/down infinitely even though the game is not open-world.

This is cool and all, but remember that our cylinder only had that property for right/left and not up/down. So, what object would have that property? Imagine taking our wireframe cylinder again. Now, imagine you take every vertical line (that is perpendicular to the circle you started with) and radially extruding it outwards to turn it into circles. If you do this in three-dimensional space it gives you a wireframe of a torus. It looks like a donut :)

This basically has the property we want! If we move along the surface of this torus, we can freely move right/left/up/down and wrap back around. However, you will notice one slight issue: If you move around the inner ring of this torus you will wrap back around faster than if you moved around the outer ring of that torus. This is not what we want. So, what can we do about it? Well, in three-dimensional space, we’re stuck.

However, this is where the Clifford torus comes in. Imagine taking our three-dimensional cylinder from previously again. Now, we will do a similar line-to-circle extruding like previously; however, this time around, we will extrude the lines into the fourth dimension. Now, these newly created circles exist in a completely different plane (ZW) to the circle we started with (XY). Essentially, we took a circle, and for every point on this circle added another circle that exists in a different plane to our original. What we get is a Clifford torus, which, when projected, looks a little bit similar to a three-dimensional torus but with a volume in four-dimensional space. This object is, in other words, the simplest and most symmetric flat embedding of the Cartesian product of two circles (yes, i stole that sentence from wikipedia).

This object has the property we want! It still has a two-dimensional surface, however since the circles exist in completely independent planes, we don’t have the issue of moving along the inner ring being shorter than moving along the outer one anymore.

Note that this is different to a 3-torus, a torus with a three-dimensional “surface”. A 3-torus could exist in 4D space, but just like how a 2-torus (torus with two-dimensonal surface) requires a “stretched” surface (as you see can see here) in 3D, a 3-torus would require a “stretched” surface in 4D; you’d need 6D space to show the product of three circles without any stretching, since each have to exist in independent planes. A Clifford torus still just has a 2D surface like a normal torus we’re used to, however it can exist without the issue of the inner ring having different radius to the outer one. It’s kind of like taking a flat, 2D piece of paper and folding it in a special way in 4D space.

Note that this is an explanation coming from someone who has no background in mathematics and has even less idea about commonly used terminology, so take the things I’ve said here with a grain of salt. It makes sense in my head, though reading about what a torus even is, I realized that I only understand, like, 80% of the terms used.

In the FixedCliffordTorus scene, you first see a cylinder, and using the slider on the bottom you can turn all the vertical lines of that cylinder into circles to create a Clifford torus. It’s fun to watch it spin around!