Now we can use this XYZW coordinate system for something more fun: visualizing 4D shapes!

Let's start with some simple patterns. To make a 1-dimensional line, you take a point, make a copy that's moved along some direction, and connect the two.

To make a 2-dimensional square, you take a 1-dimensional line, make a copy moved along a second direction, and connect the two.

To make a 3-dimensional cube, you take a 2-dimensional square, make a copy moved along a third direction, and connect the two.
So what happens if we continue the pattern?

Let's continue the pattern into the 4th dimension: to make a 4-dimensional hypercube, also called a "tesseract", you take a 3-dimensional cube, make a copy in a fourth direction, and connect the two.
Looks cool, right?

(If it helps, you can think about a hypercube as two cubes with every point connected. I've faded out some of the edges to make it easier to see.)

This type of 4D visualization, where we move points a distance along some direction depending on their fourth coordinate, is called an orthographic projection.

Of course, there was nothing special about the random direction we chose to represent the fourth dimension. If we change the direction we chose, we get a different orthographic projection. The hypercube isn't moving in 4D space - we're only changing the the way we're projecting four dimensions to three dimensions.

The second common type of 4D ➔ 3D rule is called perspective projection. In our 3D world, if something is moving away from us, that thing looks like it's shrinking. Perspective projection tries to do the same thing in 4D: if a shape's fourth coordinate increases, the shape shrinks. Here's one way make this work using math: we divide the first three XYZ coordinates by the fourth coordinate.
Perspective projection is super popular for drawing pictures of 4D shapes. If you look up the word "hypercube", you'll probably see pictures that look like what you're seeing now: two connected cubes, one inside another. Don't forget that the "inner cube" and outer cube are the same size in 4D - the inner cube just has a bigger fourth coordinate, so when we visualize the hypercube with perspective projection, it shrinks.

One fun thing that we can do with four-dimensional shapes is rotate them! It's going to look very cool.

We'll use the same strategy as before: First, we'll think about rotations in 3D, then we'll try to follow the same pattern in 4D.

One way to describe a rotation in 3D is by choosing a 2D plane and rotating something in that plane. For example, if we rotate around, say, the XY plane, a rotation would move the X axis to where Y axis is and back again.

Let's take the same rotations we did on the cube, and rotate a hypercube in the same way, around the same planes of rotation. It looks mostly similar to the 3D case, but now the two cubes which are attached to form a hypercube look like they're intersecting one another as they rotate.
That's because every plane we can rotate around in 3D only involves the X,Y, and Z coordinates, not the W coordinate. This means that none of these rotations change the W coordinate of points in this hypercube. The points in the hypercube with a nonzero fourth coordinate form a 3D cube which rotates in the same way as the original 3D cube does, but moved in the direction of the w axis.
...so what happens if we rotate along a rotation plane involving the W axis?

If we rotate along a rotation plane which does include the w axis, such as the ZW plane (the plane formed by the z axis and w axis), we get this: a beautiful rotation. See if you can choose a cube inside the hypercube and follow it with your eyes as its fourth coordinate goes up and down, going in and out of the fourth dimension. I think it's pretty mesmerizing.

Here's the same hypercube and rotation, but visualized with perspective projection. WHOA OH JEEZ WHAT'S HAPPENING EVERYTHING'S GOING CRAZY
All this craziness is because we're dividing by zero. Remember how perspective projection works? To choose where to put a point in 3D, we divide the first three XYZ coordinates by the W coordinate. But this 4D rotation of our hypercube means the w coordinates of points in the hypercube are gradually changing from w=1 to w=0 to w=-1. When a point's w coordinate becomes really small, such as w=0.001, perspective projection means we need to divide each coordinate by 0.001 - which is the same as multiplying by 1000. Because dividing by small numbers is the same as multiplying by big numbers, the hypercube's points zoom off to infinity.

We can get rid of the distortion by moving the hypercube in the +w direction so that it doesn't pass through w=0 anymore. And now we get this amazing-looking rotation. Follow a cube with your eyes as it goes in and out - I think it's so fun to watch.

There are many other 4D shapes out there beyond hypercubes. One way to build 4D shapes is to take 2D and 3D shapes we're familiar with, and find a pattern which we can continue into 4D. For example, we took the pattern Line➔Square➔Cube, and invented a 4D hypercube by continuing the pattern. We can do this with other shapes, too!
For example, one of the simplest 2D shapes is a triangle. In 2D, an (equilateral) triangle has 3 points, each the same distance apart from all the others. In 3D, a tetrahedron has 4 points, each the same distance apart from all the others. Can we extend this pattern to 4D?

If we continue the pattern, we get a new 4D shape: a hypertetrahedron, which has 5 points, each the same distance apart from all the others.

Just like before, we can rotate these shapes too. Here's what a 4D rotation of a hypertetrahedron looks like in orthographic projection. I think it's kind of relaxing.

And here's what that same 4D rotation of a hypertetrahedron looks like in perspective projection. The same distortion from before appears whenever a point's w coordinate passes near w=0.
Finally, there's one more shape I want to talk about. Remember the 3-torus from before? Now that we know how to visualize things in four dimensions, we can try visualizing it! (It might lag your browser, however, so only advance to the next slide if your device is powerful enough.)

Here's what a 3-torus looks like in perspective projection. You can see multiple copies of the normal 2-torus we're used to inside it. It's a big tangle.

This, to me, shows why studying manifolds is so interesting - you don't need to work with this ugly mess. Every point on the surface of that 3-torus corresponds exactly to a point in that cube-shaped 3D coordinate system we saw earlier. Instead of imperfect 4D visualizations, manifolds let us use more familiar Cartesian coordinate systems to study spaces we can't imagine.

Thanks for reading!
If you want to explore one more weird thing about knots in four dimensions, click here!