Topology of the 4th Dimension
I came across this fascinating video on YouTube, and wanted to share it with you.
7/9/20261 min read
A couple of days ago I came across this video on YouTube. Maggie Miller, a mathematician, talks about dimensional manifolds. Her explanations were excellent. She explained how a 1-dimensional manifold could be a circle, and a 2-dimensional manifold could be a torus. (I happen to love the torus). To get the 1-dimensional manifold she starts with a string, and connects both ends of it. To get the 2-dimensional manifold she starts with a square and connects the square in such a way so that if you go off the right side, you come back in on the left, and if you go off the top, you come back in on the bottom. That gives the shape of a torus. She then begins to describe a 3-dimensional manifold. She starts with a cube. So then in my mind, I really liked taking that to the next step. To have the right face of the cube connect with the left face of the cube, you will end up with something like a torus, except for it has a flat top and bottom. The next step would be to stretch it out so that the top connects to the bottom. Well, from the outside it then looks like a larger torus, and there would no longer be any flat sides. But the torus would be hollow on the inside. We can't, in 3-dimensional space, see what the next step looks like though, because the inside of the hollow torus would have to connect with the outside of the torus. Thus, that would happen in 4-dimensional space. What a fun thought experiment though! It reminds me of the time I played with a hypercube in VR. Just like we can draw 3-dimensional items on a 2-dimensional surface, in 3-dimensional VR space, I was able to play with, and observe, the 4-dimensional cube.
Anyhow, I wanted to share this video with you in case you might find it interesting too!
