If you have taken any math classes, which I assume you have, you are most likely familiar with the basics of geometry. The usual stuff, like the area of a square always being equal to the square of its length, or a triangle always having an area that is half of its base multiplied by its width. Going further, you could even go into the third dimension. Give everything a length, and find new and interesting shapes. Adding more sides, and then stellating those. But does it end here? It may seem so at first. But, in fact, it does not. Now, I will introduce you to objects that break this barrier, and even become impossible ones. The first one on our list, to the contrary of what I just said, you can actually make at home. All you need is a strip of paper and a single piece of tape. First, take one end of the paper in one hand, and the other end in the other hand. Rotate one of the ends 180 degrees in any direction, and then bring the two ends together without changing the orientation. Take the piece of tape, and tape the ends of paper together.
This is called a Möbius strip, and it has a very special property: it only has one edge and one side. Glancing at it, this seems wrong, as you can see that the paper has two sides, but try this: If you take a pen and draw along the center of the strip, continuously, you will find that the pen comes back to exactly where you started. You didn’t lift the pen off the paper; you kept it on the same side, the only side. Because of these properties, the Möbius strip is a very important part of three dimensional geometry. Wait. Three dimensions? That seems rather limiting. Let’s try four! A tesseract, otherwise known as a hypercube or a 4-cube, is a four dimensional cube. That doesn’t make sense though, does it? Think of it like this: a 0D shape has a single vertex, and a 1D line has two vertices and one edge. A 2D square has four vertices, two edges, and one face. A 3D shape has eight vertices, 12 edges, six faces, and one cube.
So, a tesseract, following the pattern, has 16 vertices, which is double the previous amount. It has 32 edges, which is double the previous amount plus the previous amount of vertices. It has 24 faces, which is double the previous plus the previous of edges. It has eight cubes, which is double the previous plus the previous of faces. And finally, it has one tesseract, which once again is double the previous plus the previous number of cubes.
The only way to even try to visualize this somewhat properly is to use time as a fourth dimension. But there is still one more object to talk about, and it is called a Klein bottle. It is like a Möbius strip, as it only has one side, and thus, no inside. If you were to look it up online, you will see 3D versions people have made. But these versions are all wrong. Every one of those models bisects themselves, which ruins the whole point. Again, like the Möbius strip, this special bottle uses the dimension above its original form to achieve a shape otherwise impossible. You could never, ever, fill one.