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Maxime Lucas
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The mathematical answer to that question is that you can define a manifold without embedding it into an n-dimensional vector-space. But this may be a bit too mathematical, so I will elaborate.

The usual example is to look at the circle: if you live inside the circle, you can go forward (or backward) as much as you like and you will never find a boundary. You will tell me: "Yes, but when I look at the circle from the outside, I can see that there are boundaries."

The reason why you think that is that you are used to see every object inside our usual 3-dimensional world. But mathematically speaking, there is nothing stopping a ring to exist "on its own", without a 3-dimensional space around it.

Let's go up by one dimension. Let's say you live on a sphere. You can go as far as you want forward, backward, left or right. Once again there are no boundaries. The boundaries only exist if you try to embed the sphere into our usual 3-dimensional world. [Note that you can replace the sphere with a more complicated example (like a torus, or the surface of the Klein bottle) if you like.]

What happens if I add again an other dimension? Well now you have what is called a 3-sphere. If you go in the same direction long enough you will end up in the same spot. If you have ever played Portal, you can imagine what it is like: get a big room, place portals on the floor, on the ceiling and on every wall and you obtain almost the same thing. Why do I say almost? Well by doing so you are using the fact that there are walls. However, just like in the sphere or the circle example, mathematically speaking you can define such a geometry without any reference to an outside world.

About the distance problem: It can be confusing to talk about distance in such a geometry ("I walked 200km and I end up in the same spot. How can I be both 200km away and in the same spot?"). But if you think about it, we already are doing this every day. Indeed we are living on a sphere: Earth. The good thing is that the Earth is big enough that as long as we don't move much, it doesn't really matters if the Earth is round or flat, and distances make sense. However if you walk far enough (ie. 45 000 km) you do end up in the same place.

The mathematical answer to that question is that you can define a manifold without embedding it into an n-dimensional vector-space. But this may be a bit too mathematical, so I will elaborate.

The usual example is to look at the circle: if you live inside the circle, you can go forward (or backward) as much as you like and you will never find a boundary. You will tell me: "Yes, but when I look at the circle from the outside, I can see that there are boundaries."

The reason why you think that is that you are used to see every object inside our usual 3-dimensional world. But mathematically speaking, there is nothing stopping a ring to exist "on its own", without a 3-dimensional space around it.

Let's go up by one dimension. Let's say you live on a sphere. You can go as far as you want forward, backward, left or right. Once again there are no boundaries. The boundaries only exist if you try to embed the sphere into our usual 3-dimensional world. [Note that you can replace the sphere with a more complicated example (like the surface of the Klein bottle) if you like.]

What happens if I add again an other dimension? Well now you have what is called a 3-sphere. If you go in the same direction long enough you will end up in the same spot. If you have ever played Portal, you can imagine what it is like: get a big room, place portals on the floor, on the ceiling and on every wall and you obtain almost the same thing. Why do I say almost? Well by doing so you are using the fact that there are walls. However, just like in the sphere or the circle example, mathematically speaking you can define such a geometry without any reference to an outside world.

About the distance problem: It can be confusing to talk about distance in such a geometry ("I walked 200km and I end up in the same spot. How can I be both 200km away and in the same spot?"). But if you think about it, we already are doing this every day. Indeed we are living on a sphere: Earth. The good thing is that the Earth is big enough that as long as we don't move much, it doesn't really matters if the Earth is round or flat, and distances make sense. However if you walk far enough (ie. 45 000 km) you do end up in the same place.

The mathematical answer to that question is that you can define a manifold without embedding it into an n-dimensional vector-space. But this may be a bit too mathematical, so I will elaborate.

The usual example is to look at the circle: if you live inside the circle, you can go forward (or backward) as much as you like and you will never find a boundary. You will tell me: "Yes, but when I look at the circle from the outside, I can see that there are boundaries."

The reason why you think that is that you are used to see every object inside our usual 3-dimensional world. But mathematically speaking, there is nothing stopping a ring to exist "on its own", without a 3-dimensional space around it.

Let's go up by one dimension. Let's say you live on a sphere. You can go as far as you want forward, backward, left or right. Once again there are no boundaries. The boundaries only exist if you try to embed the sphere into our usual 3-dimensional world. [Note that you can replace the sphere with a more complicated example (like a torus, or the surface of the Klein bottle) if you like.]

What happens if I add again an other dimension? Well now you have what is called a 3-sphere. If you go in the same direction long enough you will end up in the same spot. If you have ever played Portal, you can imagine what it is like: get a big room, place portals on the floor, on the ceiling and on every wall and you obtain almost the same thing. Why do I say almost? Well by doing so you are using the fact that there are walls. However, just like in the sphere or the circle example, mathematically speaking you can define such a geometry without any reference to an outside world.

About the distance problem: It can be confusing to talk about distance in such a geometry ("I walked 200km and I end up in the same spot. How can I be both 200km away and in the same spot?"). But if you think about it, we already are doing this every day. Indeed we are living on a sphere: Earth. The good thing is that the Earth is big enough that as long as we don't move much, it doesn't really matters if the Earth is round or flat, and distances make sense. However if you walk far enough (ie. 45 000 km) you do end up in the same place.

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Maxime Lucas
  • 2.1k
  • 1
  • 14
  • 20

The mathematical answer to that question is that you can define a manifold without embedding it into an n-dimensional vector-space. But this may be a bit too mathematical, so I will elaborate.

The usual example is to look at the circle: if you live inside the circle, you can go forward (or backward) as much as you like and you will never find a boundary. You will tell me: "Yes, but when I look at the circle from the outside, I can see that there are boundaries."

The reason why you think that is that you are used to see every object inside our usual 3-dimensional world. But mathematically speaking, there is nothing stopping a ring to exist "on its own", without a 3-dimensional space around it.

Let's go up by one dimension. Let's say you live on a sphere. You can go as far as you want forward, backward, left or right. Once again there are no boundaries. The boundaries only exist if you try to embed the sphere into our usual 3-dimensional world. [Note that you can replace the sphere with a more complicated example (like the surface of the Klein bottle) if you like.]

What happens if I add again an other dimension? Well now you have what is called a 3-sphere. If you go in the same direction long enough you will end up in the same spot. If you have ever played Portal, you can imagine what it is like: get a big room, place portals on the floor, on the ceiling and on every wall and you obtain almost the same thing. Why do I say almost? Well by doing so you are using the fact that there are walls. However, just like in the sphere or the circle example, mathematically speaking you can define such a geometry without any reference to an outside world.

About the distance problem: It can be confusing to talk about distance in such a geometry ("I walked 200km and I end up in the same spot. How can I be both 200km away and in the same spot?"). But if you think about it, we already are doing this every day. Indeed we are living on a sphere: Earth. The good thing is that the Earth is big enough that as long as we don't move much, it doesn't really matters if the Earth is round or flat, and distances make sense. However if you walk far enough (ie. 45 000 km) you do end up in the same place.