I can’t visualize them right now and I need someone to explain them to me simply
In: Mathematics
A 1-sphere is a circle, a 2-sphere is the sphere in the usual vernacular.
There is a simple way to visualize a 3-sphere. What is it and how does it work?
Let’s start with a 2-sphere, like the surface of Earth. If you have ever seen the United Nations logo, then you have an idea what the [Azimuthal equidistant projection](https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection#/media/File:Azimuthal_equidistant_projection_SW.jpg) is. It’s not the Earth’s surface, but it’s its *map.*
As you would expect from a map, each point on the map corresponds to a real point on the Earth. Also the map preserves “nearness” in the sense that points that are near in real world are also near on the map, but with one crucial exception. The south pole does not appear on this map as one point, rather it’s the whole boundary circle.
To get the original sphere from the map, you would need to replace the bounding circle with a single point. Imagine cutting off this boundary circle and enclosing the remaining interior by gluing it to a point representing the south pole. You get the shape of a sphere.
Now, repeating this process by starting with a solid ball, being a *map* of a 3-sphere, where the boundary of this ball, the 2-sphere, represents the south pole, you can get a good idea what kind of shape a 3-sphere is.
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