I can’t visualize them right now and I need someone to explain them to me simply
In: Mathematics
Specifically for the hypersphere:
Imagine a circle. It is essentially a closed 1d space. A line that closes in on itself. You can move in any direction on the line and you will end up where you started. All points are connected. It is the collection of points at an equal distance from a set center. You could call a circle a ‘2d sphere’.
A sphere is a closed 2d space. A surface that closes in on itself. You can go in any direction on the surface of a sphere and end up where you started. It is a collection of points at an equal distance from a set center, like a circle, but in 3d. You could call a sphere a ‘3d circle’. You can also imagine it as an infinite collection of same-sized circles with the same center, rotated in 3d space.
A hypersphere, then, is a closed 3d space. You can move in any direction on the three dimensional axes and you will end up in the same spot. It is also an infinite collection of spheres with the same center, rotated in 4d space.
We do not live in 4d space so we have a hard time imagining this kind of shape. We are very used to 3d shapes being represented in 2d, so much so that we hardly realise it. But these are optical illusions, like any hypersphere or tesseract video you’ve seen, but less obvious.
This video helps to make it clear, imho. French accent is an added bonus: https://youtu.be/RFK2_dAZvLo
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