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I am really struggling with maths at the moment. People are telling me a sphere is just as abstract as a Klein bottle.

Im not comfortable with the way I think about math anymore, I think I am confusing it a bit with reality and physics.

For example, I watched “thinking outside the 10-d box” by 3blue1brown and it’s just not sitting with me. I don’t understand the extensions of Pythagorean distance to higher dimensions. If you claim the spheres act differently in higher dimensions why define fundamental properties about distance using a 3 dimensional framework?

Any books that teach me how to think about it proper and discuss its abstract mess might just save me. Otherwise I’m going to have to end my time studying the subject. Thank you.

Hasan Zaeem
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    Flatland is a classic. – John Douma Sep 16 '23 at 07:20
  • What also will contribute to "save" you is to avoid online math videos , they are at best entertaining but useless to make progress. I know what I am talking about , for the sake of curiousity I watched many , really many of them. None convinced me concerning teaching , the best I remember was a journey through the Mandelbrod-set with fascinating pictures. – Peter Sep 16 '23 at 07:46
  • I have read flatland but I am still not comfortable. I want rigorous mathematics and a mathematician simply explaining their perception. I know I’m probably looking for a lost cause but it’s really the only thing that can save me from giving up. – Hasan Zaeem Sep 16 '23 at 08:02
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    "a sphere is just as abstract as a Klein bottle." - I disagree – Peter Sep 16 '23 at 08:03
  • Mathematically , it is no problem to generalize the Pythagorean theorem to higher dimensions. But visualization is only possible for the cases of two and three dimensions. So my question : Are you rather interested in the links between mathematics and physics ? Or just how we can describe things (like a 4-dimension hypercube) which do not exist in reality ? – Peter Sep 16 '23 at 08:06
  • Hi Peter, all I want is some kind of text that teaches me how I should be interpreting higher dimensional space purely in terms of mathematics. When I think of a hypersphere do I think of the 16 vertices in R4. I want something that says to me there is no visualisation at this point. It talks about what the hell im actually doing and discusses the abstraction of it, compared with the abstraction of say a sphere – Hasan Zaeem Sep 16 '23 at 08:17
  • You want a document showing you to "think well/have good representations". Otherwise said, you want advices whether either 1) such and such concept persist, is still a good blindstick, can be maintained for $n > 3$, 2) or 2) doesn't persist. Examples of 1) : the notion of vertices, edges, faces generalized with hyperfaces (what about the generalization of Euler relationship F-E+V=2 for a polyhedron ?). (Counter-)examples for 2) : the notion of interior and exterior which is meaningless for example for an hypercube. – Jean Marie Sep 16 '23 at 08:25
  • This kind of question has already been asked in different places. See for example here or there. – Jean Marie Sep 16 '23 at 08:33
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    My answer to Where can I start learning about higher dimensions in mathematics? lists a lot of references, many of which could be read by high school students (e.g. The Fourth Dimension Simply Explained edited by Henry Parker Manning). – Dave L. Renfro Sep 16 '23 at 13:30

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