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What is the geometric intuition of projective plane and space?

I can understand affine plane and 3 dimension affine space, for higher dimension, at least I can imagine it similarly as the 2,3 dimensional case.

Please feel freely on commenting and helping me. Thank for reading my question.

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    Look at [this answer](http://math.stackexchange.com/a/13767/742).2012-02-22
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    In "[How not to be wrong](http://www.jordanellenberg.com/how-not-to-be-wrong/)", Ellenberg gives a beautiful description of the projective plane: https://books.google.com/books?id=2k9KAgAAQBAJ&pg=PT190 .2016-06-05

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