A Riemann sphere or complex projective line is a complex 1-dimensional (1D) manifold that consists of the complex line plus a point at ∞. Points on its surface represent extended complex numbers, the points near ∞ are numbers with larger absolute values, the points near ∞'s antipode 0 are numbers with smaller absolute values, and points on the equator are numbers with an absolute value of 1, numbers that satisfy where is a real number.
The extended complex line projected on the sphere is very useful to complex analysis because they allow division by zero to be well-behaved.
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Real dimensionality | 0 | 1 | 2 | 3 | ... |
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Real space
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Point
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Real line
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Real plane
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Real realm
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Real projective space
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Point pair
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Real projective line
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Real projective plane
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Real projective realm
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Complex space
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Point
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Complex line
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Complex plane
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Complex realm
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Complex projective space
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Point pair
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Complex projective line
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Complex projective plane
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Complex projective realm
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Quaternionic space
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Point
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Quaternionic line
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Quaternionic plane
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Quaternionic realm
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... |
Quaternionic projective space
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Point pair
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Quaternionic projective line
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Quaternionic projective plane
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Quaternionic projective realm
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Octonionic space
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Point
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Octonionic line
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Octonionic plane
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Octonionic realm
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Octonionic projective space
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Point pair
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Octonionic projective line
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Octonionic projective plane
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Octonionic projective realm
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