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Inversive geometry
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=== Higher geometry === As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated β or 1/0 . In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the [[complex projective line]], often called the [[Riemann sphere]]. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by [[Eugenio Beltrami|Beltrami]], [[Arthur Cayley|Cayley]], and [[Felix Klein|Klein]]. Thus inversive geometry includes the ideas originated by [[Nikolai Lobachevsky|Lobachevsky]] and [[Bolyai]] in their plane geometry. Furthermore, [[Felix Klein]] was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the [[Erlangen program]], in 1872. Since then many mathematicians reserve the term ''geometry'' for a [[space (mathematics)|space]] together with a [[group (mathematics)|group]] of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group. For example, Smogorzhevsky<ref>A.S. Smogorzhevsky (1982) ''Lobachevskian Geometry'', [[Mir Publishers]], Moscow</ref> develops several theorems of inversive geometry before beginning Lobachevskian geometry.
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