the Greeks
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A Russian mathematician named Nikolai Ivanovich Lobachevsky is the man credited with inventing hyperbolic geometry. Nikolai lived from 1792 to 1856.
Yes Thales did in fact he was credited for it "swagg boo"
Euclidean geometry, non euclidean geometry. Plane geometry. Three dimensional geometry to name but a few
There are different kinds of geometry including elementary geometry, Euclidean geometry, and Elliptic Geometry.