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What do we owe to Riemann?

After graduating, Riemann spent two years as an unpaid lecturer at Göttingen. In 1854, he prepared to give a lecturer to the faculty in the hopes of attaining a paid position. His topic was non-Euclidean geometry, which greatly appealed to Carl Gauss (shown below), who happened to be one of the most senior members of the committee listening to the lecture.

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Riemann searched to develop a system of geometry that didn’t deal with points, lines or shapes in the way they are normally thought about. One example that’s mentioned in the Biography in Context database is a Riemann hypothesis that “every line through a point not on a given line meets that given line”. In other words, it is impossible to construct a truly parallel line – it will always intersect with the original line. Riemann also believed in taking all laws governing points and transforming them to the plenum, which means continuously filled space – essentially predicting dimensions higher than three or four. This is now called Riemannian Geometry. Apparently Riemann’s thinking about curved space even influenced Albert Einstein’s theory of relativity.