Riemann was a student of Gauss, and extended his work on spherical, and other non-flat, geometries, extending them to greater numbers of dimensions. He showed that for a 2D curved space a single number (one could think of this as the radius of curvature) is needed to specify the curvature in the neighbourhood of a point. If one has 3D space, then 6 numbers are needed to specify the curvature, while for 4D curved geometries one needs 20 numbers.
In general spaces are not uniform, with changes of curvature from one point to another. Therefore Riemann's numbers are to be considered as local measures of curvature. This was a critical foundation for Einstein's idea that space was curved by different amounts (according the the presence of mass) and that one should, in general relativity, only look at ideas locally.
The so-called Turing machine that he envisioned is essentially the same as today's multi-purpose computer.
During World War II, he used his mathematical skills to decipher German military codes. The Germans had developed a type of computer called the Enigma. It was able to generate a constantly changing code that was impossible to decipher until Turing solved the problem.
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