Non-Euclidean geometries
American author and science explorer Clifford A. Pickover explains non-Euclidean geometric systems.
What you will learn in the article
- What the Euclidean parallel postulate says and why it seemed to describe three-dimensional space
- How non-Euclidean geometry violates the parallel postulate and changed scientific thinking
- Why Einstein considered non-Euclidean geometry necessary for the general theory of relativity
- How Lobachevsky, Bolyai and Riemann contributed to consistent non-Euclidean geometric systems
Table of Contents
Since Euclidian times (ca. 325 BC— ca. 270 BC) it seemed that so-called parallel postulate is describing our three-dimensional world adequately. According to this postulate, in a plane through a point not lying on a given line, only one line can be drawn, which will never intersect with the given line. The creation of systems of non-Euclidean geometry in which this postulate is not observed has led over time to colossal consequences. Einstein said the following about non-Euclidean geometry: ‘I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity.’ Essentially, in Einstein’s general theory of relativity, spacetime is a non-Euclidean geometric system in which space-time is deformed or warped near gravitating bodies such as the sun and planets. And so does it happen: imagine placing a bowling ball on a thick sheet of rubber. If you then place a small marble in the resulting depression and push it, for a time the small marble will orbit the bowling ball, much like the planets orbit the sun. In 1829, the Russian mathematician Nikolai Lobachevsky published his work “On the Principles of Geometry”, in which he presented a fully consistent geometric system based on the proposition that the parallel postulate is false. A few years earlier, the Hungarian mathematician János Bolyai had developed a similar non-Euclidean geometry, but his results were not published until 1832. In 1854, the German mathematician Bernhard Riemann summarised the findings of Bolyai and Lobachevsky by showing that, given the appropriate number of dimensions, different non-Euclidean geometries could exist. Riemann once remarked, ‘The value of non – Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than Euclidean.’ With the advent of general relativity, this prediction had come true.
Source: Pickover, C. “The Math Book”
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Key takeaways
- Euclid's parallel postulate says that through a point outside a line only one non-intersecting parallel line can be drawn
- Non-Euclidean geometry created systems where the traditional parallel postulate does not hold
- Einstein said he could not have formulated relativity without this view of geometry
- Lobachevsky published a consistent non-Euclidean system in 1829
- Riemann showed that different non-Euclidean geometries can exist with an appropriate number of dimensions
Published
July, 2024
Duration of reading
1 or 2 min
Category
Math
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