56 HomeYamamoto and the Secret Admirers
Neal Stephenson

Misc

Non-Euclidean geometry
In the 19th century it was proposed that by replacing two of Euclid's postulates it was possible to produce equally valid geometries that were non-Euclidean.

The postulates of Euclidean geometry are.

 1.  Exactly one straight line can be drawn between any two points.
 2.  A straight line can be continued indefinitely.
 3.  With any point as center, a circle with any radius may be described.
 4.  All right angles are equal.
 5.  Through a given point outside a given straight line, there passes only one line  parallel to the given line; that is, such a line does not intersect the given line.

The proposal was to replace the second and fifth postulates with new postulates. These new postulates combined with the original three and Euclid's five common notions would create new geometries that were non-Euclidean. Two new geometries theorised being Hyperbolic and Elliptic.

Hyperbolic
This geometry substitutes the parallel postulate with one that states.

Through a given point outside a given straight line pass more than one line not intersecting the given line.

The Russian mathematician Nikolai Ivanovitch Lobachevski in 1829 was the first to publish a work on hyperbolic geometry, and was soon followed by a publication by the Hungarian János Bólyai. But it was not until the death of the prominent German mathematician Carl Friedrich Gauss in 1855 when some unpublished work of his was discovered that dealt with hyperbolic geometry did it become widely accepted.

Elliptic
The other form of non-Euclidean geometry is Riemann's elliptic geometry, which is the basis upon which Einstein's General Relativity Theory was based. This geometry rejects the parallel postulate on the assumption that there are no parallel lines and any lines if extended far enough will eventually meet. It was initially conceived by the German mathematician Bernhard Riemann and so it is commonly referred to as Riemann geometry.

An important concept introduced with this geometry is the idea that boundlessness need not imply infinitely long lines. Picture the equator it has no beginning or end and so can be considered boundless and yet it is clearly of finite length. Albert Einstein adopted this idea of unbounded space not needing to be infinite in his General Theory of Relativity.

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