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# The Applications Of Non-Euclidean Geometry

Table of Contents |
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1.Where Euclidean Geometry Is Wrong |

2.Cosmology & The Geometries |

3.The Theory of General Relativity |

4.Spherical Geometry |

5.Celestial Mechanics |

# Where Euclidean Geometry Is Wrong

Since Euclid first published his book

*Elements*

in 300 B.C. it has remained remarkably correct and accurate to real

world situations faced on Earth. The one problem that some find with it

is that it is not accurate enough to represent the three dimensional

universe that we live in. It has been argued that Euclidean Geometry,

while good for architecture and to survey land, when it is moved into

the third dimension, the postulates do not hold up as well as those of

hyperbolical and spherical geometry. Both of those geometries hold up to

a two dimensional world, as well as the third dimension.

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# Cosmology & The Geometries

Cosmology - Cosmology is the study of the origin, constitution, structure, and evolution of the universe.

The

recognition of the existence of the non-Euclidean geometries as

mathematical systems was resisted by many people who proclaimed that

Euclidean geometry was the one and only geometry. To try and 'validate'

the geometries to Euclid believers the truth of the geometry was

presented in the sense of better representing our universe, through

observation. At the present time mathematicians are still not sure which

of the three geometries provides the best representation of the entire

universe. While Euclidean geometry provides an excellent representation

for the part of the universe that we inhabit, like Newton's Laws of

physics, they break down when placed in situations that their

originators could not have imagined. Most cosmologists believe that

knowing which geometry is the most correct is important. This stems from

the belief that the future of the universe is expected to be determined

by whatever is the actual geometry of the universe. According to

current theories in the field of cosmology, if the geometry is

hyperbolic, the universe will expand indefinitely; if the geometry is

Euclidean, the universe will expand indefinitely at escape velocity; and

if the geometry is elliptic, the expansion of the universe will coast

to a halt, and then the universe will start to shrink, possibly to

explode again. This is analagous to one of the quirks of each geometry;

in hyperbolic geometry the sum of the angles of a triangle is greater

than 180 degrees, while Euclidean has the sum of a triangles angles to

be 180 degrees exactly. Elliptic geometry has the sum of the angles of a

triangle to be less than 180 degrees.

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# The Theory of General Relativity

Einstein's Theory Of General Relativity is based on a theory that space is curved. The cause is explained by the theory itself.

- Einstein's General Theory of Relativity can be understood as saying that:
- Matter and energy distort space

- The distortions of space affect the motions of matter and energy.

hyperbolic geometry which is a 'curved' one. Many present-day

cosmologists feel that we live in a three dimensional universe that is

curved into the 4th dimension and that Einstein's theories were proof of

this. Hyperbolic Geometry plays a very important role in the Theory of

General Relativity.

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# Applications Of Spherical Geometry

Spherical Geometry is also known as hyperbolic geometry and has many

real world applications. One of the most used geometry is Spherical

Geometry which describes the surface of a sphere. Spherical Geometry is

used by pilots and ship captains as they navigate around the world.

However, working in Spherical Geometry has some nonintuitive results.

For example, did you know that the shortest flying distance from Florida

to the Philippine Islands is a path across Alaska? The Philippines are

South of Florida - why is flying North to Alaska a short-cut? The answer

is that Florida, Alaska, and the Philippines are collinear locations in

Spherical Geometry (they lie on a "Great Circle"). Small triangles,

like ones drawn on a football field have very, very close to 180

degrees. Big triangles, however, (like the triangle with veracities: New

York, L.A. and Tampa) have much more then 180 degrees. Back To Top

# Celestial Mechanics

The Sun causes some medium-scale curvature that - thanks to planet

Mercury - we are able to measure. Mercury is the closest planet to the

Sun. It is in a much higher gravitational field than is the Earth, and

therefore, space is significantly more curved in its vicinity. Mercury

is close enough to us so that, with telescopes, we can make accurate

measurements of its motion. Mercury's orbit about the Sun is slightly

more accurately predicted when Hyperbolic Geometry is used in place of

Euclidean Geometry.

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