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The Applications Of Non-Euclidean Geometry
| Table of Contents |
|---|
| 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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