Download Non-Euclidean Geometries

What makes
geometry Euclidean
or Non-Euclidean?
I-1. Each two distinct points determine a line
I-2. Three noncollinear points determine a plane
The 5 Axioms of
Euclidean Geometry
I-3. If two points lie in a plane, then any line containing those two points
lies in that plane
I-4. It two distinct planes met, their intersection is a line
I-5 If a straight line falling on two straight lines makes the sum of the
interior angles on the same side less than two right angles, then the two
straight lines, if extended indefinitely, meet on that side on which the
angle sum is less than the two right angles.
The Most Controversial
The Parallel Postulate: In layman’s terms
* Given a line and a point not on that line,
there is exactly one line through the point
that is parallel to the line.
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Why is it Controversial?
• Many felt the parallel postulate was a theorem
and Euclid was just not clever enough to prove
• For two thousand years, people tried to deduce
the fifth postulate from the other four
• Were they successful?
*400 AD Proclus
*1616-1703: John Wallis
The Road of Controversy
*1667-1733: Saccheri
*1786: Posthumanous
*19th century: abandoned efforts
Proclus assumed that points at
constant distance from a given
line on one side form a straight
line
John Wallis assumed to
every triangle, there is a given
triangle of each given size
Saccheri considered quadrilaterals with
base angles equal to a right angle and
vertical sides having equal length deduced
the possibility that the remaining two
angles were not right angles
 Posthumanous publication of Lambert’s
work which was similar to that of Saccheri
Lambert noticed that, in this new
geometry, the angle sum of a triangle
increased as the area of the triangle
decreased.
*19th century: abandoned efforts
to find a contradiction to the fifth
postulate
Different Types of
Non-Euclidean
Geometry
Creation of hyperbolic geometry
*2000 years were spent trying to prove the 5th postulate
Hyperbolic
*Started to wonder could a system of plane geometry be created with
more than one line parallel to a given line
*Gauss never published his work
*Bolyai constructed the foundations of hyperbolic geometry
*Lobachevsky first to publish his results of a hyperbolic
geometry
*Saddle
*Hyperbolic paraboloid
Hyperbolic points and lines
through any two points in hyperbolic space there is a line
*A point which passes through the center is a Euclidean segment, Ie diameter
Properties of hyperbolic lines
What
does hyperbolic geometry look like?
1. Two distinct hyperbolic lines meet at most once
2. Two distinct hyperbolic lines may have one common boundary point (along the
perimeter of Poincaré disc)
Parallels
1. Two hyperbolic lines do one of the following
-Intersecting: intersect at one point
-Parallel: Share a common boundary point
-ultra parallel: neither intersect nor share a boundary point
Example: Poincare disc
*Lines are arcs of circles
*Two arcs that do not meet are parallel
*Two arcs that meet orthogonally are perpendicular (I don’t understand
what this means but maybe you do?)
*Can see in the works of Escher
Also known as spherical geometry or
Riemannian geometry
Elliptic/Spherical
Treats lines as great circles on the surface of a
sphere
In elliptical geometry, Euclid's parallel
postulate is broken because no line is parallel to
any other line.
Parallel lines?
In spherical geometry any two great
circles always intersect at exactly two points.
Example: shortest flying distance from Florida
to the Philippine Islands is a path across Alaska
Application of
 Even though the Philippines are at a more
Elliptical Geometry
southerly latitude than Florida! The reason is
that Florida, Alaska, and the Philippines lie on
the same great circle and so are collinear in
spherical geometry.
Still not
convinced?
Try This!
•In elliptic geometry, the sum of angles of a triangle is >180
The Effects of Non-Euclidean
Geometry on Triangles
References
•http://mathworld.wolfram.com/Non-EuclideanGeometry.html
•http://www.learner.org/courses/mathilluminated/images/units/8/1812.png
•http://www.daviddarling.info/encyclopedia/E/elliptical_geometry.html
•http://www.answers.com/topic/elliptic-geometry-1
•http://www.gap-system.org/~history/PrintHT/Non-Euclidean_geometry.html