Download Non-Euclidean Geometry

Non-Euclidean Geometry
Presentation by Doyle, Krauss and May
Euclid's original formulation
“If a straight line falling on two straight lines makes the
interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely,
will meet on that side on which the angles are less
than the two right angles.”
~The Elements
http://aleph0.clarku.edu/~djoyce/java/elements/logo.gif
http://www.biographyonline.net/wp-content/uploads/2015/03/euclid-e
lements.jpg
Euclid's Postulates
1.
Given two points there is one straight line that joins them.
2.
A straight line segment can be prolonged indefinitely.
3.
A circle can be constructed when a point for its centre and a distance for its radius are given.
4.
All right angles are equal.
5.
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right
angles.
https://decodedscience.org/wp-content/uploads/2012/07/euclids-postulates.png
Euclid's Axioms (Common Notions)
1.
Things which are equal to the same thing
are also equal to one another.
2.
If equals are added to equals, the wholes
are equal.
3.
If equals are subtracted from equals, the
remainders are equal.
4.
Things which coincide with one another
are equal to one another.
5.
The whole is greater than the part
http://www.math.cornell.edu/~mec/Winter2009/Mihai/section4.html
https://plus.maths.org/content/sites/plus.maths.org/files/blog/102014/axioms2_web.jpg
Parallel Line Theory
In the Euclidean geometry, make a
straight line AB in two parallel lines,
circle in a counterclockwise
direction with a straight line AB, and
then make a circle in a clockwise
direction with a straight line AB.
From the intersection of the two
circles If the vertical CD is
perpendicular to the straight line
AB, if the angle of the CD and AB is
90 degrees, then the two parallel
lines will not intersect.
But Euclid did not dare to think
about when the two parallel lines
infinitely long...
http://image.mathcaptain.com/cms/images/67/sp-geo2.png
http://www.kshitij-school.com/Study-Material/Class-9/Mathematics/Euclids-geometry/Equivalent-versions-of-eucli
ds-fifth-postulate/2.jpg
Nikolas lvanovich Lobachevsky
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December 1, 1792 - February 24, 1856
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Russian mathematician
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One of the Early Discoveries of Non-Euclidean
Geometry
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known primarily for his work on hyperbolic geometry,
otherwise known as Lobachevskian geometry.
http://news.xinhuanet.com/science/2015-10/26/134736133_14454140823881n.jpg
Lobachevsky’s Conclusions
1.
The fifth postulate cannot be proved.
2.
A series of reasoning in the new axiom
system has been a series of new theorems
that are logically contradictory and form a
new theory. This theory is like the Euclidean
geometry is perfect, tight geometry.
https://media1.britannica.com/eb-media/52/2352-004-94
2D34C9.jpg
https://www.learner.org/courses/mathilluminated/images/
units/8/1811.png
Bernhard Riemann
-September 17, 1826 – July 20, 1866
-A German mathematician
-Made contributions to analysis,
number theory, and differential geometry.
“any two straight lines on the same plane
must be intersecting” ~Riemann
https://en.wikipedia.org/wiki/Bernhard_Riemann
https://en.wikipedia.org/wiki/Bernhard_Riemann#/media/File:Georg_Friedrich_Bernhard_Riemann.jpeg
János Bolyai
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December 15, 1802 – January 27 1860
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A Hungarian mathematician
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One of the founders of non-Euclidean geometry
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Correspond to the structure of the universe helped
to free mathematicians to study abstract concepts
irrespective of any possible connection with the
physical world.
https://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai#/media/File:Bolyai_J
%C3%A1nos_(M%C3%A1rkos_Ferenc_festm%C3%A9nye).jpg
https://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai
Carl Friedrich Gauss
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April 30, 1777 Braunschweig – February 23, 1855
Göttingen
A German mathematician
Contributed significantly to many fields, including
number theory, algebra, statistics, analysis,
differential geometry, geodesy, etc.
Sometimes referred to as the Princeps
mathematicorum and "greatest mathematician
since antiquity.
Gauss had an exceptional influence in many fields
of mathematics and science and is ranked as one
of history's most influential mathematicians.
https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#/media/File:Carl_Fri
edrich_Gauss_1840_by_Jensen.jpg
https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
Euclidean VS Non-Euclidean Geometry
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Euclidean Geometry
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Geometry that we are most familiar with.
Named after Euclid, a Greek mathematician who lived in 300 BC.
Had a book called “The Elements” which is a collection of axioms,
theorems, and proofs.
Most theorems taught in high school today can be found in this 2000 year
old book.
Euclidean geometry was used by the Greeks to design buildings, predict
the location of moving objects and survey land.
Euclid’s Postulates
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Euclidean Geometry deals with points, lines and planes and how they
interact to make more complex figures.
Euclid’s Postulates define how the points, lines, and planes interact with
each other.
Euclid’s First Postulate
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A straight line segment can be drawn joining any two points.
Euclid’s Second Postulate
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Any straight line segment can be extended indefinitely in a straight line.
Euclid’s Third Postulate
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Given any straight line segment, a circle can be drawn having a segment
as radius and one endpoint as center.
Euclid’s Fourth Postulate
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All right angles are congruent.
≅
Euclid’s Fifth Postulate (Parallel Postulate)
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If a line segment intersects two straight lines forming two interior angles
on the same side that sum to less than two right angles, then the two
lines, if extended indefinitely, meet on that side on which the angles sum
to less than two right angles.
Non-Euclidean Geometry
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Any geometry that isn’t Euclidean.
Each non-Euclidean geometry is a consistent system of definitions,
assumptions and proofs that describe such objects as points, lines, and
planes.
The two most common non-Euclidean geometries are spherical and
hyperbolic.
The essential difference between Euclidean geometry and these two
non-Euclidean geometries is the nature of parallel lines.
Non-Euclidean Geometry
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In Euclidean geometry, given a point and a line, there is exactly one line
through the point that is in the same plane as the given line and never
intersects it.
In spherical geometry there are no such lines.
In hyperbolic geometry there are at least two distinct lines that pass
through the point and are parallel to the given line.
Let's solve the following problem:
A fellow took a morning stroll. He first walked 10 mi South, then 10 mi West,
and then 10 mi North. It so happened that he found himself back at his house
door. How can this be?
Two Main Types of Non-Euclidean Geometries
● Spherical
● Hyperbolic
Spherical Geometry
Spherical Geometry:
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Spherical geometry is a plane geometry
on the surface of a sphere.
Points are defined in the usual way, and
lines are defined so that the shortest
distance between two points lies along
them.
Lines in spherical geometry are great circles.
A great circle is the largest circle that can be drawn on a sphere.
-
-
Longitude lines and the equator are great circles of the earth.
Spherical geometry is used by pilots and ship captains to navigate around the
globe.
Odd property: Sum of the angles of a triangle is always greater than 180º.
Longitude & Latitude
-Longitude lines make up great circles on Earth
-Latitude lines, except for the equator, are just
parallel circles to the equator
5 Axioms for Spherical Geometry
1.
Any two points can be joined by a straight
line.
2.
Any straight line segment can be extended
indefinitely in a straight line.
3.
Given any straight line segment, a circle
can be drawn having the segment as
radius and one endpoint as center.
4.
All right angles are congruent.
5.
There are NO parallel lines.
Hyperbolic Geometry
Hyperbolic Geometry:
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Hyperbolic geometry is a “curved” space.
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The non-euclid software is a model.
Plays an important role in Einstein's general theory of relativity.
Used to model space since Einstein's theories imply that space is curved.
Hyperbolic geometry has many applications within the field of topology.
Shares many proofs and theorems with Euclidean geometry, but also has
many differences from Euclidean geometry.
Hyperbolic surfaces have constant negative curvature.
5 Axioms for Hyperbolic Geometry
1.
Any two points can be joined by a straight line.
2.
Any straight line segment can be extended indefinitely in a straight line.
3.
Given any straight line segment, a circle can be drawn having the segment
as radius and one endpoint as center.
4.
All right angles are congruent.
5.
Through a point not on a given
straight line, infinitely many lines can
be drawn that never meet the given
line.
Models of Hyperbolic Geometry
Klein-Beltrami
Model
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Poincaré Disk Model
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hyperbolic surface
is mapped to the
interior of a circle
preserves
“straightness” but at
the cost of
distorting angles
hyperbolic surface is mapped to
the interior of a circular disk, with
hyperbolic geodesics mapping to
circular arcs (or diameters) in the
disk that meet the bounding circle
at right angles
distort distances while preserving
angles as measured by tangent
lines
Poincaré Upper
Half-Plane Model
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hyperbolic surface is
mapped onto the
half-plane above the
x-axis, with hyperbolic
geodesics mapped to
semicircles (or vertical
rays) that meet the x-axis
at right angles
distort distances while
preserving angles as
measured by tangent lines
The Applications of Non-Euclidean Geometry
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Cosmology (study of the origin, construction, structure, and evolution of the universe)
The Theory of General Relativity
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Celestial Mechanics (measuring curvature and motion based on gravitational fields, etc.)
Spherical Geometry
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Theory that matter and energy distort space & the distortions of space affect the motions
of matter and energy
Used by pilots and ship captains to navigate around the world.
Hyperbolic Geometry
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Topology
Map making
NASA space exploration
Practical application: GPS
Euclidean, Spherical, or Hyperbolic?
Spherical
Geometry
Hyperbolic
Geometry
Euclidean
Geometry
Hyperbolic Geometry in Art
Hyperbolic Geometry in Art
Review
Given a line and a
point not on the line,
there exist(s)
______________ through
the given point and
parallel to the given
line…
a) Exactly
one line
(Euclidean)
b) No lines
(Spherical)
c) Infinitely
many
lines
(Hyperbolic)
Review
Euclid’s fifth
postulate is
__________________.
a) True
(Euclidean)
b) False
(Spherical)
c) False
(Hyperbolic)
Review
The sum of the
interior angles of a
triangle ______________
180 degrees.
a) =
(Euclidean)
b) >
(Spherical)
c) <
(Hyperbolic)
References
http://mathworld.wolfram.com/Non-EuclideanGeometry.html
http://www.cut-the-knot.org/triangle/pythpar/NonEuclid.shtml
http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html
https://www.britannica.com/topic/non-Euclidean-geometry
http://mathstat.slu.edu/escher/index.php/The_Three_Geometries
https://www.youtube.com/watch?v=Jvs_gTrP3wg
http://noneuclidean.tripod.com/applications.html
plaza.ufl.edu/youngdj/powerpoint/noneuclidean.ppt