Transcript Non-Euclidean Geometries

Non-Euclidean Geometries
Steph Hamilton
The Elements: 5 Postulates
1. To draw a straight line from any point to any
other
2. Any straight line segment can be extended
indefinitely in a straight line
3. To describe a circle with any center and distance
4. That all right angles are equal to each other
5. That, if a straight line falling on two straight lines
makes the interior angles on the same side less than
two right angles, if produced indefinitely, meet on
that side on which are the angles less than the two
right angles
Parallel Postulate
5. That, if a straight line
falling on two
straight lines makes
the interior angles on
the same side less
than two right angles,
if produced
indefinitely, meet on
that side on which are
the angles less than
the two right angles
•Doesn’t say || lines
exist!
Isn’t it a Theorem?
• Most convinced it was
• Euclid not clever enough?
• 5th century, Proclus stated that Ptolemy (2nd
century) gave a false proof, but then went
on to give a false proof himself!
• Arab scholars in 8th & 9th centuries
translated Greek works and tried to prove
postulate 5 for centuries
Make it Easier
• Substitute statements:
 There exists a pair of similar non-congruent triangles.
 For any three non-collinear points, there exists a circle
passing through them.
 The sum of the interior angles in a triangle is two right
angles.
 Straight lines parallel to a third line are parallel to
each other.
 There is no upper bound to the area of a triangle.
 Pythagorean theorem.
 Playfair's axiom(postulate)
John Playfair-18th century
•
Through a point not on a given line,
exactly one line can be drawn in the plane
parallel to the given line.
Proclus already
knew this!
Most current
geometry books
use this instead of
the 5th postulate
Girolamo Saccheri – early 18th century
• Italian school teacher & scholar
• He approached the Parallel Postulate with these 4
statements:
1. Axioms contain no contradictions
because of real-world models
2. Believe 5th post. can be proved, but
not yet
3. If it can be, replace with it’s negation, put
contradiction into system
4. Use negation, find contradiction, show it
can be proved from other 4 postulates w/o
direct proof
2 Part Negation
1. There are no lines parallel to the given line
•Euclid already proved that parallel lines exist
using 2nd postulate
2. There is more than one line parallel to the
given line
•Weak results, convinced almost no one
•Prove by contradiction by
denying 5th postulate
•So, 3 possible outcomes:
•Angles C & D are right
•Angles C & D are obtuse
•Angles C & D are acute
Died thinking he proved 5th postulate from the other four 
New Plane Geometry
Can there be a system of plane geometry in which,
through a point not on a line, there is more than one
line parallel to the given line?
• Gauss was 1st to examine at age 15.
“In the theory of parallels we are even
now not further than Euclid. This is a
shameful part of mathematics.”
• Never published findings
Can there be a system of plane geometry in which,
through a point not on a line, there is more than one
line parallel to the given line?
• Gauss worked with Farkas Bolyai who also made
several false proofs.
• Farkas taught his son, Janos, math, but advised
him not to waste one hour’s time on that problem.
• 24 page appendix to father’s book
• Nicolai Lobachevsky was 1st to publish this
different geometry
• Together they basically came to the conclusion
that the Parallel Postulate cannot be proven from
the other four postulates
Lobachevskian Geometry
•Roughly compared to looking down in a bowl
•Changes 5th postulate to, through a point not on a
line, more than one parallel line exists
•Called hyperbolic geometry because its playing
field is hyperbolic
•Poincare disk
•Negative curvature: lines curve in opposite
directions
•Example of this geometry
2 Points determine a
line
Given a point and a distance
a circle can be drawn with
the point as center and the
distance as radius
A straight line can be
extended without limitation
All right angles
are equal
The Parallel
Postulate
Riemannian Geometry
• Bernhard Riemann – 19th century
• Looked at negation of 1st part of Parallel Postulate
“Can there be a system of plane geometry in
which, through a point not on a line, there are no
parallels to the given line?
• Saccheri already found contradiction, but based on
fact that straight lines were infinite
• Riemann deduced that “extended continuously”
did not mean “infinitely long”
Riemannian Geometry
• Continue an arc on a sphere – trace over
•New plane is composed great circles
•Also called elliptical geometry
•Positive curvature: lines curve in same direction
Triangles
Euclidean, Lobachevskian, Riemannian
Fact: Euclidean geometry is the only geometry
where two triangles can be similar but not congruent!
Upon first glance, the sides do not look straight,
but they are for their own surface of that geometry
Riemannian Geometry
C/D
• Euclidean geometry, it is exactly pi
• Lobachevskian, it is greater than pi
• Riemannian, it is less than pi
Pythagorean Thm
•Euclidean: c2=a2 + b2
•Lobachevskian: c2>a2 + b2
•Riemannian : c2< a2 + b2
Which one is right?
Poincaré added some insight to the debate between
Euclidean and non-Euclidean geometries when he said,
“One geometry cannot be more true than another; it can
only be more convenient”.
• Euclidean if you are a builder, surveyor, carpenter
• Riemannian if you’re a pilot navigating the globe
• Lobachevskian if you’re a theoretical physicist or
plotting space travel because outer space is
thought to be hyperbolic
“To this interpretation of geometry, I attach great importance, for
should I have not been acquainted with it, I never would have
been able to develop the theory of relativity.”
~Einstein
Timeline
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Euclid’s Elements – 300 B.C.E.
Ptolemy’s attempted proof – 2nd century
Proclus’s attempted proof-5th century
Arab Scholar’s translate Greek works – 8th & 9th
centuries
Playfair’s Postulate – 18th century
Girolamo Saccheri – 18th century
Carl Friedrich Gauss – 1810
Nicolai Lobachevsky – 1829
Janos Bolyai – 1832
Bernhard Riemann – 1854
References
• http://members.tripod.com/~noneuclidean/hyperbolic.html
• http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/NonEuclidean_geometry.html
• http://www.geocities.com/CapeCanaveral/7997/noneuclid.html
• http://pegasus.cc.ucf.edu/~xli/non-euclid.htm
• http://www.mssm.org/math/vol1/issue1/lines.htm
• http://www.princeton.edu/~mathclub/images/euclid.jpg
• http://www.daviddarling.info/encyclopedia/N/nonEuclidean_geometry.html