Transcript The Strange New Worlds: The Non

The Strange New Worlds:
The Non-Euclidean Geometries
Presented by:
Melinda DeWald
Kerry Barrett
Euclid’s Postulates
1.
To draw a straight line from any point to any
point.
2. To produce a finite straight line continuously in
a straight line.
3. To describe a circle with any center and
distance.
4. That all right angles are equal to one another.
And the fifth one is …
Euclid’s fifth postulate:
• 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.
Euclid’s Work
• For 2000 years people were uncertain of
what to make of
Euclid’s fifth postulate!
About the Parallel Postulate
•It was very hard to understand. It was not as simplistic as
the first four postulates.
•The parallel postulate does not say parallel lines exist; it
shows the properties of lines that are not parallel.
•Euclid proved 28 propositions before he utilized the 5th
postulate.
•Once he started utilizing this proposition, he did so with
power.
•Euclid used the 5th postulate to prove well-known results
such as the Pythagorean theorem and that the sum of the
angles of a triangle equals 180.
The Parallel Postulate or Theorem?
• Is this postulate really a theorem? If so, was Euclid
simply not clever enough to find a proof?
• Mathematicians worked on proving this possible
“theorem” but all came up short.
• 2nd century, Ptolemy, and 5th century Greek
philosopher, Proclus tried and failed.
• The 5th postulate was translated into Arabic and
worked on through the 8th and 9th centuries and again
all proofs were flawed.
• In the 19th century an accurate understanding of this
postulate occurred.
Playfair’s Postulate
• Instead of trying to prove the 5th postulate
mathematicians played with logically equivalent
statements.
• The most famous of which was Playfair’s Postulate.
• This postulate was named after Scottish scientist John
Playfair, who made it popular in the 18th century.
• Palyfair’s Postulate:
Through a point not on a line, there is exactly one
line parallel to the given line.
• Playfair’s Postulate is now often presented in text books
as Euclid’s 5th Postulate.
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Girolamo Saccheri
Saccheri was an 18th century Italian teacher and scholar.
He attempted to prove the 5th postulate using the previous 4
postulates.
He tried to find a contradiction by placing the negation of the 5th
postulate into the list of postulates.
He used Playfair’s Postulate to form his negation.
• Through a point on a line, either:
1. There are no lines parallel to the given line, or
2. There is more than that one parallel line to the given
line.
The first part of this statement was easy to prove.
The second part was far more difficult using the first four
postulates, he found some very interesting results but never found
a clear contradiction.
He published a book with his findings: Euclides Vindicatus
Revelations of Euclid Vindicated
• In the 19th century Euclid Vindicated was dusted off and
revisited by four mathematicians.
• Three of whom started by considering:
• 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?
• Carl Friedrich Gauss (German) looked at the previous
question, but did not publish his investigation.
• Nicolai Lobachevsky(Russian) produced the first
published investigation.
• He devoted the rest of his life to study this
different type of geometry.
• Janos Bolyai (officer in Hungarian army) looked at the
same question and published in 1832.
Revelations Continued
• These three came up with the same result:
• If the parallel postulate is replaced by part 2 of
its negation in Euclid’s postulates, the resulting
system contains no contradictions.
• They settled once and for all that the parallel
postulate CAN NOT be proved by the first four
postulates.
• This discovery lead to a new type of plane
geometry, with an entire new theory of shapes
on surfaces….
Non-Euclidean
Geometry
Riemann Geometry
• The fourth person to take a look at Euclid Vindicated was
Bernhard Riemann.
• He was looking at part one of the negation of the parallel
postulate he wondered if there was a system when you are given a
point not on a line and there are NO parallel lines.
• He found a contradiction but it depended on the same
assumptions as Euclid, that lines extend indefinitely.
• Riemann observed that “extended continuously” did not
necessarily mean they were infinite.
– Ex: Consider an arc on a circle: it is extended continuously but its length is
finite.
• This contradicted Euclid’s idea of using the postulate.
• Amounted to alternate version of the postulate.
• THERE IS A NEW SYSTEM!
Differences from the Euclidean System
• After Non-Euclidean geometry was discovered other
types of geometry were distinguished by how they used
parallel lines.
– In Reimann’s geometry for instance parallel lines did not
exist.
• New systems of Lobachevsky and Riemann were
formally called Non-Euclidean geometry.
• The differences about parallelism produces vastly
different properties in the new geometric systems.
– Only in Euclidean geometry is it possible to have two
triangles that are similar but not congruent.
– In Non-Euclidean geometries if corresponding angles of two
triangles are equal then the triangles must be congruent.
More Differences
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The sum of the angles of a triangle:
Euclid: exactly 180 degrees
Lobachevsky: less than 180 degrees
Riemann: greater than 180 degrees
Euclid
Lobachesky
Riemann
Another Difference
• The ratio of the circumference “C” of a
circle to its diameter “D” depends on the
type of geometry being used.
– Euclid: exactly Pi
– Lobachevsky: Greater than Pi
– Riemann: Less than Pi
Balloon Activity
• Get in pairs.
• Draw two dots (at least an inch
apart) on a balloon.
• Connect them with a straight-line.
• Blow up the balloon to a relatively large size.
• Have string go from one point to the other over
the originally drawn line.
• Try to find a line from one point to the other that
is shorter than the string.
Geometry as a Tool
• Geometry should be used as a tool to help deal with our
world.
-The type of geometry a person uses depends on the
situation that person is faced with.
• Euclidean geometry “makes sense” in most people’s
minds because that is what we are taught as children.
• Euclidean works well for the construction world.
• Riemann’s geometry is good for astronomers because of
the curves in the atmosphere.
• Lobachevsky’s geometry: Theoretical physicists use this
system.
Timeline
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2nd century: Ptolemy tried and failed to proved the Parallel postulate
5th century: Greek philosopher, Proclus tried and failed to prove the Parallel
Postulate.
8th and 9th centuries: The 5th postulate was translated into Arabic and worked
on, but again all proofs were flawed
18th century: Playfair’s Postulate was made popular by the Scottish scientist
John Playfair.
18th century: Girolamo Saccheri published his work on “proving” Euclid’s
Fifth Postulate in Euclides Vindicatus
19th century Euclid Vindicated was dusted off and revisited by four
mathematicians (Bernhard Riemann, Carl Friedrich Gauss, Nicolai
Lobachevsky, and Janos Bolyai)
19th century mathematicians played with logically equivalent statements to
Euclid’s Fifth Postulate (thus coming to the accurate conclusion that this
statement was indeed a postulate).
19th century: Non-Euclidean Geometry was created.
References
Heath, Thomas L., Sir. The Thirteen Books of
Euclid’s Elements. New York: Dover Publications,
Inc., 1956
Non-Euclidean Geometry. 13 November 2006
<http://www-groups.dcs.stand.ac.uk/~history/HistTopics/NonEuclidean_geometry.html>.
Rozenfeld, B. A. Istoriia Neevklidovoi Geometrii
(English). New York: Springer-Verlag, 1988