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NON-EUCLIDEAN GEOMETRY
NON-EUCLIDEAN GEOMETRY: A HISTORY AND A BRIEF LOOK
Lisa K. Clayton
1. INTRODUCTION
High school students are first exposed to geometry starting with Euclid's classic postulates:
1. It is possible to draw a straight line from any one point to another point.
2. It is possible to create a finite straight line continuously on a straight line.
3. It is possible to describe a circle of any center and distance.
4. That all right angles are equal to one another
5. That if a straight line falling on two straight lines make 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 are the angles less than the two right angles.
Postulate #5, the so-called “parallel postulate” has always been a sticking point for
mathematicians. Historically, mathematicians encountering Euclid's beautiful work wonder
why #5 is a postulate instead of a proven theorem. There have been many, many
attempts to prove #5; there have been many, many failures. One such failure was that of
Jesuit priest and mathematician, Girolamo Saccheri. His failure sparked ideas that led to
what is now known as "Non-Euclidean geometry", a branch of geometry that discards #5
and finds out where the geometries lead them. In particular, two Non-Euclidean branches
will be discussed: that of Nikolai Lobachevsky and Bernhard Riemann.
2. LEARNING OBJECTIVES AND MATERIALS REQUIRED
There are two main objectives: one, to introduce the concept of non-Euclidean
geometries to high school geometry students who have examined Euclidean
geometry at length, including some basic worksheets so they can study the
concept for themselves. Two, to introduce students to the rich history of
mathematics and mathematical ideas.
To accomplish this, the lesson will include:
2.1
A 15 to 20 minute Powerpoint slide show on Euclid, Saccheri,
Lobachevsky and Riemann, to be broken up by:
2.2 A worksheet on Saccheri quadrilaterals, spherical and hyperspherical
geometries.
The time required will be 30-45 minutes total.
5. REFERENCES
Jacobs, Harold. Geometry: Seeing, Doing, Understanding. New York:
W.H. Freeman and Company, 3rd Edition, chapter 16: Non-Euclidean
Geometries.
Hawking, Stephen: God Created the Integers. London: Running Press,
2nd Edition, chapters 1 (Euclid), 11 (Lobachevsky) and 15 (Riemann).
The MacTutor History of Mathematics Archive, http://www-history.mcs.standrews.ac.uk/Biographies/Saccheri.html
Polking, John : The Geometry of the Sphere.
http://math.rice.edu/~pcmi/sphere/
3. LESSON OVERVIEW
The idea is to illustrate why non-Euclidean geometry opened up rich avenues in
mathematics only after the parallel postulate was rejected and re-examined, and
to give students a brief, non-confusing idea of how non-Euclidean geometry
works. The Powerpoint slides (attached) and the worksheet (attached) will give
the students both the basics of non-Euclidean geometry and the history behind it.
These slides give both the background, definitions and the information for the
student to understand the material.
4. REFLECTION
This will be taught during the 2nd semester, when students have a stronger
grasp of Euclidean geometry.
EUCLID (325 BCE - 265 BCE)
• Taught geometry in Alexandria, Egypt
• Put together his teacher’s theorems, wrote
many of his own, and published The
Elements, easily the most studied text in all
of mathematics and second only to the Bible
as the best selling book of all time.
• Not much else is known about his life other
than his work on The Elements.
THE FIVE 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
5. That, if a straight line falling on two straight lines
make the interior angles on the same side less than
two right angles, the two straight lines, if produced
indefinitely, meet on the side on which are the angles
less than the two right angles.
THE FIFTH POSTULATE
Although this postulate does not talk
directly about parallel lines, it states that
two lines where the transverse interior
angles are exactly 180o will not meet if
drawn indefinitely.
Many mathematicians were bothered by
the fact this was a “postulate” and left
unproven. Many mathematicians tried
to prove the parallel postulate, and they
all failed.
There is a very difficult proof as to why
you cannot prove the parallel postulate.
However, once mathematicians started
asking themselves “is this really true?”
then some very interesting results
happened.
SACCHERI’S QUADRILATERAL
The sides are called “legs”, and are congruent. The base is
perpendicular to the legs. The top is called the “summit.”
Try the first problem on the worksheet:
1. Given the parallel postulate, prove that the summit angles of
Saccheri’s Quadrilateral must be right angles.
2. Now try proving this without the parallel postulate. If you
can, you will become a famous mathematician.
IS THE WORLD FLAT?
Definition: A “Great Circle” is a set of points
that is the intersection of the sphere and a
plane containing its center.
Definition: “Antipodal points” are points
that are the intersection of two points on a
Great Circle and a line through its center.
Do two points determine a line, if we think
of a “Great Circle” as a line?
Is there such thing as a parallel to a Great
Circle?
SPHERICAL GEOMETRY
Spherical geometry is incredibly important in
transcontinental navigation, both on water
and in air. Menelaus (100 CE) wrote a treatise
on spherical geometry, in particular on
triangles on spheres.
Take a look at problem #2a on your worksheet.
What can you conjecture about perpendicular
lines (great circles) on a sphere? What can
you conjecture about the sum of the angles
of a triangle on a great sphere? Why?
Menelaus proved that if the angles of one triangle
are equal to the angles of another triangle on
a sphere, then the triangles are congruent. Is
this true in Euclidean geometry?
BERNHARD RIEMANN (1826-1866)
• Riemann developed a geometry
similar to spherical geometry that
postulated no parallels to any line.
• He is best known for “Riemann
manifolds”, a type of geometric space
that can extend beyond three
dimensions and has curvature.
• It takes an advanced degree in
mathematics to understand most of
Riemann’s theorems.
MORE THAN ONE PARALLEL TO THE LINE?
The Lobachevskian Postulate:
“The summit angles of a Saccheri quadrilateral are acute.”
Lobachevskian Theorem #1:
“The summit of a Saccheri quadrilateral is longer than its base”
Worksheet question #1c: If Theorem #1 is true, what happens to the
triangle midsegment theorem? Explain why. (Hint: draw a triangle using
the summit as a base. What does that “extra length” tell you?)
NICOLAI LOBACHEVSKY (1793-1856)
• The history of attempted proofs of the
parallel postulate sparked Lobachevsky’s
interest in the question. He was the first to
follow through on the question “what if it is
not true?”
• He published the first work ever on nonEuclidean geometry in 1829.
• Janos Bolyai (1802-1860), a Hungarian
mathematician, independently came to
same conclusions Lobachevsky did about
the potential of non-Euclidean geometry.
• Unfortunately, his legacy as a
mathematician came after his death. He
died in near-poverty, his genius not yet
recognized by his contemporaries.
FURTHER THOUGHTS
Mr. Cohen always emphasizes WHY things are rather than
HOW things are. Here are some final WHYs (and a
WHAT) to consider:
• Why did Euclid’s fifth postulate stay unchallenged until
Lobachevsky? Even he tried to prove its truth until he
realized that it may not be the case.
• As it turns out, the universe itself is NOT flat. We don’t
know exactly what kind of geometry (yet), but we do
know it isn’t Euclidean. What is the geometry of the
universe?
• Nonetheless, Euclidean geometry worked, and worked
well, for centuries. Why?
NON-EUCLIDEAN ALGEBRA WORKSHEET
Saccheri's Quadrilateral
1. Given the parallel postulate, prove that the summit angles of Saccheri’s Quadrilateral must be right angles.
a. Now try proving this without the parallel postulate-- don't worry if you can't. If you could, you would become a world-famous
mathematician.
b. Lobachevskian's postulate: “The summit angles of a Saccheri quadrilateral are acute.” Lobachevskian Theorem #1: “The
summit of a Saccheri quadrilateral is longer than its base”
c. If Theorem #1 is true, what happens to the triangle midsegment theorem? Explain why. (Hint: draw a triangle using the
summit as a base. What does that “extra length” tell you?)
Great Circles and Spherical Geometry
2. Let P be a plane and S be a sphere with center C. There are three possible ways P and S can interact in space. What are they?
a. The most interesting case is where P intersects C. This creates a great circle on S. They are analogous to lines on a plane. Can
two points on S always determine a great circle? Why or why not?
b. Triangles on a sphere have the property that if all three angles are equal, then the triangles are congruent. We know that
"Angle-Angle" (AA) works for similar triangles only. ? Why must all three angles be equal for spherical triangles instead of
two? What property of parallel lines make it possible to have similar triangles on a plane but not on a sphere?