NON-EUCLIDEAN GEOMETRY - Mathematicsmath.sfsu.edu/cm2/papers/Clayton_Paper.pdf · NON-EUCLIDEAN GEOMETRY: ... High school students are first exposed to geometry starting with Euclid's

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.st-

andrews.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 non-Euclidean 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?