MATH325: Lecture 15


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Introduction to Non-Euclidean Geometries

We have seen how Euclid's five axioms (or something similar) describe plane figures. In this sense, Euclid's axioms are the "foundations of geometry".

To better understand these foundations, we'll spend most or all of the remainder of the semester investigating what can be built on other foundations. By the final exam you should be able to answer an essay question along the lines of "Describe one way in which the theorems we studied would change if Euclid's first axiom stated that you could construct exactly two segments joining any pair of points?"

In today's class we discuss what Euclid's axioms would look like if we wished to describe figures on a spherical, rather than planar, surface.

Spherical Geometry

Segments and Lines

Our first axiom says that we can connect two points by a segment. Given two points on a sphere, draw a segment joining them.

Is there just one way to do this, or can you join the points with more than one segment? If there's only one way, what is it? Can you describe what makes that segment "right" and the others "wrong"?

The lesson plan buried under the adds at http://www.docstoc.com/docs/22102188/Spherical-Geometry/ suggests that your students use balls and rubber bands to describe lines or segments on the surface of a sphere. Some teachers use balls purchased from a crafts supply store.

Using rubber bands to define the lines implicitly assumes the following definition: A line is the shortest distance between two points. In general, the shortest curve connecting two points is called a geodesic.

The second axiom says you can extend a segment indefinitely into a line. Given two points on a sphere, draw a line containing both points.

Is there more than one way to draw this line?

Can this line be extended indefinitely? Each line has finite length, which means that if you have two points on the line you can get from one to the other going in either direction along the line. Does this mean that each pair of points determines two different segments? Or should we only "count" the shorter one? What if the two points are at the north and south poles? Then both segments have equal length!

Wait a second. What if the two points are the north and south poles, or any pair of antipodal (opposite) points? Then there are infinitely many segments joining them and so those two points deterimine infinitely many lines.

Axiom one in the text says that any two points determine a unique line. This isn't true when the points are antipodal points on a sphere. In Theorem 1.7.1, if there are two different lines B'C' must they both be parallel to line BC? Is Theorem 1.7.1 not true on the surface of the sphere, or is it true except when B'C' are antipodal points, or does everything magically work out so that B' and C' are never antipodal points?

Changing our axioms slightly (removing the word "unique" from axiom one) forces us to doubt and recheck every theorem which used this axiom, or which used a theorem, lemma or definition which used this axiom. In other words, all but the most simple theorems may not be true on the sphere.

Circles

Axiom three states that a circle of any radius and center can be drawn. Is this still true on the sphere? Draw a circle on the surface of the sphere.

How will we define circles in spherical geometry? Axiom three says "any radius", but distances on the sphere only make sense up to a given distance (half way around). Is this a problem? Is it possible to have a radius longer than the longest segment in our space?

Angles

Axiom four says that any two right angles are congruent. How will we measure angles on the surface of the sphere? Will axiom four still be true? Draw two lines on the sphere that meet at right angles.

Parallel Lines

Axiom five says that given a line l and a point P not on l, there exists a unique line through P which does not intersect l. Is that true on a sphere? Try to draw two lines on the sphere that do not intersect.

If you define lines on the sphere to be geodesics -- shortest paths -- then any pair of distinct lines intersects at two antipodal points. If we define two lines to be parallel only if they never meet, there is no such thing as a pair of parallel lines on the sphere.

Try to draw three lines L1, L2 and L on the sphere so that L1 and L2 are both perpendicular to L.

Euclid's fifth postulate talks about two lines meeting if the sum of two angles is less than ninety degrees. The postulate is still true on the sphere, but the implication that the two lines do not meet on the other "side" of L becomes false. Any pair of lines will meet no matter what angles they form with line L.

(Since Euclid's fifth postulate seems to be true for lines on the sphere as well as in the plane, we might hope that almost everything proven based on Euclid's postulates is true on the sphere as well as in the plane. In practice this isn't true -- the sum of the angles of a spherical triangle is never 180°. Euclid must have made some assumptions about geometry that didn't appear in his axioms. This may be why the author chose to use a simplified version of the fifth postulate for axiom five.)

Homework, due 4/12: In the plane, the perpendicular bisector of segment AB is the set of points P for which |AP| = |BP|. (Lecture 14 includes a brief proof of this.) Try to prove something similar for points on the surface of a sphere. If you get stuck, explain what difference between the plane and the sphere caused the problem and why you can't surmount the problem.