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MAT102 LESSON 15 Date: Oct 25

Topic: Non- Euclidean geometry and Topology.

Read pages section 9.7 about Non-Euclidean Geometry and Topology. For the first time you will be acquainted with geometries that are different from the geometry we use in the classroom: the geometry of Riemann and the geometry of Lobachevski. You will be completing a writing assignment on them (check at the end of this lesson).

You will also get acquainted with a science that studies space figures without being concerned with their specific sizes. This science is called Topology. Two figures are said to be topologically equivalent if one can be transformed into the other one with only stretching, bending and other distortions without tearing or scattering. Of course, in order to consider such manipulations we will assume that all figures are made of a special rubber that can be bent or stretched very thin without tearing. In our course it is only required that you are able to recognize when two figures are topologically equivalent. If you want to learn more about topology I suggest you go to http://en.wikipedia.org/wiki/Topology

By doing HW #15 on MyMathLab (due date June 22) you will learn to recognize figures that are topologically equivalent.

Important: Writing assignment to be deposited in Blackboard.

Assignment on Non-Euclidean Geometries:

Please copy the questions below into your document and type the answers in. Answers should be short and to the point

•When (approximately) were The Elements written and by whom? (not to be confused with the date when The Elements were translated into English)

•Euclid’s fifth postulate was initially written in a somewhat cumbersome manner? Who improved on it and how?

•How many versions are there of Euclid’s fifth postulate? Who wrote them?

•Why was the parallel postulate controversial? Can you state a theorem from Euclidean geometry that does not hold in the geometries of Lobachevski and Riemann? How is this theorem different in these Non-Euclidean geometries?

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