Assignment #5

Due Friday, March 30, 2001 Read Section 5.3 from page 229 - 240. Send me an email addressing the following questions with the subject line: your name here: 341, RA5 What is a transformation of the Euclidean plane? What is an isometry of the Euclidean plane? Is every transformation an isometry? If not, give an example of a transformation of the Euclidean plane which is not an isometry. Is every isometry a transformation? If not, give an example of an isometry of the Euclidean plane which is not a transformation.

Assignment #4

Due Friday, 9 February 2001 Read Section 3.4 (you have already seen a lot of this material) Send me an email addressing the following questions with the subject line: your name here: 341, RA4 Suppose that two parallel lines are cut by a transversal such that a pair of alternate interior angles are equal. Must we be in a geometry in which the Euclidean parallel postulate is assumed? Explain why or why not? The Euclidean parallel postulate is logically equivalent to many statements you have probably assumed in previous geometry courses. State two of them.

Assignment #3

Due Wednesday, January 24, 2001 Read Sections 2.3 and 2.6 (don't worry about comparisons to Hilbert's and Birkoff's axiom systems) Send me an email addressing the following questions with the subject line: your name here: 341, RA3 Briefly discuss the rationale of the SMSG Axiom system. The rationale of the SMSG Axiom system was to provide an axiom set that was complete and teachable. They wanted students to be able to use and understand this system at the beginning of their formal study of geometry. Are the axioms in the SMSG system independent of one another? If not, why is this a useful axiom system for us? The authors of the SMSG decided to sacrifice independence because independent axiom sets require the proof of many "smaller" theorems before proofs of the major results can be made. Furthermore, without independence, the axioms will be more accessible to students beginning their study of geometry.

Assignment #2

Due Wednesday, 17 January 2001 Read Section 2.1 and 2.2 Send me an email addressing the following question with the subject line: your name here: 341, RA2 Briefly discuss two problems in Euclid's axiom system for plane geometry. Euclid did not realize the importance or usefulness of undefined terms. Euclid thought that he was able to give meaning to all terms so that he could use them. Euclid did not see that in his definition of point, he really introduced new ideas that he failed to define. Thus Euclid created more undefined terms even though he did not [address] it. Another property of Euclid's system I noticed was his dependance on diagrams (models). It seemed as if the proofs and postulates came from the models instead of the model coming from the proof.

Assignment #1

Due Wednesday, 10 January Read Section 1.2 Send me an email addressing the following questions with the subject line: your name here: 341, RA1 Why are undefined terms needed in geometry? In order to prove something clear definitions of the terminology used must be provided for the reader. However, each definition typically requires another definition and so on. At some point it must be left that there undefined terms in the proof, otherwise this process would go on forever. Is an axiom a true statement? The absolute truth of an axiom is not important, what is important is that the axiom must be taken to be true in order to be useful. Why are models useful? Models are useful in that they can make an abstract axiomatic system more clear and "real" to us by bringing into the situation concrete examples. Several of these models can lead us in the right direction with our proofs.