Change in Format
I flipped things around so that the most recent items appear at the top. Earlier stuff is still there, page down to find it.
Test 4 and Course Grades Posted
Here find the Test 4 and course letter grades posted by the Test ID number on your Test 4 paper.
Homework 9 Solutions and Review Sheet
Here are the solutions to Homework 9 and some review guidance for Test 4. Some proofs of mine have not been added yet; I say that they will be up by Sunday evening but I intend to have them up sooner. I indicate what these items are: they are things I did in class so you may have notes.
Solutions to Test III
Here are solutions to Test III with some indication of the grading rubric. Here is the diagram for problem 1.
Grades on Test III
The average on the test was 76, with no adjustments other than dropping the lowest question as stated on the test paper. Here are the grades posted by the ID number on your paper.
Here find a document with more proofs from the chapters we are currently working on. Update April 20th at noon; I am not planning to make any major additions (unless someone requests something that I think is reasonable). Make sure that you study the definition of congruent triangles and the two proofs that base angles of isosceles triangles are equal; this is also content relevant to the test. The coverage of the exam is chapter 3, except for the brief last section which talks about parallel postulates. Stuff in the notes is obviously fair game (I have added new comments and study questions throughout the notes, make sure you read them through again!). Anything you did in homework is fair game. Of course, anything in the book is, too, but we do have to take time limitations and level of difficulty into account. Happy studying! Tentatively, I am planning to be generous about supplying axioms and less generous about supplying definitions. Make sure you are familiar with important definitions. It would not hurt at all to review the axioms too.
Grades on Test II
Here are the grades on Test II posted by the ID number on your test paper.
Homework 6 (mostly a practice test!)
The handout is here. Thanks to the alert student who noticed the typo in the last question (now corrected): you are proving H=J not I=J! Please notice that I am accepting this up until Tuesday after the break: there is no reason not to, as I am out of town from Thursday morning until late Sunday.
Notes on Neutral Geometry
Here find the notes on neutral geometry. These will be updated frequently.
Test I grades posted by Test ID number
Test I grades posted by Test ID number
Here is the handout for Homework III. The assignment is due next Thursday.
Here is the manual of logical style. This is a work in progress; there will be changes and additions.
ADDITIONAL EXERCISE for assignment 2
Find a model of incidence geometry in which none of the three parallel postulates hold. The idea is that more than one of the three situations can hold: for example, you could build a model in which there is a line L1 and a point P1 with just one line through P1 parallel to L1 and a line L2 and a point P2 with two distinct lines through P2 parallel to L2. (The third situation is to have no parallels to a given line through a given point).
Postulates and definitions for section 2.3
The undefined notions of incidence geometry are point, line, and lies on (as in “point P lies on line L”).
postulate 1: for each pair of distinct points P and Q there is exactly one line L such that P and Q both lie on L.
postulate 2: for any line L, there are at least two distinct points P and Q such that P and Q both lie on L (there may be more!)
postulate 3: there are three distinct points P, Q, R such that there is no line L such that P, Q, and R all lie on L.
Those are the axioms. We also give a definition of parallel lines and three possible parallel postulates.
Definition: lines L and M are parallel iff there is no point P such that P lies on L and P lies on M.
Postulate 4a (Playfair's Postulate): for any point P and line L such that P does not lie on L, there is exactly one line M such that P lies on M and M is parallel to L.
Postulate 4b (Hyperbolic Parallel Postulate): for any point P and line L such that P does not lie on L, there are at least two lines M such that P lies on M and M is parallel to L.
Postulate 4c (Elliptic Parallel Postulate): for any point P and line L such that P does not lie on L, there is no line M such that P lies on M and M is parallel to L. (this is equivalent to saying that there are no parallel lines at all).
Geometry with three-point lines
Here is the description of the geometry with three-point lines that we developed in class:
line 1 ABC
line 2 ADE
line 3 BDF
line 4 FEC
line 5 GDC
line 6 AFG
line 7 GBE
Questions to think about before the next class: is this the same as Fano's geometry given in the book? Which if any parallel postulate does it satisfy?
Some of you may already have noticed that I misstated the second postulate of incidence geometry (Tuesday the 24th): the second postulate actually says that each line has at least two points on it (but allows lines to have more points!). The statement that two lines can intersect in no more than one point is actually a logical consequence of the first postulate. A list of postulates from the Tuesday lecture is given above.
Take a look at this online presentation of Euclid's Elements. I'm also planning to put one or more paper copies on reserve in the library. UPDATE: volume 1 of the Heath translation of the Elements is on 3 hour reserve in the library, if you prefer to handle a paper book. The online version should be adequate for any class purposes, though.