Project 2 details

It is time for a second project. I would prefer to have it before Thanksgiving, but will take it by the Monday after. Here are three projects for you to choose from. Whichever one you choose, it will be submitted as a maple worksheet, with text and formatted inline mathematics. You will develop, define and use at least 2 Maple words (procedures) in your worksheet. Your project will be individually prepared and submitted by email attachment by Monday 11/29. Alternatively, you can post it to your web page and send me the hyperlink (I have a worksheet detailing how to do this on your sac account or Chris Rakes can show you how to do it).

1. Make up an exposition and assignment for your advanced algebra class in high school about complex numbers. Pick at least 5 of the concepts we discussed and explain them in your own terms. Include the fundamental theorem of algebra. Make up and solve at least 5 sample problems (including some word problems) involving complex numbers. Make up at least 8 problems for your students to work on, including at least one 'challenge' problem. Put these at the bottom of your worksheet, which will be at least 4 pages 12 point with Maple output removed. (you can check this using the print preview under the file menu.)

2. Suppose we extend the group of Moebius maps on the extended plane by throwing in the conjugation map . In other words, we look at the set EM of all maps obtained by composing translations , rotations , radial stretches (r a positive real), inversion , and conjugation .

a) Show that EM is a group under composition by showing the following.

i) Each basic map in EM (translation, rotation, radial stretch, inversion, and conjugation) has an inverse of the same type: for example, the inverse of the translation is the translation .

ii) If is in EM where each is a basic map in EM, then the map is the inverse of f, if is the inverse of for i from 1 to n.

Definition. Let L be a line and r be a map of the plane. r is the reflection about L provided for each z in C, is on L.

b) Show that reflection about the imaginary axis ( ) is in EM by showing that it is conjugation followed by muliplication by -1; also show that reflection about the line y = x is in EM by showing that it is conjuation followed by multiplication by I.

c) Fix . Show that the EM map is the reflection about the line , t real.

d) Show that the EM map is the reflection about a line if . What is the line?

Definition. Let S be a circle with center and radius R. A map r of the plane is the reflection through S provided for each z in C (except ),

e) Show that the map is a reflection through a circle (what circle?)

f) Show that all reflections through circles are EM maps.

g) Show that every EM map is the composition of a radial stretch with at most 4 reflections, at most one of which is reflection through a circle.

3. A model for hyperbolic geometry: When the parallel axiom of Euclidean plane geometry is replaced with one of its denials (namely, there is a line and point not on it such that there are at least two lines through the point which are parallel with the line), a perfectly consistent set of axioms for a different geometry (called hyperbolic geometry) results. One the models for this geometry is the Poincare circle model. a point is defined as a complex number of absolute value less than 1, a line is defined as the 'points' on a circle orthogonal to the unit circle or as the 'points' on a diameter of the unit circle. Develop a small Maple vocabulary for drawing figures in this geometry. Here are the minimal number of words I will accept in the vocabulary.

> restart;

> hyperb := table([]):

> hyperb[drawpts] := proc(lst) print(`have this proc plot a list of points in the model`) end:
hyperb[midpt] := proc(z1,z2) print(`this proc returns the midpoint of the segment from z1 to z2 in the model`) end:
hyperb[seg] := proc(z1,z2) print(`this proc should call midpt several times, then draw a broken line segment approximating the segment from z1 to z2 in the model`) end:
with(hyperb);

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