Math 110-02, Spring 2014
Information on the Third Test

The third test for this course takes place in class on Monday, April 14. The test counts for 15% of the final grade for the course. The format will be similar to the first two tests. There won't be any calculation problems, and hardly any problems involving numbers since we have done very little with numbers recently.

The test covers everything that we have done in class since the second test. We started with Newcomb's Paradox, from the end of Section 10.1 in the textbook. That was followed by Section 3.5, on geometric infinities. The next topic was symmetry and symmetry groups, which is not covered in the textbook (except for brief mention in Section 4.4). Finally, we have been looking at fractals, the Mandelbrot Set, the Chaos Game, and fractal dimension; except for the Mandelbrot Set, this material is from Sections 7.1, 7.2, and 7.3.

Here are some of the things that you should know about for the test:

Newcomb's Paradox
    -- the general setup; what are the two choices?
    -- the argument for taking both boxes (you get $1000 more)
    -- the argument for taking one box (the expected value is greater)
    
"geometric infinities" -- cardinality of geometric sets of points
    -- finding one-to-one correspondences between points on geometric objects
    -- all line segments have the same cardinality
    -- a line segment has the same cardinality as the entire real line
    -- stereographic projection of a circle minus its top point onto a line
    -- a big circle has the same cardinality as a little circle
    -- a line segment has the same cardinality as a square
    
symmetry -- something is left unchanged after a transformation is applied
bilateral symmetry (basic reflection symmetry, the D1 dihedral group)
possible symmetry operations on patterns in the plane:
    -- reflection
    -- translation
    -- rotation
    -- glide reflection
rosette patterns
    -- patterns with rotation symmetry only (Rn rotation groups)
    -- patterns with rotation and reflection symmetry (Dn dihedral groups)
frieze patterns (translation symmetry in one direction only)
wallpaper patterns (translation symmetry in two independent directions)
identifying the symmetries of rosette, frieze, and wallpaper patterns
what it means to be a symmetry "group"
    -- "doing nothing" is a [boring] symmetry operation, called the "identity"
    -- one symmetry operation followed by another symmetry operation is a symmetry operation
    -- the inverse of a symmetry operation is a symmetry operation

fractals
self-similarity
classic fractal:  a figure is made up of identical reduced-size copies of itself
Sierpinski triangle; how to construct a Sierpinski triangle
Koch curve; how to construct a Koch curve
Sierpinski carpet; how to construct a Sierpinski carpet
Chaos Game; "maps" in the Chaos Game
how to get various fractals using the Chaos Game
dimension
how dimension relates to the number of copies needed to build a larger copy of an object
fractal dimension
the formula  d = ln(N)/ln(S) for dimension of classic fractals
how to apply the dimension formula to specific fractals

the Mandelbrot set
zooming in on the Mandelbrot set to experience its infinite complexity

Here are some exercises from the textbook that you could look at:

10.1.37
3.5.6 through 3.5.10
7.2.21, 7.2,22, 7.2.25
7.3.8, 7.3.11, 7.3.13, 7.3.15, 7.3.16

You should be able to identify the rosette symmetry groups of patterns like these, where the answers could be stated for example as R3 or D4:

You should be able to identify symmetries of wallpaper and frieze patterns. For example, you should be able to identify centers of rotation and whether it's a 2-, 3-, 4-, or 6-fold rotation. You should be able to identify lines of reflection symmetry. And you should be able to find translation and glide reflection symmetries.