Let's wrap things up a little...

Here's what we studied this semester:

Fun and Games
- Some of those games were dramatic foreshadowing... e.g. dodge ball
- Others merely taught us important life lessons: e.g., when things get complicated, try a simpler example (e.g. the shakey situation); or just do it.
- Generalize interesting problems.
Estimation
- You need useful referents. There's no substitute for a little knowledge socked away....
  - How many people in the world?
  - How many Americans?
- Approximation by mathematical objects
- Pidgeonhole Principle (remember musical chairs)
- Exaggeration may prove useful.
- Don't forget bounds (e.g. polls that are given with a "plus or minus 3%"). Estimates aren't always useful: bounds give limits on the possible.
- Pay attention to units.
Number Systems
- Numbers have personalities.
- There are many different ways to express a number:
  - Using powers: choose a given base in which to write it as a decimal (e.g. base 8, base 2 -- binary);
  - Using prime numbers and products (that is, prime factorization); or
  - Using Fibonacci numbers and sums.
- Nature is full of Fibonacci numbers and patterns
- Fibonacci Nim (how to play, and how to win!)
Infinity
- We've seen some infinite sets: the natural numbers, primes, Fibonaccis, etc.
- One-to-one correspondence is essential for measuring the "sizes" of sets, and particularly infinite sets.
- Infinity is a subtle concept. The smallest one is the countable infinity, an example of which is the natural numbers.
- The power set (set of all subsets of a set) shows (and diagonalization -- dodge ball -- shows) that there are small infinities and larger ones.
- There are, in fact, an infinite number of successively larger infinities (measured by cardinality).
- Hotel infinity examples; ping pong ball conundrum.
Dimension
- 0, 1, 2, 3, 4-dimensional objects and beings
- Learning about complicated things (the 4th dimension) by analogy -- by considering something simpler first
- How do we mess with each others' dimensions?
- What do things from higher dimensions look like in lower dimensions?
Geometry
- Golden Rectangles and golden ratio (Fibonacci again!)
- Logarithmic (golden) Spirals
- Regular polygons (e.g. hexagons)
- Platonic solids -- all five of them! Symmetry, duality
- Borromean rings (bringing it all together: icosahedra, golden rectangles, and interlocking rings)
Lying with Statistics and Graphs
- Measures of central tendency: Mean, median
- When averages are not typical: "The average American has one testicle and one ovary."
- Measures of spread: range, deviations, standard deviation; spread in polls.
- Histograms -- visualizing mean, median, spread, skew (mean chases extreme values)
- Distribution of data (e.g. normal distribution, or bell-shaped curve)
- Using randomness to our advantage: answering sensitive questions (one- and two-toss methods -- lying to protect anonymity).
Dynamics
- Simple rules and Iterations may lead to complicated dynamics
- Social Dynamics are incredibly complex
- Chaos: sensitive dependence on initial conditions.
- Population Dynamics
- The Game of Life (know the rules and how to play!)
Fractals
- Simple rules lead to complicated pictures.
- "Worlds within worlds" (self-similarity).
- Nature must use the principles behind fractals, as evidenced by broccoli, coastlines, or ferns.
- Do it once; then do it again! We made fractals starting from a single straight line segment, or by dividing up geometric figures (e.g. squares or triangles), or even donuts!
- The collage-making process, and fractal dust.
- The chaos game.
- The Fibonacci Spiral as a fractal.
Financial Math
- Compound interest problems (dynamics are the same as simple population dynamics) -- simple rules, applied over and over.
- How do Mortgages and loans work?
- What is inflation? How does it impact your money?

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