Day 21: Heart of Mathematics

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Announcements:
- Homework for Section 7.6 returned (graded 9, 22)
- We have an exam Wednesday. Details on the types of problems you're likely to encounter from three units:
  - Dimension
    - 0, 1, 2, 3 dimensional objects
    - Impressions of higher dimensional objects in lower dimensional spaces.
    - Slicing and Dicing
    - Time as a fourth dimension
    - Grappling with the fourth dimension
    - How angels and ghosts and monsters from the fourth dimension could be impacting us....
  - Geometry
    - Regular Polygons
    - Platonic solids
      - Duality
      - Their names and numbers
      - Uses of platonic solids
    - Golden Rectanges
    - Fibonacci spirals
    - Borromean Rings
  - Lying with Statistics and Graphs
    - Histograms
    - Measures of central tendency (means versus medians)
    - Problems and advantages with each
    - Standard deviation
    - Using randomness to answer embarrassing questions
      - one-toss (we lose half the data)
      - two-tosses (we don't toss out any data!)
- Remember: your assignment from section 7.5: pp. 581-: #1, 3, 4, 8, 12. will be attached to your exam, and constitute 10% of the exam. There will likely be an additional problem on the exam like these, to show me if you really understand the procedure.
  From an email to a student:
  For "short test" (#3), you should think about a couple of things:
  1. if all the students know the answers to the same 80% of the material, then it depends on which question the professor asks;
  2. if students know different parts of the material, then it doesn't depend so much on which question the professor asks.
  For number 4 (computer polls), there's no formula to use. You're just supposed to reason this through.
Section 7.5: Collecting Data rather than dust (the power and pitfalls of statistics) -- aka "Lying with Statistics (for good!;)"
- Activity: Let's answer an embarrassing question: are you male?
  - We'll attempt to calculate the true rates of maleness and femaleness in class...
  - Each person will flip a coin:
    - If it's heads, answer "Male"
    - If it's tails, answer truthfully.
  - How do we now estimate the true rate?
    - Guess that we'll have half the people say "Male" because they tossed a heads. The excess (above half) are truthful "male" answers.
  - We guessed:
    - M=7+8 (7 "true males", 8 "false males"))
    - F=4 (4 "true females")
    - True rate: 7/19
    - Estimated rate: (15-9.5)/9.5=5.5/9.5=11/19
    - Where did the 9.5 come from?
- More examples:
  - Mindscape #2, p. 581.
  - How likely are you to use your cell phone while driving?
  - Now: what embarrassing question should we now attempt to answer? Who's got something they really want to know?
- How could we adapt this process to lose less of our data (generalize, when you get a good idea!)
  - with a die
  - with two tosses of a coin?
Dynamics:
- For today: you were to read section 6.2: The Dynamics of Change
- Today we continue our unit on dynamics. The word "dynamics" has two slightly different senses:
  - changes in time, and
  - how one thing leads to another.
  Now, the latter happens in time, so it seems to be just an example of the former. But there are certain examples which we will consider which don't quite seem to fit the former sense.
  We started with examples of what I call "social dynamics", that come in three flavors:
  - Good dynamics
  - Neutral dynamics
  - Bad dynamics
- Last we took a look at some examples, and discussed them. We you able to think of examples (of each kind, relationships between them, or other examples)? E.g. MAD -- Mutually Assured Destruction.
- In section 6.2: The Dynamics of Change, our authors give examples of the use of dynamics:
  - Personal Finance (more about this later, in our personal finance unit).
    Housing prices as a dynamical system
  - Biological populations. Example: How many people do you suppose will be on the earth in 2109 (in a hundred years)?
    - Start Somewhere (collect data!)
      World Population clock
    - Apply a process and get a result.
    - Apply the same process to that result to get a new result.
    - Apply the same process to the new result to get a newer result.
    - Repeat patiently and persistently, forever.
  - The Game of Life:
    1. Is a poster child to prove something important: simple rules, repeated over and over, can give very complicated dynamics.
      Why are simple rules that result in complex phenomena important? Here's an important example:
      - Some people doubt the "theory of evolution", because they don't believe that simple rules can result in the complexity we see around us.
      - If scientists can demonstrate that complex phenomena can arise from simple phenomena, then that argument holds less water.
    2. Is generally played on an infinite checkerboard, or on a world that wraps back around on itself.
    3. The rules are:
      - A living square will remain alive in the next generation if exactly two or three of the adjoining eight squares are alive in this generation; otherwise, it will die.
      - A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.
    4. the website's version
    5. I.12, p. 423
    6. Create your own creatures!
      - Can you think of one that will lead to extinction?
      - Can you think of a new one that will lead to a stable pattern?
Links:

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