Day 28 in Mat115H

Today:

• Announcements:
  • Let's finish what we were working on: red and rusty bikes

  • Return of your Assignment on logic: your write up (and resolution) of some syllogisms of Lewis Carroll.

    In every case we want to think about the universe (the single universe!) of discussion.

    1. Dragons and "Scotchmen": universe=creatures. Given, that
       1. All Dragons are uncanny;
       2. All Scotchmen are canny.
       What can we conclude?
       • No dragon is a Scotchman.
       • No Scotchman is a dragon.
       • All Scotchmen are not dragons.
       While we may conclude that
       • No dragon is canny, or
       • No Scotchman is uncanny,
       These are simply restatements of the original propositions, and don't resolve the two propositions into a common conclusion.

    2. Harder! Universe=children of mine
       1. All my sons are slim.
       2. No child of mine is healthy who takes no exercise.
       3. All gluttons, who are children of mine, are fat.
       4. No daughter of mine takes any exercise.

       Possible Resolutions (but there are more!):

       • All my daughters are unhealthy.
       • No daughter of mine is healthy.
       • My sons are not gluttons.

  • Assignment due next time: Test of trios

  • Last new assignment (due Thursday, 12/10): Find two new items to add to the Lewis Carroll timeline. Items relevant to the mathematics you're presenting is best....

  • Assignment of days for presentations: coming Tuesday!
    • Dylan
    • Jillian
    • Daniel
    • Hannah
    • Holly
    • Annna
    You'll have 10 minutes, and should try to use all that time. There will be a brief period for questions between presentations.

    I'm distributing copies of the rubric which each person will use to evaluate your presentation.

  • A word on the final (Thursday, 12/17, 1:00). It will be comprehensive, but will emphasize the material from Fit the Seventh to the Conclusion. Mathematically, you can expect to see some problems involving or resembling
    • Knot the First: Excelsior
    • River-Crossing problems
    • Calculating the Day of the Week
    • The Game of Logic
    You should also expect to see some specific questions about Lewis Carroll's life.

• The Game of Logic, Fit the Eighth:
  • Do you have any more questions?

• Conclusion: Math and Aftermath
  • Events in Lewis Carroll's life? What do we learn?

  • "Pillow Problems" (p. 199):
    • Curiosa Mathematica: Pillow-problems thought out during wakeful hours. (check out the page before the title page in this volume)
    • Appeared 1893
    • #29 (p. 204) is not original with Dodgson: it's the Brahmagupta-Fibonacci identity
    • Amusing #72 (p. 203).

  • More "number guessing", and divisibility: he liked those topics!
    • You know the divisibility rule for 2.
    • Do you know the rule for 3?
    • Do you know the rule for 9?
    • Do you know the rule for 11?

  • Right-angled Triangles -- his last mathematics (December 19, 1897)

    These are called Pythagorean Triples, and we're looking for Pythagorean triples which represent triangles having the same areas.

    It turns out that Pythagorean triples all have the form

    

    for arbitrary integers s and t.

  • The end comes... Falls sick January 6th, and dies January 14, 1898

  • The series

• His own mathematical epitaph: Yet what are all such gaieties to me
  Whose thoughts are full of indices and surds?
  

  (who but Lewis Carroll would put a fraction into a poem?!).