February 21, 2006

From MathWiki

  • There was an editorial in the Toronto Star (http://garsia.math.yorku.ca/~zabrocki/math1590w06/articles/algebrasucks.pdf) which was syndicated from the Washington Post. The point of this article is that there are some people (like the author of the editorial, but also a woman named 'Gabriela') who have a hard time with algebra. The author argued that one could live a full life without knowing algebra.
  • We had a discussion of the editorial as well as some of the responses to that editorial.
    • A blog on the subject (http://scienceblogs.com/pharyngula/2006/02/richard_cohen_advocate_for_ign.php)
    • two editorial responses (http://garsia.math.yorku.ca/~zabrocki/math1590w06/articles/twoanswers.pdf) that read what I did in that article, namely, it is OK to go through life without knowing basic mathematics and you will survive fine.
  • We talked about the homework problems and I gave you an extra week on them. The problem that I gave you about the maximum number of regions can you get when you divide space with n planes is called 'the lazy caterers problem' or the 'cake numbers (http://www.research.att.com/~njas/sequences/A000125)' because it can be phrased in terms of what is the maximum number of pieces of cake you can get if you make n cuts with a knife. This is hard to solve but worth thinking about.
  • We talked about quadrilaterals. We started by listing all the classes of quadrilaterals and giving a definition of each. ( I am not sure I got the definitions exactly as we had named them ). You will remember that I said that Wikipedia was a good reference for correct definitions of these shapes and then we found that it had a mistake on the picture of 'isoscelese trapezoid.' I put my foot in my mouth about that one. However, within the week the mistake had been corrected and I didn't contact them.
    • quadrilateral (http://en.wikipedia.org/wiki/Quadrilateral) - a polygon with 4 sides
    • square (http://en.wikipedia.org/wiki/Square_%28geometry%29) - a quadrilateral with 4 equal sides and 4 equal angles
    • rectangle (http://en.wikipedia.org/wiki/Rectangle) - quadrilateral with 4 equal angles
    • parallelogram (http://en.wikipedia.org/wiki/Parallelogram) - quadrilateral with 2 pairs of parallel sides
    • trapazoid (http://en.wikipedia.org/wiki/Trapazoid) - quadrilateral with 1 pair of parallel sides
    • rhombus (http://en.wikipedia.org/wiki/Rhombus) - quadrilateral with all 4 sides are equal
    • isoscelese trapazoid (http://en.wikipedia.org/wiki/Isosceles_trapezoid) - quadrilateral with 1 pair of parallel sides and two adjacent angles which are equal.
    • kite (http://en.wikipedia.org/wiki/Geometric_kite) - a quadrilateral with two pairs of adjacent equal sides.
  • We remarked that
    • every square is a rectangle
    • every square, rectangle, rhombus are parallelograms
    • every square is a rhombus
    • every square, rectangle, parallelogram and rhombus are trapazoids
    • every square and rectangle is an isosceles trapazoid
  • Put it together and what do you get? The following picture.

Image:poset.jpg

  • For the discussion next week I would like you to look at each class of quadrilateral that we have listed here and see what symmetries they have (of the following list):
    • rotation by 90 degrees
    • rotation by 180 degrees
    • flip across a diagonal (or both)
    • flip across a horizontal/vertical (or both)


Previous class : February 7, 2006

Next class : February 28, 2006

Main class page : Mathematics 1590 Nature of Mathematics II