March 7, 2006

From MathWiki

  • We spent most of the time talking about quadralaterals. We also discussed for a little while the homework problems on cube painting and milk cartons.
  • REMEMBER: Part of your assignments are to bring in news articles. Please do this soon. You should write a short paragraph that explains what your news story has to do with mathematics.
  • We discussed some issues with the definitions that I had posted on this site earlier. We changed the definitions that I had written to bring them in line with wikipedia. These new definitions are shorter and contain a minimal amount of information to define the class clearly.
    • old definition: rhombus - quadrilateral with 2 pairs of parallel sides and all 4 sides are equal
    • new definition: rhombus - quadrilateral with all 4 sides are equal (e.g. we don't need to say we have 2 pairs of parallel sides because that will be forced on us).
    • old definition: square - a quadrilateral with 4 equal sides and 4 right angles
    • new definition: square - a quadrilateral with 4 equal sides and 4 equal angles
    • old definition: rectangle - quadrilateral with 2 pairs of equal sides and 4 right angles
    • new definition rectangle - quadrilateral with 4 equal angles
  • We talked a little bit about why one definition would be better than another. We tend to prefer definitions which are shorter but which contain just enough information to define the class. For instance in a rectangle we don't need to say th the quadrilateral has "2 pairs of equal sides and 4 right angles" because it follows from the fact that it has 4 equal angles that those angles must be right angles (90 degrees) and because those angles are right that opposite sides will have equal length.
  • If you look in books and on web pages you will often find seriously different definitions for some quadrilaterals (in particular trapazoid: is a parallelogram a trapazoid?). Consider the following list of links:
    • MathWorld (http://mathworld.wolfram.com/Trapezoid.html) - A trapezoid is a quadrilateral with two sides parallel
    • UIUC (http://www.geom.uiuc.edu/~dwiggins/conj19.html) - A trapezoid is a quadrilateral with exactly one pair of parallel sides
    • id.mind.net (http://id.mind.net/~zona/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html) - The trapezoid is a quadrilateral with one pair of parallel sides.
    • Wikipedia (http://en.wikipedia.org/wiki/Isosceles_trapezoid) - you can either define it with EXACTLY one pair of parallel sides or AT LEAST one pair of parallel sides. FYI, these definitions are not the same but they are both used. In the case of the link at id.mind.net we see that it is not even clear which one is being used.
  • Depending on which definition you take above, there are either two or three trapazoids in the picture below.

Image:trapazoids.jpg

  • I asked you what symmetries each of the types of quadrialaterals had. Of course it depends on the orientation of the shape, but assuming that the long edge is along the horizontal then which of the quadrilaterals have
    • H - horizontal, V - vertical
    • D1, D2 - flip across the diagonals
    • R90, R180, R270 - rotate 90, 180, 270
    • R360 - essentially do nothing
  • We noticed that
    • squares are left invariant under { H, V, D1, D2, R90, R180, R270, R360 }
    • rectangles are left invariant by { H, V, R180, R360 }
    • rhombi (rhombuses?) are left invariant by { D1, D2, R180, R360 }
    • kites are left invariant by { D1, R360 } or { D2, R360 } (but not both unless it is also a rhombus)
    • isoscelese trapazoids are left invariant by { V, R360 }
    • parallelograms are left invariant by { R180, R360 }
    • trapazoids and quadrilaterals don't have any symmetries in general
  • We then considered what types of symmetries are possible. There are natural restrictions of what subsets of symmetries will ever arise.
    • We showed that if a shape is left invariant under H and V, then it must also be invariant under R180
    • We showed that if a shape has H and D1 symmetry then it will also have R90
    • If a shape has R90 symmetry then it will also have R180 and R270
    • There were all types of restrictions of this sort. We constructed a diagram of all possible sets of symmetries
  • We did an exercise where we fold paper and ask what shapes we see (and why).

Previous class : February 28, 2006

Next class : March 14, 2006

Main class page : Mathematics 1590 Nature of Mathematics II