I think it's healthy and productive in mathematics to find these areas where people agree to disagree, or else go off the deep end and get full blown religious, mount some kind of jihad.
The trapezoid example is especially valuable because it's so accessible. To find some inclusive / exclusive dialectic in topology more generally might require too much background to let the reader even care.
In the programming world I frequent, we have similar vicious debates between camps which boil down to who is most willing to see it as either/or.
"Complementary" and "Oppositional" are different concepts for a reason.
Remember "opposites attract": sometimes a divergence is a basis for strong bonding. The difference is treasured.
On Thu, May 2, 2013 at 1:48 PM, Wayne Bishop <firstname.lastname@example.org> wrote: > But he is also right. Although favored in some circles, it is one of those > (rather few) situations where the inclusive definition is not universal. In > fact, I think the majority of us prefer his, exactly 1 pair of parallel > sides. Moreover, convictions border on the religious. You know, God is on > our side whichever side that is. > > Wayne > > > At 08:57 PM 5/1/2013, kirby urner wrote: >> >> Yes, parallelograms are quadrilaterals with both pairs of sides parallel, >> meaning they include rhombi (all sides same length) and rectangles >> (all angles same size). >> >> A square is a "rhombic rectangle". >> >> Trapezoids have at most one pair of parallel edges according to the >> exclusive definition (versus "at least one"). >> >> The trapezoids we don't see so often are the ones where the base >> angles are not both < 90 degrees, or both > 90 >> >> >> http://zonalandeducation.com/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html >> (second from last example) >> >> This source makes it clear that the inclusive definition is favored in >> some circles. >> >> >> http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html >> >> This inclusive definition does allow us to speak of parallelograms >> as a subspecies of trapezoid. >> >> Kirby >> >> On Wed, May 1, 2013 at 1:57 PM, Dongwahn Suh <email@example.com> >> wrote: >> > I remember teaching quadrilaterals and creating a tree diagram to >> > differentiate and connect the characteristics of quadrilaterals. Stemming >> > out of quadrilaterals, the parallelogram then breaks up into rhombii and >> > rectangles, which then combine to form a square. The trapezoid drops down >> > into its own stem and then from the trapezoid was the special isosceles >> > trapezoid. Since parallelograms must have two pairs of parallel sides, the >> > trapezoid only has two parallel sides and no more. Otherwise we would be >> > able to categorize some trapezoids as parallelograms.