rectangle vs. square

[victor steinbok]

I disagree in the strongest possible terms with at least one of these
assertions. To be honest, I have no idea what "oblong" is--in all my
mathematical education, I have never heard this even remotely
approaching any technical meaning. More precisely, I have never heard
it used to any mathematical object or property. But I've also never
have heard of anyone seriously contesting that a square is a
rectangle.

But a circle is an extreme or a degenerate case of an ellipse and
there are reasons to classify circles as a special subset. There are
obvious dynamic implications when looking at conic sections. In a
sense, a circle is also a transitional point between an ellipse and a
hyperbola, but I am not going to get into details why this is the
case. The transformational properties of ellipses are also represented
in a circle, although somewhat trivially so.

Squares are a subset of rectangles, and, in fact, the justification
for classifying them this way is even stronger. Mathematically,
pretending that these are distinct categories makes no sense at all.
That is, they are semantically distinct--the definition of a square is
not the same as a definition of a rectangle. On the other hand,
logically, they are not distinct--every definition that describes all
rectangle also describes all squares.

A square is a rectangle for which the two pairs of edges have the same
length. To make matters even more fun, it's also a rhombus. And these
two descriptions are very informative--they specify that a square has
all the properties of a rectangle and all the properties of a
rhombus--something that the remaining rectangles and rhombuses cannot
claim. The reason I stated that the square-rectangle relationship is
somewhat stronger is because the only distinguishing characteristic of
squares compared to the other rectangles is the congruence of adjacent
edges. The distinction between a circle and an ellipse is a bit more
robust--the two foci of the ellipse collapse at the center of the
circle, so there are some distinct properties there.

The bizarre recategorization occurs because of elementary school text
written by illiterates (some of whom also teach in their spare time).
For this reason one may find new quadrilateral poseurs in textbooks,
such as "diamonds", "kites", etc. These are given formal but
meaningless definitions (diamonds are really just rhombuses, excluding
the squares, kites are two isosceles triangles joined at the base,
but, again, not including the squares, i.e., those with the two
triangles non-congruent). But these definitions are mathematically
vacuous, just as the definition of a rectangle that does not include
the square (i.e., the usual definition with an additional proviso that
the two pairs of edges are not congruent to each other). Generally,
you will find few exclusions in traditional definitions--you're more
likely to find /additional/ definitions that define special cases. The
goal is to generalize, not to exclude.

VS-)

On Sat, Jun 26, 2010 at 8:17 PM, Dan Goncharoff <thegonch at gmail.com> wrote:
>
> First, when did "rectangle" ever include "oblong"?
>
> Second, what would be the justification for using "ellipse" when describing
> a circle, or rectangle for square? It may be correct, but it's not
> informative.
>
> DanG

------------------------------------------------------------
The American Dialect Society - http://www.americandialect.org