rectangle vs. square

Laurence Horn laurence.horn at YALE.EDU
Mon Jun 28 00:50:09 UTC 2010

At 5:55 AM +0100 6/27/10, Robin Hamilton wrote:
>>---------------------- Information from the mail
>>header -----------------------
>>Sender:       American Dialect Society <ADS-L at LISTSERV.UGA.EDU>
>>Poster:       victor steinbok <aardvark66 at GMAIL.COM>
>>Subject:      Re: rectangle vs. square
>>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
>Yeah, I think "oblong" is the crunch (or one of the crunches) in this
>{Apropos of nothing, thinking about Victor's "What is an oblong?" it struck
>me that I'd term the shape of a coffin "oblong" rather than "rectangular".
>For what that's worth.}
>But coming back to ....
>        Polygon > Quadrilateral > Rectangle > square + oblong.
>A Quadrilateral is a subset of the set:  Polygon
>A Rectangle is a subset of the set:  Quadrilateral
>A Square is a subset of the set:  Rectangle

But as Victor points out below, a square is a also a rhombus, which
is a quadrilateral but not necessarily a rectangle.  (A square is an
equilateral rectangle and an equiangular rhombus.)  And just as a
"prototype" rectangle is an oblong, with unequal-length sides, so too
a "prototype" rhombus is a non-rectangle, with unequal angles.


>An Oblong is (also) a subset of the set:  Rectangle -- *but it's defined as
>"All those members of the set:  Rectangle which are not members of the set:
>So "oblong" is where it goes drifty.  Inelegant to say the least, whether or
>not it makes sense in the domain of mathematics or symbolic logic.
>Except that we need the word in general language usage, even if only to
>apply to a coffin.
>>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.  [...]

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