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Topic: Inclusive and exclusive definitions... again!
Replies: 17   Last Post: Mar 29, 2010 10:12 PM

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Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05
Re: Inclusive and exclusive definitions... again!
Posted: Mar 15, 2010 12:52 PM
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Allan and others,

All these questions about geometric shapes and their names is
interesting and worth considering. A related issue about whether
equilateral triangles should be considered isosceles as well,
which I had begun a discussion about last summer on mathedu,
showed me that there is no consensus--even among mathematicians--
on this issue.

I too prefer an inclusive approach to most of these definitions
since many theorems that apply to one of these shapes will hold
for other shapes if we use these inclusive definitions. And
many mathematical definitions in other areas are also inclusive.
For example, groups are semigroups as well (at least according to
the definitions of semigroup that I have seen). Inner product
spaces are also vector spaces. Vector spaces over a field F
are F-modules. Differentiable functions might be twice-
differentiable, thrice-differentiable, etc. or even infinitely
differentiable (not necessarily, of course).

I see nothing wrong with using the term "oblong" for a
rectangle that is not a square.

"Oblique rhombus" works well for a rhombus that is not a
square because "oblique" often is used to describe a geometric
figure without any right angles or one having an axis not
perpendicular to the base. And these descriptions fit these
kinds of rhombuses. The term "diamond" does not sound good
to me--at least not as a formal mathematical term because it sounds
too casual to me--and is misleading because a baseball diamond is
actually a square. It is called a diamond only because of
its position relative to the viewers. But geometric shapes
such as these are not named for the position they are in.

I have seen the term "rhomboid" in several books, and this
word is used in the sense you are using it here. Wikipedia
labels this term as a traditional term. Since it is an
acceptable term already, why not use it? The only
problem I have with it is that the term suggests that such
shapes ought to be related to rhombuses in some way.
But certain other words in mathematics are not accurate either:
Examples include "imaginary number," "real number," and "negative
number." If I had to reject "rhomboid" for this reason, I would
have to reject these other words as well, but I have no viable
alternatives for these words.

I would support Conway's use of the word "strombus." I don't know
if he has included this term in his books yet.

The term "scalene trapezoid" sounds good to me since it is an
extension of the term "scalene triangle" for a triangle that is
not an isosceles triangle. And I see nothing wrong with using
the term "oblique scalene trapezoid" as well.

How about the term "oblique isosceles trapezoid" for isosceles
trapezoids that are not rectangles?

Jonathan Groves

On 3/7/2010 at 7:47 pm, Allan Turton wrote:

> After wrestling with definitions and hierarchies for
> quadrilaterals yet again, I'd like to get some other
> opinions on the matter. I adopt inclusive definitions
> for the good reasons offered by John Conway, Walter
> Whiteley, et al. To avoid all the arguments about
> what is included in an inclusive definition of
> quadrilaterals, I've included a link to a picture
> that should show it clearly enough:
> -shapes/ A word of warning: it is written for UK and
> US audiences so UK trapezium = US trapezoid. As a
> refresher to the inclusive definition of trapezoid,
> it means that it has *at least* (not *exactly*) one
> pair of parallel sides.
> What I think compounds the problem of teaching an
> inclusive approach is what to call the "endpoints".
> For example, following a line of descent down from
> trapezoids, through parallelograms then rectangles,
> we come to two end points - either a "square" or a
> "non-square rectangle". Now, as John Conway noted,
> "oblong" is a perfectly acceptable term to use
> instead of "non-square rectangle". So, we now have
> two very clear terms to describe related (but
> different) shapes that we can say are both
> rectangles.
> But what names do we have for the other end points?
> Looking at the class of shapes we call rhombuses, we
> have endpoints of "squares" and "non-square
> rhombuses". The latter term is a bit of a mouthful
> and the only alternatives I've found are "oblique
> rhombus", "lozenge" and "diamond". You'll see in the
> diagram that it is just labelled "other". The term
> "diamond" has become dreadfully unfashionable for
> reasons I haven't been able to fathom, yet in
> everyday use it almost always refers to a non-square
> rhombus. It seems perfectly suited to the job! Anyone
> know of the sources that put the death warrant on
> "diamond"?
> Stepping up a level to look at parallelograms, we
> have ones that are either rhombuses, rectangles or
> "non-rhombic, non-rectangular parallelograms". Awful.
> A suitable replacement is "rhomboid" which is
> underused.
> John Conway, in looking at the class of kites,
> suggested that kites could be divided into the end
> points of "rhombuses" and "strombuses", "strombus"
> being used for the toy shape (two adjacent short
> sides, two adjacent long sides). He mentioned that he
> hoped to include the term in books he was writing -
> does anyone know if this happened?
> As for trapezoids, well, what should we call
> "non-isosceles trapezoids"? I know there are "right
> trapezoids", but what about the ones that are
> non-right and non-isosceles? "Scalene trapezoid" is
> about the best I can think of, but even this isn't
> truly accurate for the whole shape, just the "sides".
> Isosceles trapezoids branch into "rectangles" (then
> onto "squares" or "oblongs") and "non-rectangular
> isosceles trapezoids". I am at a complete loss as to
> what to call the latter. This naming problem would
> provide the only reason I'd have to exclude
> trapezoids from having more than one set of parallel
> sides. What do we call it apart from non-rectangular?
> For such an omnipresent shape (kids see them all the
> time in pattern blocks) it is no wonder that it is
> often seen as the only representative of the term
> "trapezoid" - there simply doesn't appear to be any
> other name for it!
> In outside-school use, and especially in the early
> years of school, we use the "end points" more often
> than the "classes" - they have more utility. I think
> that once there are names established for the end
> points then the higher classes of shapes can be more
> easily introduced. For example, using "rhomboid" to
> label that particular type of shape would have to be
> better than labelling it with the higher order name
> of "parallelogram", then trying to convince kids
> later that sometimes when we talk about
> "parallelograms" we're actually meaning a whole bunch
> of shapes, some equiangular, some equilateral, some
> both and some neither!
> I hope I've made sense. Can anyone provide some words
> of wisdom apart from "relax"?
> P.S. This has also been posted in the
> "geometry.pre-college" discussion list too, so
> apologies if you're reading this twice.

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