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

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GS Chandy

Posts: 7,738
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05
Re: Inclusive and exclusive definitions... again!
Posted: Mar 28, 2010 7:15 PM
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Kirby Urner's fascinating post of Mar 29, 2010 12:34 AM:
> On Mon, Mar 15, 2010 at 9:52 AM, Jonathan Groves
> <JGroves@kaplan.edu> wrote:

> > Allan and others,
> >
> > All these questions about geometric shapes and
> >their names is interesting and worth considering.  A
> >related issue

<snip>
> Ah, nomenclature...
>
> I'd cut and paste the Japanese I've got, for lots of
> these words.
>
> In the international curriculum, we use geometric
> terms as an excuse to go multi-lingual, to start using
> Google Translator or some such perhaps.
>
> Poly: many
> Hedra: faces
> Gon: knees
>
> Interesting that polygon relates to the body part
> "knee" and hedron to "face".
>
> In a graph-like polyhedron, the faces become
> "windows" or "openings".
>

<snip>
> > 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.

>
> That's the point of namespaces. You get to be up
> front with your preferences, give the definitions right
> there and then.
>
> The mistake too many curriculum writers make is
> thinking they need to define "global variables" e.g.
> "from now on, everyone in the world
> will mean X by Y, and here are my arguments why that
> should be the case..." -- the premise is bogus to begin
> with, so why work through the arguments.
>
> On the contrary, if you have a preferred meaning for
> "oblong" or "skew" or "rhombus" or "trapezoid", then
> just spell it out right there and then in your math
> text. That's what maths encourage: up front
> treatment of definitions, no relying in "implicit
> globals". These are strong ideas in computer science as
> well -- must mean we share a lot of the same heritage,
> eh?
>

> > 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).

>
> Everyone writing a math book is going to produce a
> bunch of definitions.
>
> Here's a geometry of lumps:
>
> Topological features:
>
> node (vertex, point)
> edge (vector, segment)
> window (face, opening)
> graph (network, polyhedron)
>
> Are those going to be the same associations you get
> in every math language under the sun? No, of course
> not.
>
> I call it a geometry of lumps after an essay by
> dimension theorist Karl Menger.
>

As an aside, it was Karl Menger I believe that conceived of a 'monstrous structure' that came to be called the "Menger sponge", which is created by iteratively cutting away inner one-third sections of the original cube...forever. You finally end up with a astonishing 'thing' of zero volume and infinite surface area! I can't recall what number of 'dimensions' this sponge has - but it's NOT three. I once did a calculation of how the volume and the surface area of the Menger sponge approached zero and infinity respectively - and it literally made my head spin. Here a link to some fascinating pictures of 'Menger sponges' and structures developed from that concept: http://images.google.co.in/images?hl=en&q=menger+sponge&um=1&ie=UTF-8&ei=Jd2vS5SIJIPGrAe91KFB&sa=X&oi=image_result_group&ct=title&resnum=5&ved=0CCgQsAQwBA - tinyURL: http://tinyurl.com/ybh5fvb ).

(The Menger sponge is, probably, a development in 3-D space of the idea of the 'Sierpinski triangle' in which something similar is done to a triangle in a 2-D plane - you get a 'thing' consisting of zero area and infinite length of sides!)
>
> We introduce it as an example of a non-Euclidean
> geometry in that all the above features are lump-like
> and live in a tank or space we consider 4D instead of
> 3D.
>
> Kids dig it.
>

Indeed, they should. As they do the 'Menger sponge' and suchlike, I've found!
>
> If you don't get it at Cleveland high school, or
> Grant, maybe you'll get it at Reed College, from the
> Literature department. Or from Lewis & Clark,
> philosophy department.
>
> I've shared it through Saturday Academy and the Linus
> Pauling campus (also local).
>
> There's a link here to "place based" education, i.e.
> in being explicit about "local variables" (the
> definitions we wish to use for the purposes of this
> course) we're somehow asserting a locales
> prerogatives.
>

> >
> > 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.
> >

>
> Math languages often impress "informal words" to do
> "formal" work. Consider the kites and darts of Roger
> Penrose.
>
> Mathematicians have been known to get playful in
> their inventing of terms.
>

(As do physicists - see "quark")
>
> So what's "oblique" in Arabic?
>

> > 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 understand the phrase "imaginary numbers" has caused innumerable real headaches for students through the ages!
>
> Viable in what sense? If you mean to conquer the
> world with some new way of talking, then no, that's
> probably not a viable goal. But if you want to come up
> with some new terms locally, for the purposes of
> some class, I don't see why you couldn't. Use
> "complex number" in place of imaginary. Maybe explain
> we won't be using "real numbers" at
> all in this course, only floating point type and
> decimal type (along with integer type, Q type, C type or
> whatever).
>
> "Real numbers" have limited appeal and one may define
> a logical mathematics without them (they were only
> invented relatively recently,
> lets remember -- the greeks had no such notion, in
> the sense it's defined today in rarefied academic
> circles).
>

> >
> > 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

>
> These are matters of personal taste and design.
> Great curriculum writers do good local work. Students
> are glad for the up front definitions and clear use of
> them, operationally, in a consistent manner.
>
> What to avoid is enshrining one's taste in some "thou
> shalt" language, although it's certainly OK to advise
> students as to what are the more globally accepted
> meanings of this and that. If there's already a
> widely accepted usage for a term, it certainly makes
> sense to share that, before introducing a more local
> meaning in contrast.
>
> When categorizing polyhedra, some people take issue
> with some of the enshrined decisions. I've seen
> alternatives, deviations. I don't tend to castigate the
> geometers making these proposals, saying: thou
> shalt toe the party line. I respect them as math
> teachers, and listen to their arguments, which are
> sometimes quite cogent, quite thoughtful.
>
> Remember: place-based education is your friend.
> Feel free to localize. The emperor has no clothes
> after all.
>

Thanks for that most illuminating discussion, Kirby and Jonathan! Helped changed a few notions of mine.

GSC



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