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Re: Inclusive and exclusive definitions... again!
Posted:
Mar 28, 2010 3:04 PM
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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 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.
>
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".
Cromwell, in his book 'Polyhedra' describes how they've become more
gossamer, more web-like, less "solid", over the centuries.
When doing flash card Kanji, you get these kinds of associations, e.g.
KYO for capital city, has a hat glyph on top, and shows up in words
like KYOto and ToKYO.
In other words, we don't just want to say "Hedron" and be done with
it. We want to look somewhat explicitly at the etymologies, going
across languages as necessary.
This helps root geometry in the humanities more, a radical concept to
some no doubt, but how we do it in the better schools.
> 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.
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.
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.
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.
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.
Kirby
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