GS Chandy
Posts:
7,980
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05


Re: Inclusive and exclusive definitions... again!
Posted:
Mar 28, 2010 7:15 PM


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 multilingual, 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 graphlike 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 Fmodules. Differentiable functions might be > > twicedifferentiable, thricedifferentiable, 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 onethird 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=UTF8&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 3D space of the idea of the 'Sierpinski triangle' in which something similar is done to a triangle in a 2D plane  you get a 'thing' consisting of zero area and infinite length of sides!) > > We introduce it as an example of a nonEuclidean > geometry in that all the above features are lumplike > 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 meat least not > > as a formal mathematical term because it sounds > > too casual to meand 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: placebased 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

