Re: Isn't it the limit?

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

On Sat, 29 Mar 1997, Fred Bucheit wrote:

> It isn't the limit because the triangles when flattened out are no longer
> tirangles!

The weren't tirangles before flattening, Fred. (Couldn't resist. But my
own proofreading has been so shabby lately that I shouldn't criticize.)

Seriously, I've been most interested in the responses so far. You are the
first to respond to the crux of the problem, rather than peripheral and
unimportant matters. In fact that was the reason I chose the subject
header line for this problem. I had said that the figure bounded
by two longitude lines from pole to equator, and an arc at the equator (a
spherical triangle) goes to an ordinary plane triangle in the limit as the
arc goes to zero. Now here comes the crucial step deliberately left
unstated in the original post, and which seems to students as intuitively
correct as the other area subdivision methods which *were* successful.

	In the limit this spherical triangle becomes infinitesimally
	skinny, and can then be peeled away from the sphere without
	changing its shape being infinitesimally close in all respects to
	a plane triangle with straight sides. 

Well, isn't it the limit? No, but for a more subtle reason than has been
revealed so far. 

My question centered on the apparent paradox *as seen by the student*. The
failure of this limit makes all previous area subdivision tricks suspect,
for example the ribbon around the sphere along a latitude line, used to
(correctly) calculate the area of a sphere. The student has trouble at
first realising that you must use the strip width, not the height measured
along the polar axis, but does appreciate that the two choices lead to
different answers. Choosing dw gives the correct answer (agrees with that
given in all the books!) and choosing dh gives a wrong answer. Still the
student wants *guidance* to avoid making wrong choices in the future. A
reasonable enough request, to which we ought to respond. 

To explain the correct method of choosing the ribbon area, you can show a
triangle with sides dw, dy, and dr, where w is the arc along the sphere
surface, y is height measured parallel to the polar axis, and dr is the
distance from strip to polar axis, measured perpendicular to the polar
axis. We treat this as a triangle, and derive the relation between dw and
dy. But it isn't a triangle, is it? One side is an arc when we use finite
differences instead of differentials. We wave hands and say "In the limit
that arc becomes straight". So, the student asks, "Why didn't that
spherical triangle become a plane triangle with straight sides when made
skinny and flattened out? Explain it in plain language, please." 

I'll stop here and let you savvy pedagogues take over. And there's much
more to be said.

BTW, on proofreading the above I noted that I had originally said "gives
the wrong answer." I changed it to "gives *a* wrong answer". Any student
can give more than one wrong answer to any problem, and any mathematician
can prove conclusively that for any problem there an infinity of wrong
answers. The mathematician might even be able to specify the order of that
infinity. Well, maybe not. This might be akin to Goedel's proof which
proves that in any mathematical system there are always true theorems
which can't be proven. I'll pass this on to a mathematician friend I know
who could come up with an informed opinion, probably humorous. There could
be a publishable paper in this.

And another. "Apparent paradox." Isn't this redundant? From a
philosophical point of view, aren't *all* paradoxes only apparent? 

	-- Donald

......................................................................
Dr. Donald E. Simanek                            Office: 717-893-2079
Prof. of Physics                    Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745          CIS: 73147,2166 
Home page: http://www.lhup.edu/~dsimanek            FAX: 717-893-2047
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