I thought this would best be kept offline, but I disagree with
most of these points, where I could see many in your first
private email.
If you read the article, you will see that there is basically
no use of diameter in mathematics or physics, that pi is
an invention of the 1700's, not the greeks, and that
in fact, regarding the area formula, we have noticed that you gain by
the factor of 1/2 natural for all other quadratics - 1/2 m v ^2 . 1/ 2
g t^2, 1/2 k x^2. p^2 / 2 m, etc.
I apologize to the unicode people and realize that here is
NOT the best place for these aspects of the discussion.
Without doubt, 2 \pi is the most widely used form of use
of \pi, with no valid purpose except historical convention.
Many numerical analysts/scientific computers begin their codes
by defining a constant equal to 6.28... to simplify their
codes, and I thought typographers might see the value of
such an option for simplicity as well. Apparently I was
wrong. Rather than use a new character, using the \TeX macro
is easy enough for most of those in the mathematical community
who use some version of \TeX.
On Wed, 16 Jan 2002, Barry Caplan wrote:
> At 11:33 AM 1/16/2002 -0700, Robert Palais wrote:
> >is at the same time somewhat a Catch-22. Nelson Beebe recommended it since
> >he figured unicode 3.2 would be the make or break for "getting it in use".
> >I'd be curious if you disagree with the thesis that a symbol for
> >6.28 has scientific/mathematical merit (in comparison 3.14...), and if so
> >why?
>
>
> My guess is that since pi is the ratio of the circumference to the
> diameter, that the diameter is a more natural conception of the size of a
> circle than the radius. Of course mathematically, it doesn't matter other
> than the factor of 2. But other geometrical shapes, particularly polygons,
> are measured by line segments that extend from one point to another on the
> same shape, or series of shapes. A radius just sort of ends in the middle,
> while a diameter or other chord begins and ends on the circle.
>
> I can't quote the history, but if I imagine back to the Greek days, I bet
> the diameter was the primary measure. Other polygonal shapes with which
> they were familiar had their measures in terms of a line segment crossing
> the entire shape and touching the boundaries, or coincident with the boundary.
>
> Mathematicians pondering the circle for the first time, there probably was
> no reason to think otherwise. How to proceed from there to figure the area
> of a circle or the ratio of the diameter to the circumference were
> probably some of the greatest challenges of the day. They wanted to know
> the circumference and area, same as they had calculated for other shapes.
>
> I would guess that since pi is the ratio of the circumference and diameter,
> that this problem was solved first. Had it been the other way around, our
> formulas might look the way Dr. Palais suggests.
>
> Now that I think about it, I wonder if the very concept for "radius" grew
> out of the solution to the area of the circle: was the original formula A =
> pi * (d over 2)squared? If so, then maybe a conceptual leap was made to
> simplify it, thus inventing the radius.
>
> Why simplify the d/2 part and not the other way (pi/4)? Probably because pi
> is just a number, while d/2 turned out to have some connection to the
> physical world - the distance from the edge of a circle to the center.
>
> But this is just idle lunchtime speculation on my part.
>
> Note that using the new symbol the circumferance of a circle is simply
> "tri"*r, but the Area changes form pi*r(squared) to tri *(1/2) times r
> squared, so you lose as much as you gain it seems to me.
>
>
> Barry Caplan
>
>
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