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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