🚀 go-pugleaf

In article <3kl17u$k1c@utdallas.edu>,  <son3776@utdallas.edu> wrote:
>DaveHatunen (hatunen@netcom.com) wrote:
>> In article <3kab8t$h4l@utdallas.edu>,  <son3776@utdallas.edu> wrote:
>
>> [...]
>
>> >Draw a circle on the earth, cut through the ground to get the diameter, 
>> >and you'll get C/D and pi.
>
>> Wrong. For two basic reasons.
>
>> 1. The earth is not a perfect sphere, so you cannot even theoretically
>> draw a perfect circle on even it's ideal surface.
>
>Yeah -- it's an "squashed sphere" . . . but considering the fact that it 
>has a positive (and gradual) curvature, you CAN theoretically draw a 
>perfect circle on it (that is if you neglect valley/mountain combinations 
>and artificial constructs).

No, it's not simply an oblate spheroid, althugh that's a first
approxiation to the deviation from sphericity. And you cannot, in
general, draw a true circle even on the idealized (mountains excluded)
lumpy spheroid (I don't mean the approximately perfect oblate
spheroid). Unless, of course, you use the definition that a circle is
the locus of all points equidistant (along the geodesic) from a given
point. But that will not yield a true circle as measured from the
"third" dimension. In fact, once you admit the third dimension, the
circle you ahve drawn won't even lie in a Euclidean plane. Of course,
the deviation will be so small as to be unmeasurable by any practical
instruments.

>> 2. The earth is a gravitating body, and the space in, around and near
>> it is not perfectly Euclidean.
>
>> Not even perfect measurements to infinitesimal precision will yield
>> exactly pi. But the mathematical computation of one of the several
>> infinite series will, carried out to infinity, yield pi.
>
>Considering that we can't detect any such curvature in spacetime because 
>we ourselves are affected by the same curvature to such an extent that if 
>we travel with a relative velocity of zero in relation to the drawn 
>circle there won't be any noticeable distortions, we will measure pi (to 
>a degree of accuracy).

A number of people have pointed out that the curvature of space can be
measured in principle. All one need do is measure the sum of the
interior angles of a triangle. If the sum deviates from pi, the space
is curved. If the sum is greater than pi, the space is curved
positively; if the sum is less than pi, the space is curved negatively.

How many times are you going to have to have this pointed out?

I really do urge you to get a copy of <1, 2, 3, ... Infinity>.

-- 


    ********** DAVE HATUNEN (hatunen@netcom.com) **********
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