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Re: Types in Mathematica thread

On Thursday 08 December 2005 19:23, Andrzej Kozlowski wrote:
> On 9 Dec 2005, at 01:01, Steven T. Hatton wrote:

> I confess I do not understand much of the above, and in particular,
> the phrase: "the value of the value Pi is unique to Euclidian
> geometry".

Actually that statement is a bit incorrect. The 2D geometry on the surface of 
a bent sheet of paper will also have the value Pi when determined by the 
traditional definition.

> I have always believed that Pi is 
> 4*Sum[(-1)^i/(2*i + 1), {i, 0, Infinity}]

Interesting. I learned that Pi = circumference/diameter long before I learned 
of the Taylor series expansion.

> What is Euclidean about that?
> If what you have in mind is the fact that the 5th postulate of Euclid
> is equivalent to  the sum of the angles in a triangle being equal to
> two right angles  (180 degrees or Pi in radians)  then this is true
> but the profound significance of this for number theory escapes me.

As I implied above, that will also apply to any surface which is curved in 
only one direction.  IOW, no stretching is involved in order to get the 
Euclidian surface to conform to the curved surface.   

> Why is not 180 unique to Euclidean geometry? Pi, being a number,
> lives just as comfortably in the Bolyai-Gauss geometry or
> Lobachevski's geometry  as in Euclidean geometry.

Pi also has a geometrical significance as I explained above. The value 
determined for Pi using the geometric definition in a non-Euclidian geometry 
- which is what a 3-space slice in general relativity is - will in general 
not be the same as the value given by the sum you expressed above.  That 
suggests there is something a priori about spaces in which Pi as defined as 
circumference/diameter does have the traditional value.


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