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MathGroup Archive 2005

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Re: Types in Mathematica thread

  • To: mathgroup at smc.vnet.net
  • Subject: [mg62971] Re: Types in Mathematica thread
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 9 Dec 2005 06:19:40 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

On 9 Dec 2005, at 15:13, Steven T. Hatton wrote:

> 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.
>
> Steven

It seems to me that you are simply confused and have not studied  
enough mathematics, and these two facts together make you see  
profound ideas where there are none. You are also jumping form one  
topic (transcendental numbers)  to a quite different one  (non- 
Euclidean geometries) hinting that they are somehow related, although  
of you never explain how.
  Pi of course occurs in formulae in Lobachevsky's geometry in  
exactly the same places as in  Euclidean geometry. Have you ever seen  
any of these formulae?  For example the formula for the are of a  
circle of radius r is, if I have not made a mistake,

4 *Pi * Sinh^2(r/2)

Of course it is different from the Euclidean one, but there is Pi in  
it. You will find Pi in all the other formulae corresponding to those  
where there is PI in Euclidean geometry - so in what sense is Pi  
uniquely Euclidean? Indeed the formulae of Euclidean geometry are  
unique - in the sense that they are not like the non-Euclidean ones -  
so what is profound or surprising about that?

Andrzej Kozlowski

  


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