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MathGroup Archive 2005

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Re: Types in Mathematica thread

  • To: mathgroup at smc.vnet.net
  • Subject: [mg62977] Re: Types in Mathematica thread
  • From: "Steven T. Hatton" <hattons at globalsymmetry.com>
  • Date: Sat, 10 Dec 2005 06:02:47 -0500 (EST)
  • References: <dnbpqj$6t1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski wrote:

> 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? 

I have clarified that statement.  Please review the statement above.

> 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?

http://etext.library.adelaide.edu.au/k/kant/immanuel/k16p/
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