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Re: Types in Mathematica thread
*To*: mathgroup at smc.vnet.net
*Subject*: [mg62976] Re: Types in Mathematica thread
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Fri, 9 Dec 2005 06:47:02 -0500 (EST)
*References*: <dmp9na$hi2$1@smc.vnet.net> <200512081602.jB8G22ZA018703@ljosalfr.globalsymmetry.com> <A52B2836-ED7F-4C6B-82DE-660BF6662C08@mimuw.edu.pl> <200512090113.42310.hattons@globalsymmetry.com> <A3FA86CA-05DC-432F-B8CD-9DE2D04BAC1C@mimuw.edu.pl>
*Sender*: owner-wri-mathgroup at wolfram.com
On 9 Dec 2005, at 19:26, 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? 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
>
Since you attach so much importance to the way you were taught to
think of Pi in school perhaps instead of the formula for the area I
should have sent the one for the length of the circumference of a
circle:
2 Pi Sinh[r]
So Pi is the ration between the length of the circumference and twice
the Sinh of the radius, which still makes it a universal constant for
all circles. Thus if there were any intelligent creatures living in
Lobachevsky's space they would surely discover Pi ;-)
Andrzej
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