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Re: Types in Mathematica thread
*To*: mathgroup at smc.vnet.net
*Subject*: [mg62989] Re: Types in Mathematica thread
*From*: "Steven T. Hatton" <hattons at globalsymmetry.com>
*Date*: Sat, 10 Dec 2005 06:02:58 -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> <dnbqs8$7ic$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Andrzej Kozlowski wrote:
> 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
Series[Sinh[r], {r, 0, 1}]
\!\(\*
InterpretationBox[
RowBox[{"r", "+",
InterpretationBox[\(O[r]\^2\),
SeriesData[ r, 0, {}, 1, 2, 1],
Editable->False]}],
SeriesData[ r, 0, {1}, 1, 2, 1],
Editable->False]\)
Only after they survey their Hanover. Their first take will be 2 Pi r.
--
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