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/ -- The Mathematica Wiki: http://www.mathematica-users.org/ Math for Comp Sci http://www.ifi.unizh.ch/math/bmwcs/master.html Math for the WWW: http://www.w3.org/Math/