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