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. -- 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/