Steven D'Aprano <st...@remove-this-cybersource.com.au> writes:
> On Wed, 13 Oct 2010 16:17:19 +0200, Antoon Pardon wrote:
>
>> On Wed, Oct 13, 2010 at 01:20:30PM +0000, Steven D'Aprano wrote:
>>> On Tue, 12 Oct 2010 22:13:26 -0700, RG wrote:
>>>
>>> >> The formula: circumference = 2 x pi x radius is taught in primary
>>> >> schools, yet it's actually a very difficult formula to prove!
>>> >
>>> > What's to prove? That's the definition of pi.
>>>
>>> Incorrect -- it's not necessarily so that the ratio of the
>>> circumference to the radius of a circle is always the same number. It
>>> could have turned out that different circles had different ratios.
>>
>> If that is your concern, you should have reacted to the previous poster
>> since in that case his equation couldn't be proven either.
>
> "Very difficult to prove" != "cannot be proven".
But in another section of your previous post you argued that it cannot
be proven as it doesn't hold in projective or hyperbolic geometry.
>
>> Since by not reacting to the previous poster, you implicitely accepted
>> the equation and thus the context in which it is true: euclidean
>> geometry. So I don't think that concerns that fall outside this context
>> have any relevance.
>
> You've missed the point that, 4000 years later it is easy to take pi for
> granted, but how did anyone know that it was special? After all, there is
> a very similar number 3.1516... but we haven't got a name for it and
> there's no formulae using it. Nor do we have a name for the ratio of the
> radius of a circle to the proportion of the plane that is uncovered when
> you tile it with circles of that radius, because that ratio isn't (as far
> as I know) constant.
>
> Perhaps this will help illustrate what I'm talking about... the
> mathematician Mitchell Feigenbaum discovered in 1975 that, for a large
> class of chaotic systems, the ratio of each bifurcation interval to the
> next approached a constant:
>
> δ = 4.66920160910299067185320382...
>
> Every chaotic system (of a certain kind) will bifurcate at the same rate.
> This constant has been described as being as fundamental to mathematics
> as pi or e. Feigenbaum didn't just *define* this constant, he discovered
> it by *proving* that the ratio of bifurcation intervals was constant.
> Nobody had any idea that this was the case until he did so.
But you were claiming that the proposition "C = 2πr is the definition of
π" was false. Are you claiming that "δ is defined as the ratio of
bifurcation intervals" is false as well? If you are not, how does this
tie in with the current discussion?
Also, it is very intuitive to think that the ratio of the circumference
of a circle to it radius is constant:
Given two circles with radii r1 and r2, circumferences C1 and C2, one is
obviously the scaled-up version of the other, therefore the ratio of
their circumferences is equal to the ratio of their radii:
C1/C2 = r1/r2
Therefore:
C1/r1 = C2/r2
This constant ratio can be called 2π. There, it wasn't that hard. You
can pick nits with this "proof" but it is very simple and is a convincing
enough argument.
This to show that AFAIK (and I'm no historian of Mathematics) there
probably has never been much of a debate about whether the ratio of
circumference to diameter is constant. OTOH, there were centuries of
intense mathematical labour to find out the value of π.
--
Arnaud
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