[quote=xilman;100262]Even in pathological spaces, such as those which are everywhere discontinuous?
I think you have to include certain non-singularity conditions.
[/quote]
reading problem ? We're in normed vector spaces. (narrowed to R²...)
Also, may I cite google :
[quote]
Your search - [B]"everywhere discontinuous space"[/B] - did not match any documents.
Suggestions:[LIST] [*]Make sure all words are spelled correctly.[*]Try different keywords.[*]Try more general keywords.[/LIST][/quote]
Did you mean "disconnected"? Once again, this is rare in 2-dim normed vector spaces, according to my experience....

[quote] Anyway, I claim that \pi is a constant equal to 3.14159....[/quote]
That's what I anticipated by adding the "for the pleasure of pedants" phrase.

[QUOTE=xilman;100262]
Anyway, I claim that \pi is a constant equal to 3.14159..., independent of geometry, and therefore it's trivial that 3 <= \pi <= 4. Paul[/QUOTE]

Very very true Paul!

The ancients did better than 3<=pi <=4.

Archimedes gave the approx. 3^ (10/71) < pi < 3^ (1/7)


The Bible used the simple 10/3

The fraction 22/7 is what is normally given in high school problems on the circle and sphere. There are various approximations for pi besides 22/7.

Ramanujan came up with sq rt. of sq.rt of (2143 /22). Its equal to the figure you have quoted.

The value of pi was used in the Great Pyramid as 2 x h x pi =4b where h is the height and b as a base side . 'x' is the multiplication sign. Try this for a 3,4,5 pyramid.

Since they did not add the vertex stone the actual height is not known. Possibly they even knew that pi is transcendental but have kept us guessing the value of the exact value of pi.

In some pyramids the appx. pi =4/(sq.rt. phi) was used.

phi = ( 1 +sq.rt.5 ) / 2

It is close enough for the gigantic size of these structures.

And of course the well known e^(i.pi) + 1 = 0

The calculus can derive the value of pi to any desired accuracy by Taylor's theorem or from a variety of summations.

Leonhard Euler (pronounced 'oiler') gave the amazing formula
that the sum of the reciprocals of natural numbers is = [(pi)^2] / 6

Truly there is no limit to a conception of the mysterious pi !

Mally :coffee:

[QUOTE=cheesehead;100214]By considering a unit circle inscribed within a unit square, we see that the circumference of the circle, pi, is less than the perimeter of the square, 4.

[/QUOTE]

Ahh, I taught my two boys, who were both struggling with pi in their homework, why it was less than 4, using this. They were both convinced.

Ahh, the good old days. Neither is likely to ever use pi again, in their lives, but they were convinced !!!!!!

They both think my interest with integer maths as very strange. But they don't mind.

Neither of them use computer time for prime testing. Sigh! I paid for both computers.

So pi is less than 4, apparently, don't you know.

Also I love e^(i*pi)=-1. I just think it is so neat.

Now I need my dad to explain this. He's long gone. I need convincing.

[QUOTE=robert44444uk;100793]Ahh, I taught my two boys, who were both struggling with pi in their homework, why it was less than 4, using this. They were both convinced.

Also I love e^(i*pi)=-1. I just think it is so neat.

Now I need my dad to explain this. He's long gone. I need convincing.[/QUOTE]

:smile: Son !

Just Google 'pi and the Fibonacci numbers' [Im feeling lucky] and you will come to know more about pi than just the Euler formula

Mally :coffee: