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July 3rd, 2013, 09:35 PM   #1
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Pi is an integer

http://en.wikipedia.org/wiki/N-sphere
Quote:
The 0-sphere consists of its two end-points, -1 and 1, so S0 = 2
Circle, sphere, hypersphere, however many dimensions, its surface is defined in terms of pi.
x^2 + y^2 + z^2 = 1 //2-sphere, what we normally just call a sphere
x^2 + y^2 = 1 //1-sphere, circle
x^2 = 1 //0-sphere, what we normally call a bit

The 0 dimensional sphere with radius 1 has a surface area of 2.

2*pi*radius = 2

pi = 1

If you start with the 0 dimensional sphere, -1 and 1, and keep dividing it into 2 times as many points, you binary search a circle like Cooley Tukey Fast Fourier Transform.

If you double the number of points on a circle, the distance between each 2 adjacent points is half, unless you think you get to measure a circle from outside itself like those earlier theories of quantum physics that put the observer outside the observed. No, we're on the circle measuring the circle. Pi is exactly 1.
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July 3rd, 2013, 10:19 PM   #2
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Re: Pi is an integer

Quote:
Originally Posted by BenFRayfield
The 0 dimensional sphere with radius 1 has a surface area of 2.

2*pi*radius = 2
The former doesn't implies the later. A dim 0 sphere hasn't have the property S = 2?r. It' applicable for only a dim 2 sphere, a.k.a, a circle.

Quote:
Originally Posted by BenFRayfield
Pi is an integer
True, it acts like one is Z[?] although I am not sure how much collapsing can occur if we do that.

Quote:
Originally Posted by BenFRayfield
Pi is exactly 1.
That doesn't seems to follow.
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July 4th, 2013, 11:00 AM   #3
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Re: Pi is an integer

If you like, you can define the hypervolume of an n-dimensional sphere as in which case , and

(This was the most sensible way to interpret the OP, I thought.)
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July 4th, 2013, 11:24 AM   #4
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Re: Pi is an integer

I realize that geometers and topologists are adept at dreaming up some pretty funky objects, but how does a zero dimensional sphere have a radius of 1? Doesn't having a measure imply having a dimension in which that measure can exist? And doesn't 'radius' in turn have a meaning that requires the "sphere" to be at least two-dimensional, ie a circle?

Honest questions.
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July 4th, 2013, 11:51 AM   #5
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Re: Pi is an integer

Quote:
Originally Posted by johnr
I realize that geometers and topologists are adept at dreaming up some pretty funky objects, but how does a zero dimensional sphere have a radius of 1?
I didn't say that it did! In fact, the radius is raised to the zero power, so whatever value you assign to it is ignored. It's almost as though the concept of "length" didn't matter in 0 dimensions.

Quote:
Originally Posted by johnr
And doesn't 'radius' in turn have a meaning that requires the "sphere" to be at least two-dimensional, ie a circle?
Don't let language trick you! That's like the ancient Greeks declaring that 1 was not a number.

A sphere in 1-dimensional space is a line segment, and it has 1-volume ("length") equal to 2r.
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July 4th, 2013, 12:59 PM   #6
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Re: Pi is an integer

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by johnr
I realize that geometers and topologists are adept at dreaming up some pretty funky objects, but how does a zero dimensional sphere have a radius of 1?
I didn't say that it did! In fact, the radius is raised to the zero power, so whatever value you assign to it is ignored. It's almost as though the concept of "length" didn't matter in 0 dimensions.
Just trying to get my head around the original poster's stipulations!

I think I can learn to live with length not mattering as opposed to simply not existing in zero dimensions.

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by johnr
And doesn't 'radius' in turn have a meaning that requires the "sphere" to be at least two-dimensional, ie a circle?
Don't let language trick you! That's like the ancient Greeks declaring that 1 was not a number.

A sphere in 1-dimensional space is a line segment, and it has 1-volume ("length") equal to 2r.
Well, yes, there is that danger, but there is also a danger in Humpty-Dumptyesque linguistic anarchy. Indeed, I'm trying to navigate between these two dangers, which is why, though my questions were admittedly raised in a spirit of skepticism, I did mean them as real questions, not merely rhetorical shrieks!
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July 4th, 2013, 01:12 PM   #7
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Re: Pi is an integer

Just an observation, the 'radius' of convergence in single variable calculus is a 1 dimensional 'object'.

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July 4th, 2013, 02:01 PM   #8
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Re: Pi is an integer

Quote:
Originally Posted by agentredlum
Just an observation, the 'radius' of convergence in single variable calculus is a 1 dimensional 'object'.

So of the radius of that object is 1, it's diameter is ...?
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July 4th, 2013, 06:56 PM   #9
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Re: Pi is an integer

Quote:
Originally Posted by johnr
Quote:
Originally Posted by agentredlum
Just an observation, the 'radius' of convergence in single variable calculus is a 1 dimensional 'object'.

So of the radius of that object is 1, it's diameter is ...?
2

Actually radius and diameter are 1 dimensional 'objects' of length no matter what dimension we consider > 0 (zero scares me)

Take for examples

1) 3d sphere where we know radius is just dimension of length so 1 dimensional

2) Vector spaces of any dimension where the distance between 2 points in N dimensional space is defined as the Euclidean Norm.

http://en.m.wikipedia.org/wiki/Norm_(mathematics)

http://en.m.wikipedia.org/wiki/N-sphere

You may find the bottom link more germain to the topic at hand but both links should be useful.

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July 4th, 2013, 07:13 PM   #10
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Re: Pi is an integer

Quote:
Originally Posted by agentredlum
Quote:
Originally Posted by johnr
Quote:
Originally Posted by agentredlum
Just an observation, the 'radius' of convergence in single variable calculus is a 1 dimensional 'object'.

So of the radius of that object is 1, it's diameter is ...?
2

Actually radius and diameter are 1 dimensional 'objects' of length no matter what dimension we consider > 0 (zero scares me)

Take for examples

1) 3d sphere where we know radius is just dimension of length so 1 dimensional

2) Vector spaces of any dimension where the distance between 2 points in N dimensional space is defined as the Euclidean Norm.

http://en.m.wikipedia.org/wiki/Norm_(mathematics)

http://en.m.wikipedia.org/wiki/N-sphere

You may find the bottom link more germain to the topic at hand but both links should be useful.

OK! I of course knew that D=2r [edit!!!] for circles. I was just being cagey about asking whether 'radius' was really being used in the same sense here. But I'll check the links!

Oh, my own internet search yielded talk of 0 spheres. But these are not 0 dimensional, but 1 dimensional, consisting of the two endpoints of a line segment.
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