July 3rd, 2013, 09:35 PM  #1  
Senior Member Joined: Jan 2013 Posts: 209 Thanks: 3  Pi is an integer http://en.wikipedia.org/wiki/Nsphere Quote:
x^2 + y^2 + z^2 = 1 //2sphere, what we normally just call a sphere x^2 + y^2 = 1 //1sphere, circle x^2 = 1 //0sphere, 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.  
July 3rd, 2013, 10:19 PM  #2  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Pi is an integer Quote:
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Quote:
 
July 4th, 2013, 11:00 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Pi is an integer
If you like, you can define the hypervolume of an ndimensional sphere as in which case , and (This was the most sensible way to interpret the OP, I thought.) 
July 4th, 2013, 11:24 AM  #4 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  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 twodimensional, ie a circle? Honest questions. 
July 4th, 2013, 11:51 AM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Pi is an integer Quote:
Quote:
A sphere in 1dimensional space is a line segment, and it has 1volume ("length") equal to 2r.  
July 4th, 2013, 12:59 PM  #6  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Pi is an integer Quote:
I think I can learn to live with length not mattering as opposed to simply not existing in zero dimensions. Quote:
 
July 4th, 2013, 01:12 PM  #7 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: Pi is an integer
Just an observation, the 'radius' of convergence in single variable calculus is a 1 dimensional 'object'. 
July 4th, 2013, 02:01 PM  #8  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Pi is an integer Quote:
 
July 4th, 2013, 06:56 PM  #9  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: Pi is an integer Quote:
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/Nsphere You may find the bottom link more germain to the topic at hand but both links should be useful.  
July 4th, 2013, 07:13 PM  #10  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Pi is an integer Quote:
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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