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 October 23rd, 2013, 03:00 PM #1 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 the total number of points in n-dimensional sphere Hello, Could somebody tell me how to count the total number of points in n-dimensional sphere depending on r? I think about it more than week and I can't do this... I know that for 1-dimensional sphere it is r*2 + 1. But I can't count it for 2,3.. 10 - it's max dimension. Can somebody help me?
 October 23rd, 2013, 04:43 PM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 Re: the total number of points in n-dimensional sphere Mr. Business Man, nobody can count the number of points in the n-sphere for any $n \in \mathbb{N}$. What do you possibly want to do with this number?
October 23rd, 2013, 05:07 PM   #3
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Re: the total number of points in n-dimensional sphere

Quote:
 Originally Posted by Business Man Hello, Could somebody tell me how to count the total number of points in n-dimensional sphere depending on r? I think about it more than week and I can't do this... I know that for 1-dimensional sphere it is r*2 + 1. But I can't count it for 2,3.. 10 - it's max dimension. Can somebody help me?
There are an infinite number of points, unless you put on some restrictions, such as all coordinates are integer.

 October 23rd, 2013, 10:16 PM #4 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 Re: the total number of points in n-dimensional sphere Sorry, I've forgotten about that. Yes, all coordinates must be integer.
 October 24th, 2013, 05:34 AM #5 Global Moderator   Joined: Dec 2006 Posts: 19,505 Thanks: 1741 Try a google search for "The Sphere Problem", but results are rather limited.
 October 24th, 2013, 01:18 PM #6 Global Moderator   Joined: May 2007 Posts: 6,581 Thanks: 610 Re: the total number of points in n-dimensional sphere When you say n is the number of dimensions, your answer for n=1 is a little confusing. A 1 dimensional sphere is a straight line interval, so the number of points covered would be ~ r, not r². Also is the center of the sphere at the origin?
 October 24th, 2013, 05:43 PM #7 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 Re: the total number of points in n-dimensional sphere The center of the sphere is point 0 in n=1, (0,0) in n=2, (0,0,0) in n=3 and so on. Sorry, I didn't mean $r^2$ I mean: $2*r+1$ because for r=1 I am going one point to the right, one point to the left and the zero, so the result is 3. for r=2 I am going two points to the right, two points to the left ant the zero, so the result is 5. But I can't imagine how it looks in more dimensional sphere... I searched on google but without any success.
 October 24th, 2013, 06:19 PM #8 Global Moderator   Joined: Dec 2006 Posts: 19,505 Thanks: 1741 Try giving Google additional search terms, such as "integer coordinates".

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