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March 21st, 2017, 12:00 PM   #1
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Integer coordinates in a sphere

Given a sphere of integer radius R, how many integer coordinates (a, b, c) at most does it contain?
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March 21st, 2017, 12:09 PM   #2
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Do you want to include the points on the surface as well?
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March 21st, 2017, 01:07 PM   #3
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Yes. please.
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March 21st, 2017, 01:39 PM   #4
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It's looking like

$f(r) = 0.573291 r^{2.98494}$

is a pretty good fit
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March 21st, 2017, 02:31 PM   #5
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Many thanks, romsek, for the computation.

Can this problem be solved analytically?
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March 22nd, 2017, 02:33 AM   #6
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Quote:
Originally Posted by Loren View Post
Many thanks, romsek, for the computation.

Can this problem be solved analytically?
I would suggest a geometric approach that would use a 3d imaging program with every triple integer lattice highlighted as a dot and start with a sphere of radius 1 centered at the origin. Count the dots enclosed by the sphere , increase the radius to 2 , count the dots enclosed by the sphere , repeat a few times and check the numbers to see if you can find a constant ratio. That would at least give you an insight as to what you are up against. A good idea is to write code that counts the dots for you then visually confirm that the code is working properly by comparing it to your count. If you can do all that then you can make predictions for random radii and see how your predictions compare to the program answers.

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March 31st, 2017, 12:03 AM   #7
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Ball from cubes

Can a ball be constructed out of finite cubes?

I once read about a proof of this in the affirmative.
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