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 June 5th, 2018, 09:06 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 444 Thanks: 29 Math Focus: Number theory Pack unit circles on unit sphere What is the maximum number of non-overlapping unit circles that can be packed upon a unit sphere?
 June 5th, 2018, 09:26 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,530 Thanks: 1390 You're going to have to expand on what you mean by "packed upon". If you mean that every point on the circle intersects the surface of the sphere... wouldn't every such circle have the center of the sphere as it's center? I.e. they will all overlap.
June 6th, 2018, 11:08 AM   #3
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 You're going to have to expand on what you mean by "packed upon". If you mean that every point on the circle intersects the surface of the sphere... wouldn't every such circle have the center of the sphere as it's center? I.e. they will all overlap.
The center of the sphere is not a point on the sphere itself. Indeed, every point on the circles, but not their centers, intersects the sphere -- though the circles themselves do not overlap.

Since a sphere is a surface, this situation (the maximum number of unit, nonintersecting circles upon a unit sphere) is akin to other packing problems (circles within a circle, spheres within a cube, etc).

By "packed upon," here I mean filling (in three dimensional space) a two-dimensional boundary (a unit sphere) with two-dimensional shapes (unit circles) that do not overlap (i.e., are mutually exclusive).

Do I use the word "upon" incorrectly? Maybe "resting upon" is better.

June 6th, 2018, 12:17 PM   #4
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Quote:
 Originally Posted by Loren The center of the sphere is not a point on the sphere itself. Indeed, every point on the circles, but not their centers, intersects the sphere -- though the circles themselves do not overlap. Since a sphere is a surface, this situation (the maximum number of unit, nonintersecting circles upon a unit sphere) is akin to other packing problems (circles within a circle, spheres within a cube, etc). By "packed upon," here I mean filling (in three dimensional space) a two-dimensional boundary (a unit sphere) with two-dimensional shapes (unit circles) that do not overlap (i.e., are mutually exclusive). Do I use the word "upon" incorrectly? Maybe "resting upon" is better.
When you say unit circle, you mean distance on the sphere? Or distance in 3-space?

June 6th, 2018, 08:02 PM   #5
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 When you say unit circle, you mean distance on the sphere? Or distance in 3-space?
A "unit circle," of area 1, here subtends one steradian from the center of the unit sphere to the sphere itself, of surface area 4*pi.

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