June 23rd, 2018, 01:16 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory  Area 1 circles on radius 1 sphere
What is the maximum number of circles of area one packable, without overlapping, onto a sphere of radius one? (I couldn't figure out StackExchange's use of tags.) 
June 23rd, 2018, 01:44 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,586 Thanks: 612 
Do you mean on the surface of the sphere? For stack exchange "geometry" would be a tag. 
June 23rd, 2018, 03:44 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory  
June 24th, 2018, 06:41 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,542 Thanks: 1752 
Interesting question! Is each circle's area the area of the smaller part of the sphere's surface that is enclosed by the circle? If so, the answer must be less than 4pi (the surface area of the sphere), and so is at most 12. I would imagine that 7 nonoverlapping circles could be accommodated quite easily. I think the answer is 10.

June 24th, 2018, 07:30 AM  #5 
Senior Member Joined: Sep 2016 From: USA Posts: 444 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics 
The question is very ambiguous. What does it mean to have a circle on the boundary of a sphere? To start with, you probably don't mean circles since they are 1 dimensional and have no area. I guess you mean disc? But this also has problems since strictly speaking, no disc of positive radius exists on the surface of a sphere. The only way I can parse this is that you mean a topological disc (i.e. a set which is homeomorphic to a disc) of area 1. In this case however, the answer is pretty trivial. Either you allow them to intersect in which case the answer is clearly infinite. Or, they may not intersect and skipjack's upper bound of 12 is obviously attainable. Since I doubt the latter interpretation is what you have in mind, but I don't know how else to even interpret this, you should ask the question more clearly. 
June 24th, 2018, 09:30 AM  #6  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory  Quote:
 
June 24th, 2018, 10:02 AM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 Math Focus: Number theory 
My general question might loosely be stated as "What is the maximum number of circles of area one which do not overlap, and pack onto a spherical surface of radius N, where N is a nonzero integer?" My correspondent says: "how do you define the circle area? Is it the area of a "flat" circle with its circumference on the surface of the sphere, or is it the (larger) curved area of a spherical cap bounded by a circle on the sphere surface? Of course, the difference between the two definitions becomes negligible for large sphere radii, but for the small values of N the results are different. In general, it seems that you are looking for the results of the "Tammes" problem, for which the best known, most likely optimal solutions up to 130 circles are tabulated on Neil Sloane's web page http://neilsloane.com/packings/index.html#I (results for D=3) http://neilsloane.com/packings/dim3/ For larger sphere radii (larger already means N>3), good approximations would be the packings from spherical codes with icosahedral symmetry http://neilsloane.com/icosahedral.codes/index.html The achievable maximum density of circles on the sphere surface would asymptotically reach the well known Pi*sqrt(3)/6 ~= 0.9069 of the hexagonal lattice packing of circles in the plane. If you are you interested in 'exact' values, then the only proven results are for N=1 (9 or 10 circles dependent on the area definition), and unproved results with good numerical evidence would be 40 or 42 circles for N=2 and 94 or 95 circles for N=3." SDK: Please see that you were right about the ambiguity of my question regarding area surrounded by a circle on the sphere; skipjack, you estimated well with the guess of 10 circles for N=1. 

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area, circles, radius, sphere 
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