May 25th, 2017, 04:52 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory  Largest square on sphere
What is the largest square that can be constructed on a unit sphere?

May 25th, 2017, 05:27 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra 
How are you defining a Euclidean shape on a nonEuclidean surface? It can't have straight sides, but do they follow great circles? What angles does your square have?

May 25th, 2017, 08:05 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
Yes, the sides follow great circles (parallel line pairs on a sphere). The angles between great circles seem to approach either zero degrees or 180 degrees for minimal square area. I can't figure out the conditions for maximizing that area, though. 
July 3rd, 2017, 08:43 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
The largest great circle on a sphere is an equator. To make the largest "square", divide that into four equal parts, each 90 degrees. The area of the interior of that square is half the surface are of the sphere, 
July 3rd, 2017, 08:53 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra 
Aren't all great circles the same size? Aren't they all equators for some orientation of the sphere?

July 3rd, 2017, 08:54 AM  #6 
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269  Seems to me that those instructions divide the surface into 1/8 areas by (spherical) triangles, but perhaps I misread them.
Last edited by studiot; July 3rd, 2017 at 09:05 AM. 
July 3rd, 2017, 10:30 AM  #7  
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269  Quote:
 
July 3rd, 2017, 12:45 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra 
We can define a square by drawing two great circles meeting at a rightangle at the poles and choosing a line latitude. The vertexes of the square are at the points of intersection of the line of latitude and the great circles. The edges of the square are segments of great circles joining vertices not on the same great circle through the poles. The area of the square is the part of the sphere's surface thus enclosed and containing the North pole.(*) The largest such square comes (after some adjustment in the definition) when all four points coincide at the South pole. It's area is the surface area of the entire sphere. This is a general construction because every square has a single centre. (*) The area outside the square is another square centered on the South pole. 
July 3rd, 2017, 02:25 PM  #9  
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269  Quote:
 

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