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 May 25th, 2017, 04:52 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 424 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: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra How are you defining a Euclidean shape on a non-Euclidean 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: 424 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, $2\pi r^2$
 July 3rd, 2017, 08:53 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 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
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 Originally Posted by Country Boy 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, $2\pi r^2$
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
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 Yes, the sides follow great circles (parallel line pairs on a sphere).
No two great circles are parallel on a sphere.

 July 3rd, 2017, 12:45 PM #8 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra We can define a square by drawing two great circles meeting at a right-angle 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. Thanks from studiot
July 3rd, 2017, 02:25 PM   #9
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 Originally Posted by v8archie We can define a square by drawing two great circles meeting at a right-angle 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.
Yes indeed the square can bend around the equator, being formed of the four arcs connecting the corners the long way round the sphere.

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