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I wonder whether presented in wikipedia square (with angles $120^{\circ}$) is the biggest (taking into account lengths of sides) possible square on a sphere?

  • If not what is the biggest?
  • If so how it can be proved?

One can assume here definition of a square on a sphere:
Square on a sphere = the 4-sided polygon with equal length sides and equal angles, sides are determined by 4 different great circles.

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    @Mathematician42 Is it not clear from the link ? See https://en.wikipedia.org/wiki/Square#Non-Euclidean_geometry for details.. – 2017-02-01
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    Ah, only saw the picture, let's read it :) – 2017-02-01
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    Fix a point to be the north pole of your sphere. If one takes four equally spaced meridians, so that adjacent meridians are separated by $90^{\circ}$, then the four intersections of the meridians with any parallel (i.e., line of latitude) will define a square, and one can show that every square arises this way. One can see that as the chosen parallel approaches the equator, the side lengths monotonically approach one-quarter the circumference of the sphere. So, if you allow the degenerate square with four $180^{\circ}$ angles, that is the solution. If you do not, there is no maximum. – 2017-02-01
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    On the other hand, the square with $120^{\circ}$ angles is the largest one that can be used to tile the sphere uniformly, assuming again that one ignores the degenerate tiling by two hemispherical 'squares', which is a consequence of the fact that there is a tiling where three squares meet at each corner (just the central projection of a cube onto a sphere). – 2017-02-01
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    @Widawensen I second Mathematician 42's original suggestion to include the definition you use here, for self-containedness (and for independence of any future edits to the wiki). – 2017-02-01
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    @Travis Square on a sphere = the 4-sided polygon with equal length sides and equal angles, sides are determined by different geodesics. Travis, Is it sufficient definition ? – 2017-02-01

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