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July 8th, 2017, 11:57 PM   #1
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Quadrilaterals on a sphere

Does there exist a mapping of equivalent quadrilaterals that covers exclusively an entire spherical surface? Without two singularities?
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July 9th, 2017, 08:53 AM   #2
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I assume you're talking planar quadrilaterals? Why not just use triangles to avoid confusion?

Pretty sure the answer is no. Even the smallest patch of sphere will have curvature to it whereas your planar tiles will be flat.
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July 9th, 2017, 12:21 PM   #3
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romsek,

Thank you for correcting my oversight.
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July 9th, 2017, 12:38 PM   #4
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If you meant spherical quadrilaterals, consider a spherical rhombic dodecahedron.
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July 9th, 2017, 05:55 PM   #5
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skipjack, why is the quadrilateral "rhombic" in this case, as opposed to "pentagonal" for a dodecahedron?
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July 10th, 2017, 04:46 AM   #6
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A dodecahedron is given that name because it has 12 faces. Its faces needn't be pentagonal (5-sided). The faces of a rhombic dodecahedron are 4-sided (hence quadrilateral). Each face is a rhombus. Each face of a spherical rhombic dodecahedron is a spherical version of a rhombus.
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