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 July 8th, 2017, 11:57 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory Quadrilaterals on a sphere Does there exist a mapping of equivalent quadrilaterals that covers exclusively an entire spherical surface? Without two singularities?
 July 9th, 2017, 08:53 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,495 Thanks: 1370 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. Thanks from Loren
 July 9th, 2017, 12:21 PM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory romsek, Thank you for correcting my oversight.
 July 9th, 2017, 12:38 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,812 Thanks: 2154 If you meant spherical quadrilaterals, consider a spherical rhombic dodecahedron.
 July 9th, 2017, 05:55 PM #5 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory skipjack, why is the quadrilateral "rhombic" in this case, as opposed to "pentagonal" for a dodecahedron?
 July 10th, 2017, 04:46 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,812 Thanks: 2154 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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