July 8th, 2017, 11:57 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 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: 2,364 Thanks: 1270 
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. 
July 9th, 2017, 12:21 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
romsek, Thank you for correcting my oversight. 
July 9th, 2017, 12:38 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,370 Thanks: 2007 
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: 397 Thanks: 27 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: 20,370 Thanks: 2007 
A dodecahedron is given that name because it has 12 faces. Its faces needn't be pentagonal (5sided). The faces of a rhombic dodecahedron are 4sided (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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