March 23rd, 2017, 10:08 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory  Flatland Platonic solids' formula
Does a modification of Euler's polyhedral formula, ce+f=2, hold when corners, edges and faces are directly projected onto a plane? Edge intersections count as corners. For a projected cube, I get 10 corners, 14 edges and 10 faces. 
March 24th, 2017, 09:27 AM  #2  
Senior Member Joined: Feb 2010 Posts: 643 Thanks: 111  Quote:
If you mean the above projection, it has 10 vertices (corners), 16 edges, and 8 faces (including the infinite region around the picture). So $\displaystyle ve+f = 1016+8=2$ and the formula works.  
March 24th, 2017, 10:13 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,232 Thanks: 2411 Math Focus: Mainly analysis and algebra 
Depends on the projection doesn't it? There are projections where points and/or edges merge.

March 24th, 2017, 10:18 AM  #4 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory 
I still get 10 corners, 14 edges and 10 faces the way I count them. Are you able to label these features on your projection of a wire frame cube? I get for a tetrahedron 4 corners, 8 edges and 4 faces. I've never seen the volume (or area) outside the figure to be included in Euler's formula, maybe in other branches of topology. My octahedron yields 8 corners, 14 edges and 9 faces. v8archie, I should have mentioned that these are maximum numbers I used. 
March 24th, 2017, 11:56 AM  #5  
Senior Member Joined: Feb 2010 Posts: 643 Thanks: 111  Quote:
10 vertices: A, B, C, D, E, F, G, H, I, J 16 edges: AB, AE, BC, ED, DC, EI, AD, DF, BH, CG, GJ, FG, GH, IJ, JH, IF 8 regions: AED, ABCD, BCGH, GHJ, CDFG, EDFI, FGJI, and ABHJIE (the outside region) Since we agree on 10 vertices, then I would be curious to know how you are defining edges and faces since I don't get 14 edges and 10 faces.  
March 24th, 2017, 01:39 PM  #6 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory 
mrtwhs, You've proven your argument and yourself to me. Thanks for the helpful diagram. It makes the calculation so much easier. Was the general 2D case (for maximal features) known before Euler? Your including the outside region reminds me of Venn diagrams, but you show it works for the projected wire cube according to the Euler formula. Does a similar inclusion for the "outside" apply for polyhedra? Maybe a 3D spatial region which negates the frame, would be topologically more difficult. 
March 24th, 2017, 02:32 PM  #7  
Senior Member Joined: Feb 2010 Posts: 643 Thanks: 111  Quote:
Quote:
Last edited by mrtwhs; March 24th, 2017 at 02:37 PM. Reason: addendum  
March 24th, 2017, 10:04 PM  #8 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 319 Thanks: 26 Math Focus: Number theory 
Much gratitude for your contributions.


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