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March 23rd, 2017, 10:08 PM   #1
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Flatland Platonic solids' formula

Does a modification of Euler's polyhedral formula, c-e+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.
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March 24th, 2017, 09:27 AM   #2
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does a modification of euler's polyhedral formula, c-e+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.
cube.png

If you mean the above projection, it has 10 vertices (corners), 16 edges, and 8 faces (including the infinite region around the picture).

So $\displaystyle v-e+f = 10-16+8=2$ and the formula works.
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March 24th, 2017, 10:13 AM   #3
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Depends on the projection doesn't it? There are projections where points and/or edges merge.
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March 24th, 2017, 10:18 AM   #4
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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.
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March 24th, 2017, 11:56 AM   #5
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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?
cube.png

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.
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March 24th, 2017, 01:39 PM   #6
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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 2-D 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 3-D spatial region which negates the frame, would be topologically more difficult.
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March 24th, 2017, 02:32 PM   #7
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mrtwhs,

Was the general 2-D case (for maximal features) known before Euler?
I'm not really certain! So much is attributed to Euler that it might be tough to know.

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Does a similar inclusion for the "outside" apply for polyhedra? Maybe a 3-D spatial region which negates the frame, would be topologically more difficult.
I don't know for sure but I would doubt it. The faces of polyhedra are 2 dimensional even though they lie in space. What I do know is that a similar formula relating vertices, edges, faces, and volume does not exist in 3 dimensions. Here is an example. A tetrahedron (not regular) with vertices at (1,0,0), (1,1,0), (0,1,0), and (0,0,n) will have 4 vertices, 4 faces, and 6 edges but will have an arbitrarily large volume (dependent on n).
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Last edited by mrtwhs; March 24th, 2017 at 02:37 PM. Reason: addendum
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March 24th, 2017, 10:04 PM   #8
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Much gratitude for your contributions.
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