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I´m trying to obtain the Euler characteristic of this polyhedron $P$, that is homeomorphic to the torus $T$ (I think):

enter image description here

So it should be $\mathcal{X}(P)=\mathcal{X}(T)=0$.

But we get $V=16, F=10, E=24$, so $\mathcal{X}(P)=2$.

However, if we consider a triangulation as this two cases:

enter image description here

it is $\mathcal{X}(P)=0$, because $V=C=16$ and $E=32$, and $V=16, F=32, E=48$, respectively.

So, what is it wrong?

Thanks for the support!

J. W. Tanner
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LH8
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1 Answers1

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The problem is that your first polygon setup uses non-simple polygons. add single edge to each donut shaped face and it will work better.

Cauchy's basic proof for Euler's formula for characteristic requires that we triangulate the polygons and takes advantage of a particular property of this triangulation: adding a single edge via splitting a polygon must also add a single face. This is true of simple polygons, yes, but complex polygons like the donut shape don't have this property: adding an edge may not split the polygon into two.

Dan Uznanski
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  • Yes, that´s what I do in the first triangulation, yeah? But what means non-simple polygons? Maybe you mean convex? In what sense it disturb a Euler characteristic? Since it´s homeomorphic to the torus, they should have same Euler characteristic, directly. Thanks! – LH8 Feb 18 '19 at 10:39
  • Your polygons on the 'top' and 'bottom' goes around the hole in the torus. This is not allowed since then they are not contractable. – An.Ditlev Feb 18 '19 at 10:48
  • So, a polyedron has different Euler characteristic depending the triangulation? I´m confuse. Thanks. – LH8 Feb 18 '19 at 11:19
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    Not at all: all triangulations of a polyhedron should have the same Euler characteristic... But in order to get an Euler characteristic that makes sense all the polygons need to be simple. A triangle within a triangle doesn't have the same layout as a hexagon, despite having technically the same number of edges and vertices. – Dan Uznanski Feb 18 '19 at 11:24
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    "Simple" here is "simply connected", meaning you can draw any loop in the polygon and shrink it down to a point without crossing any boundaries. That does not work for your top and bottom faces without triangulation, because a loop drawn around the inner boundary gets "trapped" around that boundary. Cutting the faces from inside to outside, thus excluding those offending loops, is what makes the faces simply connected again. – Oscar Lanzi Feb 18 '19 at 12:19
  • @LH8 Do you consider the top "piece" of your figure an octagon? It has eight sides, but they are not in one "cycle", four of the sides create one "circumference", and the remaining four sides create another "path". Such a figure is not usually allowed as a face in a polyhedron. You cannot calculate the Euler characteristic correctly if you include such faces. – Jeppe Stig Nielsen Feb 18 '19 at 16:36