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In my graph theory book exercise, I found a problem that:

Prove that the World is not flat using Mathematics

enter image description here

This picture is used in the exercise, but no idea of applying it.

I would have used the angles don't add to $180^\circ$, but the author is using a fourth city.

Martin Sleziak
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Hailey
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    Do they mean something like 'cover it with hexagons and twelve pentagons'? – Empy2 Jun 07 '16 at 06:03
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    @Michael There are hint in the exercise as 'Use planar graph concept',but I can't figured it out – Hailey Jun 07 '16 at 06:05
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    Wow, you seriously didn't think that figure was relevant? Please post all relevant information from the beginning. Thank you. – Em. Jun 07 '16 at 06:48
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    Can you find complete graph on 5 vertices i.e. $K_5$ inside the map of the world? – Nitin Uniyal Jun 07 '16 at 06:56
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    I think that adding the information about the book where this comes from would be useful. Is it Einstein Gravity in a Nutshell by Zee, page 66? (If it is, this seems to be closer to differential geometry than to graph theory.) – Martin Sleziak Jun 07 '16 at 14:17
  • @Hailey You have repeatedly asked why your question was put on hold - for example, in your most recent edit. I am not sure whether it is the same question, but I have tried to post what I think you are asking and include sufficient context there. – Martin Sleziak Jun 08 '16 at 06:51
  • BTW at least giving the title and author of the "graph theory book" where you found this would be, in my opinion, a good way of adding context to your question. ("In my graph theory book" sound much more vague than if you provide exact bibliographic information identifying the book.) – Martin Sleziak Jun 08 '16 at 06:54
  • Hint: Since you don't have angles at your disposal, you have to replace them by distances via the cosine rule. Assume that the world is flat and that you know the mutual distances between Berlin, Barcelona and Paris. The you can compute the distance from Rome to Paris from its distance to Berlin and Barcelona. If this does not agree with the real distance then the assumption that the world is flat was wrong. – Andreas Cap Jun 08 '16 at 07:20
  • @Hailey Even if you do not wish to add some more context to your question, could you at least say whether you consider it to be the same question as the one I posted? So that one of them can be closed as a duplicate or perhaps they could be merged. – Martin Sleziak Jun 08 '16 at 07:31
  • You wrote that the hint given there was: "Use planar graph concept." That seems strange to me for two reasons: $K_4$ is planar. And even graph which is not planar can be drawn in a plane. But only in a such way that edges will intersect. – Martin Sleziak Jun 08 '16 at 09:02

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The two diagonals $p$ and $q$ of a plane quadrilateral and the four side lengths $a$, $b$, $c$, $d$ are related by the Cayley-Menger determinant: $$\det\pmatrix{0&a^2&p^2&d^2&1\cr a^2&0&b^2&q^2&1\cr p^2&b^2&0&c^2&1\cr d^2&q^2&c^2&0&1\cr1&1&1&1&0\cr}=0$$ See https://en.wikipedia.org/wiki/Quadrilateral#Properties_of_the_diagonals_in_some_quadrilaterals

Gerry Myerson
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