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.

  • 1
    $\begingroup$ Do they mean something like 'cover it with hexagons and twelve pentagons'? $\endgroup$ – Empy2 Jun 7 '16 at 6:03
  • 1
    $\begingroup$ @Michael There are hint in the exercise as 'Use planar graph concept',but I can't figured it out $\endgroup$ – Hailey Jun 7 '16 at 6:05
  • 4
    $\begingroup$ Wow, you seriously didn't think that figure was relevant? Please post all relevant information from the beginning. Thank you. $\endgroup$ – Em. Jun 7 '16 at 6:48
  • 1
    $\begingroup$ Can you find complete graph on 5 vertices i.e. $K_5$ inside the map of the world? $\endgroup$ – Mathlover Jun 7 '16 at 6:56
  • 5
    $\begingroup$ 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.) $\endgroup$ – Martin Sleziak Jun 7 '16 at 14:17

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

  • 1
    $\begingroup$ Now all you need are the distances, which you can get from Wolfram Alpha $\endgroup$ – Robert Israel Jun 8 '16 at 7:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.