# Prove the statement 'World is not flat'

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

Prove that the World is not flat using Mathematics

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.

• Do they mean something like 'cover it with hexagons and twelve pentagons'? Jun 7, 2016 at 6:03
• @Michael There are hint in the exercise as 'Use planar graph concept',but I can't figured it out Jun 7, 2016 at 6:05
• Wow, you seriously didn't think that figure was relevant? Please post all relevant information from the beginning. Thank you.
– Em.
Jun 7, 2016 at 6:48
• Can you find complete graph on 5 vertices i.e. $K_5$ inside the map of the world? Jun 7, 2016 at 6:56
• 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.) Jun 7, 2016 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