In attemopt to understand basic principles of non-Euclidean geometry and its relation to physical space, I am reading General Relativity by Ben Crowell. On page 149 there is a discussion of hexagons on a surface of sphere which I don't understand. He writes (caption for image attached):

"Because the space is locally Euclidean, the sum of the angles at a vertex has its Euclidean value of 360 degrees. The curvature can be detected, however, because the sum of the internal angles of a polygon is greater than the Euclidean value. For example, each spherical hexagon gives a sum of 6 x 124.31 degrees, rather than the Euclidean 6 x 120. The angular defect of 6 x 4.31 degrees is an intrinsic measure of curvature."

enter image description here

As three identical angles meet at the vertex and their sum is 360 degrees, one third of which is 120 degrees, how could the angle be 124.31?

  • $\begingroup$ This question is better suited for math.SE, where it can get better answer regarding how curvature affects the angles and stuff (even though that is not the main focus of the question). $\endgroup$ – mythealias Nov 10 '12 at 2:26

It's a slightly unclear example, because there is no vertex at which three hexagons meet. The surface is covered with a mixture of hexagons and pentagons, and if you study the diagram carefully you'll see that every vertex is a meeting point of two hexagons and one pentagon. So if you take the first of Crowell's diagrams, the one he labels as d/1, the three interior angles at each point are 120$^\circ$, 120$^\circ$ and 108$^\circ$. The three angles don't add up to 360$^\circ$ because the three lines at the vertex are not coplanar.

Crowell's point is that in his second diagram, d/2, as the polygons are "curved" outwards to lie on the surface of the sphere the interior angles increase. So the interior angles in the two hexagons increase to 124.31$^\circ$ and the interior angle of the pentagon increases to 111.38$^\circ$. So if you measure the angles round a vertex you'll still get the result 360$^\circ$, but if you measure the interior angle in the hexagon you find it's 124.31$^\circ$ rather than 120$^\circ$ and that's how you tell you're on a curved surface. You could just as easily measure the interior angles in the polygon and find they are bigger than 108$^\circ$.

  • $\begingroup$ "The surface is covered with a mixture of hexagons and pentagons, and if you study the diagram carefully you'll see that every vertex is a meeting point of two hexagons and one pentagon." Thanks a lot, that's the point, I feel ashamed :-) Can we delete the question? $\endgroup$ – Leos Ondra Nov 9 '12 at 12:18
  • $\begingroup$ I think you can delete your question. Isn't there a "Delete" link. Actually I'm not sure it can be deleted now I've answered it, and even if I delete my answer that won't help because deleting an answer just hides it instead of actually deleting it. You can ask the moderators if they will delete the question and my answer. That would be fine by me. $\endgroup$ – John Rennie Nov 9 '12 at 12:21
  • 2
    $\begingroup$ Don't delete, this is a valuable question and answer. $\endgroup$ – FrankH Nov 9 '12 at 12:44
  • $\begingroup$ @FrankH Ok, let's keep it here. $\endgroup$ – Leos Ondra Nov 9 '12 at 14:15
  • $\begingroup$ @LeosOndra Nothing to be ashamed of. There are plenty of subtleties here, and pretty much everyone gets tripped up now and again with non-Euclidean geometry. If only more of them were honest enough to ask when they didn't understand... $\endgroup$ – user10851 Nov 10 '12 at 4:26

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