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May 21[edit]

Vertex angles of a convex polyhedron[edit]

As JBL mentioned in a recent thread, the sum of the face angles at a particular vertex of a convex polyhedron must be less than 360°. (1) Is there a simple proof of this? (2) Is there a formula for this sum? (Presumably it must depend on the angles between the faces or edges and some reference line.) Loraof (talk) 16:25, 21 May 2017 (UTC)[reply]

This seems relevant for 1) but I am not sure this counts as "simple". Tigraan^{Click here to contact me} 11:35, 22 May 2017 (UTC)[reply]

I suppose that, while not really simple, that may be as simple as it gets (and as far as I can see it only applies when the number of faces meeting at the vertex is 3). I was thinking in terms of an umbrella whose rigid ribs can be viewed as edges of a polyhedron. When you (or the wind) open it all the way so it makes a plane, the sum of the angles at the apex is 360°. On the other hand, if you close it completely so each rigid edge approaches the umbrella's pole, each angle (and hence their sum) approaches 0°. It seems like all the angles between the edge-ribs should be monotonic as you gradually close the umbrella. I have this feeling that a proof of this should be trivial, but I can't come up with it. Loraof (talk) 14:53, 22 May 2017 (UTC)[reply]

Here is how I would try to prove it: pick a vertex and choose a direction that points towards the interior of the polyhedron. Show that there is a small number epsilon so that taking a plane slice at distance epsilon in the chosen direction cuts off a pyramid. By construction, this pyramid has the property that the altitude from the apex lands inside the base. Cut the pyramid into tetrahedra, taking the apex, the foot of the altitude, and a pair of consecutive base vertices as the vertices. Then the desired result is the following: in a tetrahedron ABCD such that AB is perpendicular to BCD, angle CAD is smaller in measure than CBD. (Separately, the original argument can be made globally, without considering individual vertices like this -- see angular defect.) --JBL (talk) 15:04, 22 May 2017 (UTC)[reply]