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        riddles >> medium >> Missing Vertex
        
(Message started by: ThudanBlunder on Mar 23^rd, 2009, 4:24pm)

Title: Missing Vertex
Post by ThudanBlunder on Mar 23^rd, 2009, 4:24pm

The sum of the face angles at all but one of the vertices of a given simple polyhedron is 5160^o. What is the sum of the face angles at the missing vertex?

Title: Re: Missing Vertex
Post by towr on Mar 24^th, 2009, 2:50am

[hide]
polyhedral formula, V + F - E = 2
number of angles, #A = 2E

sum(180 - A_n) = F*360
#A*180 - sum(A_n) = F*360
E*360 - sum(A_n) = F*360

E*360 - F*360 = sum(A_n)
V*360 = sum(A_n) + 720
sum(A_n) = 0 mod 360
5160 = -240 mod 360

So the remaining angles sum to 240 degrees (as it has to be between 0 and 360).
[/hide]

Title: Re: Missing Vertex
Post by JohanC on Mar 24^th, 2009, 4:28am

on 03/23/09 at 16:24:44, ThudanBlunder wrote:

.... of a given simple polyhedron ...

What's your definition of a simple polyhedron?
If the polyhedron isn't convex, Towr's conclusion is only valid modulo 360, I think.

Title: Re: Missing Vertex
Post by ThudanBlunder on Mar 24^th, 2009, 6:44am

on 03/24/09 at 04:28:07, JohanC wrote:

What's your definition of a simple polyhedron?
If the polyhedron isn't convex, Towr's conclusion is only valid modulo 360, I think.

'Simple' means topologically equivalent to a sphere.