20070314, 06:54  #12  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Quote:
If they are the angles the sum around the origin will be 4 x 120 = 480* A point only has 360* around it. I will say the four angles will be 90 * each if that is possible in this rectangle. Mally Last fiddled with by mfgoode on 20070314 at 06:55 

20070314, 09:57  #13  
"Lucan"
Dec 2006
England
1933_{16} Posts 
Quote:
to a pair of vertices. So each node has three roads. If fixed width font were available, text diagrams would be easier but I'll try. :><: should have been clear enough. A B \ / ++ / \ D C Last fiddled with by davieddy on 20070314 at 10:01 

20070314, 11:44  #14 
"Lucan"
Dec 2006
England
1100100110011_{2} Posts 
Conjecture
The angle between routes in the minimum length
network problem is never less than 120 degrees. I think this applies to any number of dimensions. David 
20070314, 12:54  #15 
Jun 2003
2^{2}×3^{2}×131 Posts 

20070314, 14:04  #16 
"Lucan"
Dec 2006
England
14463_{8} Posts 
THX
I think even Mally might understand that 
20070314, 16:03  #17 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
linked nodes
Well Davie thats just what I was imagining. I am not sure of any angle being = 120* for a minimum unless it is vigorously and mathematically proved. This is not as difficult as it seems. Just get a function involving angles or sides and deriving and solving for max , min. lengths. It will give you a test of your differentiating powers. Come on Davie Im sure you can do it as I am not inclined to go to such a length. Mally 
20070314, 19:31  #18 
"Lucan"
Dec 2006
England
6,451 Posts 
That's not the way I solved your original triangle problem,
my rectangle or arrived at my conjecture. And it is not the simplest proof of my conjecture. It boils down to recognizing that 3 vertices linked by a "V" with angle <120 are linked more efficiently with a "Y" with 120 degree angles. It follows that no more than 3 routes meet at a node/vertex. The conjecture doesn't give a recipe for finding the best network though David 
20070316, 10:18  #19 
"Lucan"
Dec 2006
England
1100100110011_{2} Posts 
It's like talking to a brick wall

20070319, 15:31  #20 
"Lucan"
Dec 2006
England
6,451 Posts 
Did you approve of my construction?

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