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Math EE- Optimization

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Hi I wanted to do my math EE in optimization, geometry specifically. I found a problem similar the the "Alhazen Billiard Ball" Problem that I want to write my essay on. It is:"Road System in a sqaure - Four villages each being a vertex of a square, should be connected by a road network so that the total length of the road system is minimal." Is this a good question to write an extended essay on? And if I decide to do the problem, should I explicitely state that the essay will solve this problem like the Alhazen Billiard Ball one? My coordinator told me to do it on a general topic such as optimization is constructing infrastructure and then use this problem as an example. Which format is better to use?

Also I wanted to io an extension of this problem as part of the essay, either road system in a cube, or road system in other regular polygons. Which one is easier/better to do? Thanks for any imput. :yes:

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