There is a popular math book called “Cows in the Maze”, written by the British mathematician Ian Stewart. Stewart devoted Chapter 2 of the book to the Opaque Forest Problem. The problem can be summarized in the following manner:
You have inherited a plot of land and want to build a fence so that no neighbors’ "evil eyes" can pass across the plot. You want to minimize the cost of the fence by minimizing its length. How can you do that?
The above is an open mathematical problem. A simple solution will be to approach the problem traditionally and build a fence on every edge of your plot, but it is too costly. On page 21, Fig. 5b of his book Stewart claims to provide the best current solution for the regular pentagon case (see my Fig. 1 below, having in mind that the edges of the pentagon are of unit length and the green lines form the minimum Steiner tree for the three top vertices of the pentagon).
You have inherited a plot of land and want to build a fence so that no neighbors’ "evil eyes" can pass across the plot. You want to minimize the cost of the fence by minimizing its length. How can you do that?
The above is an open mathematical problem. A simple solution will be to approach the problem traditionally and build a fence on every edge of your plot, but it is too costly. On page 21, Fig. 5b of his book Stewart claims to provide the best current solution for the regular pentagon case (see my Fig. 1 below, having in mind that the edges of the pentagon are of unit length and the green lines form the minimum Steiner tree for the three top vertices of the pentagon).
Fig. 1
Unfortunately, Stewart’s solution is far from the optimal, but with Fig. 2 below I’ll demonstrate how some creative thinking leads to gradual improvements and a much better solution. Calculations are needless as every successive solution is obviously an improvement on the previous one.
Fig. 2
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