"What is the road system that connects all four towns using the smallest total length of road? What perhaps feels like the right answer is a network where opposing towns are connected in straight lines. If the square has side length 1km, the total length is about 2.83km In fact, the minimal network is the one below, which shaves off about 4 per cent of the length of the X solution."
"Here's a video of how it works presented by James Grime. I love this problem because it is one of the most powerful illustrations of how Nature effortlessly solves optimization problems that might otherwise require a lot of hard thinking. I gave a hint saying that the pattern resembles a geometric shape that appears in the real world in a very familiar setting."
Four towns at the corners of a unit square are to be connected by roads minimizing total length. The naive X-shaped solution joining opposite towns along diagonals has total length about 2.83 km. The true minimal network uses two interior junctions (Steiner points) where three roads meet at 120° angles, reducing total length to about 2.73 km. Proving minimality requires advanced calculus, but a soap-film experiment produces the optimal configuration naturally. A transparent sandwich model with four dowels dunked in soapy water shows the soap film forming the minimal road network. The 120° junctions connect to hexagonal packing principles used by bees to store honey efficiently.
Read at www.theguardian.com
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