How can one construct a network of streets for a city that does not lead to major detours? Ideally everyone should be able to go from any one place to another on a path along the streets which is not much longer than the airline distance between the two locations. The worst-case ratio of path length and airline distance is called geometric dilation. It measures the quality of the network. We want to construct networks of small geometric dilation. In search of solutions to this seemingly simple question, we explore various fields of mathematics and computer science such as computational geometry, number theory, differential and integral geometry, disk packing, convex geometry, knot theory, fractals, and robot motion planning. Important arguments are based on the halving distance of a closed curve like the circle, i.e. the airline distance between two points which divide such a curve in parts of equal length.

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