When Chang started to work on the Steiner problem, he moved it into a different setting. Instead of finding Steiner points on a plane in which roads can go in any direction, he looked for them on a grid, meaning that the roads can go only in certain directions. This put the problem into the realm of discrete geometry. It's like finding the shortest routes between buildings in New York City, where you have to follow the street pattern. You can't use the Pythagorean theorem to calculate the walking distance between two points.
Distance measurements take on a new meaning.
Without resorting to computers at any stage, Chang began exploring the types of situations that arise for three points on various types of two-dimensional grids, using tools from algebra and calculus to guide his efforts. He quickly found out how difficult the Steiner problem becomes in such a setting. The Steiner point problem with three points, which is a trivial plane geometry problem, becomes enormously complex in discrete geometry, as many close approximations compete to substitute for the simple optimal solution in the Euclidean plane. Chang discovered that the optimal solution to the Steiner problem typically no longer involves just a single point. That happens only under exceptional circumstances for certain grids.