Researchers kept working on the Heilbronn triangle problem over the years, despite the long wait for progress, motivated by its tangle of links to other areas of math. “I think it’s a stunning result,” said Anthony Carbery, a mathematician at the University of Edinburgh. Then this past May, three mathematicians - Alex Cohen, Cosmin Pohoata and Dmitrii Zakharov - announced a new cap on the size of the smallest triangle. The problem is simple to state, but progress on the Heilbronn triangle problem, as it came to be called, has been halting, and results dried up entirely in the 1980s. The soldiers didn’t appear to be in formation, which got him thinking: If there are n soldiers inside a square, what is the size of the largest possible smallest triangle defined by three of them? Heilbronn wondered how one might go about arranging the soldiers (or, for mathematical simplicity, points) to maximize the size of the smallest triangle. Hans Heilbronn, a German mathematician who fled his country before World War II and settled in England, thought of these triangles in the late 1940s when he saw a group of soldiers outside his window. Each of those triangles, of course, has a particular area. The numbers grow quickly from there - 100 points define 161,700 different triangles. Four points define four different triangles. Take three of those points, and you can make a triangle. Consider a square with a bunch of points inside.
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