For years, a simple question has haunted Máté Matolcsi, a professor at the Budapest University of Technology and Economics. How much of an infinite plane can you color in while making sure that no two colored points are exactly one unit of distance apart?
The question was first posed by Leo Moser, a Canadian mathematician, in the early 1960s. In 1967, Hallard Croft at the University of Cambridge came up with a construction that seemed to do a pretty good job. His shape, now called “Croft’s tortoise,” looks like a circle that met a hexagon-shaped cookie cutter. Every point inside each tortoise is less than one unit away from any other point in the same tortoise, and more than one unit away from the closest point of the neighboring tortoise.
In the half-century since, nobody has been able to find a shape that improves on the 22.936% of the plane that the tortoises cover. But could one exist, even in theory? In 1984, László Székely, a Hungarian mathematician, proved that it is impossible to find a shape that covers more than 27.91% of the plane. The next year, Paul Erdős, the prolific conjecturer (and fellow Hungarian), said he thought the upper bound was less than 25%. As with many Erdős conjectures, it attracted the attention of numerous ambitious mathematicians over the years.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου