New math revives geometry's oldest problems
In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight. Such questions, which ask for the number of solutions that satisfy a set of geometric conditions, were a favorite of the ancient Greeks. And they’ve continued to entrance mathematicians for millennia. How many lines lie on a cubic surface? How many quadratic curves lie on a quintic surface? (Twen