Using a relatively young theory, a team of mathematicians has started to answer questions whose roots lie at the very beginning of mathematics. Kristina Armitage/Quanta Magazine Introduction 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 fa...