Carbon dating tells us the age of once-living things, from ancient cloth to Ice Age mammoths,by measuring how much radioactive carbon-14 they still contain. Every carbon date quietlyassumes that time is a straight line running from the endless past to the endless future.This paper asks, in plain language, what would happen if time were instead a circle inwhich history repeats. We explain how carbon dating works, what it means for time to becircular, and why ordinary radioactive decay cannot simply continue around a circle. Wedescribe three ways the radiocarbon clock could be made to fit a circle of time and applythem to ten famous samples, including the Shroud of Turin, ¨ Otzi the Iceman, the Chauvetcave paintings and the baby mammoth Lyuba. If time repeats every 100,000 years, recentdates barely change but Ice Age dates can shift by thousands of years. If it repeats every5,000 years, every carbon date, however old, is squeezed into the last 2,500 years. We endby explaining how tree rings and other radioactive clocks can test which picture of time iscorrect.