I was wondering why Greeks chose to have Olympic games every four years. Now, since we usually every fourth year is a leap one, it makes sense; but the reform of the calendar which stated this is due to Julius Caesar, and besides (see here) Greeks had a lunisolar calendar based on Metonic cycle, and a very good appoximation of the tropical year came only with Hipparcus, who lived in the 2nd century BC. I naively thought that a 5-year period should have been more logical, but maybe they had some scientific reason to choose a 4-year period. Does any evidence exist?
It is unknown why the 4 year period, a.k.a. the olympiad, was chosen. Most of the writings concerning the origins of the games come from 2nd century AD historians, nine centuries after the fact, and are largely mythological. According to one of them, Pausanias, "dactyl" Herakles and his four brothers, Paeonaeus, Epimedes, Iasius and Idas, raced at Olympia to entertain the newborn Zeus, who then crowned the victor with an olive branch, which became a peace symbol. Each year of the olympiad (inclusively) is to celebrate each brother. This is most likely a late fable invented to "explain" traditions the true source of which was forgotten. What little can be plausibly extracted from the myths is that the games were tied to worship of the gods, and were intended to promote and maintain peace and cooperation across Greece.
The only known astronomical connection is that the games began on the full moon closest to the summer solstice, which marked the start of a new year. The date was somewhat altered after the introduction of the Metonic cycle c. 432 BC. The Metonic cycle is close to a common multiple of the solar year and the synodic lunar month, and it is 6,940 days, about 19 years. Dividing gives 365 + 1⁄4 + 1⁄76 days per year, which comes close to rounding up the fraction every four years. Whether this played a role in the choice made earlier one can only speculate. If there was something tied to gods that all Greeks could agree on it would be what is seen in the heavens. We do know that after three centuries Greek historians started using the olympiad as a universal interval for reckoning time independently of individual conventions in city-states.
The estimate of the length of the tropical year and the synodic month by Meton, Hipparchus and others has no bearing on the system of four-year Olympiads. The use of a 365-day year with a leap day every four years was introduced by Julius Caesar in the first century BC. A similar calendar had previously been decreed in Egypt in 238 BC, but never actually implemented. The dating of events according to Olympiads can be traced back at the latest to Thucydides (5th century BC).