Metonic cycle: the yearly cycle of lunar phases repeats every 19 years. Visualized as a supercoiled torus with season, lunar & daily helices

About 12.37 lunar cycles occur each year. It makes regular calendars a challenge or cultures that use 1 cycle for each month. Most years have 12 lunar months, leap years have 13. Ancient cultures from Babylon to Greece and China noticed 235 lunar cycles repeat every 19 years. Regularity allows rules for which years to add a 13th month to keep a decentralized kingdom's calendars synchronized. The first picture is the whole cycle as a torus. Each year makes a spoke with the winter solstice in the center and summer on the outside. Within each year is a helix of moon phases with new moons as a black dot and full as white. A tighter helix is wound around the phases to show the day night cycle. Sun rise and set times are based on Denver. Further from the equator, winter to summer, day to night ratios become more extreme. The second image is a cross section showing +/- 1 year.