Nine is three taken three times, and a striking number of traditions that value three arrive at nine without any contact with one another.
Greek tradition counts nine Muses, and the underworld river Styx is said to wind nine times. Norse cosmology has nine worlds held by the world tree, and the Havamal has Odin hang nine nights on that tree to win the runes, a self-sacrifice counted in nines throughout. Indian tradition counts the navagraha, the nine celestial influences of its astrology, and keeps the festival of Navaratri across nine nights.
Chinese thought treats nine as the greatest of the yang digits and therefore as the imperial number. The plan of the classical capital, the studs on palace gates and the tiers of imperial ritual are counted in nines and multiples of nine, and the dragon, itself an imperial emblem, is conventionally described as having nine attributes.
Christian angelology, following the arrangement set out under the name of Dionysius the Areopagite, ranks the angels in nine orders grouped in three triads, which is the three-times-three logic made explicit. The Bahai Faith takes nine as its own number and builds its houses of worship with nine sides and nine entrances.
Arithmetically nine has a property the modern numerological systems depend on entirely. Add the digits of any multiple of nine and the result is again a multiple of nine, which is what makes digit-sum reduction behave the way those systems need it to.