Yang Xiong, writing around the turn of the first millennium, composed the Taixuanjing, the Canon of Supreme Mystery, as a deliberate counterpart to the I Ching. Where the I Ching uses six-line figures of two possible states, giving sixty-four, Yang Xiong used four-line figures of three possible states, solid, once broken, twice broken, giving three to the fourth power, eighty-one tetragrams. The work was an intellectual construction in the fullest sense: a complete alternative divinatory system built by one man on a different arithmetic base, and it is one of the more remarkable exercises in systematic thought to survive from the Han.
The eighty-one tetragrams map onto the year in units of roughly four and a half days each, since 81 x 4.5 is 364.5, close to a solar year, and the system was designed with that fit in mind.
Elsewhere eighty-one has narrower uses. Journey to the West has the pilgrims undergo eighty-one ordeals, a number chosen to complete a set of nine nines. Chinese calendrical folk practice counts the nine nines of winter, eighty-one days from the winter solstice to the return of warm weather, marked by counting charts.
Mathematically eighty-one is three to the fourth power, nine squared, and the fourth power of the first odd prime; its square root is nine and its digits sum to nine, which is a property of every multiple of nine and not of eighty-one in particular. That last point is worth making, because writing about this number frequently presents the digit sum as a special property when it is a consequence of the base.