Fifteen is the magic constant of the smallest genuine magic square, the three-by-three grid the Chinese tradition calls the Luo Shu, in which every row, column and diagonal totals fifteen; the figure is tied to the myth of a pattern seen on the back of a turtle emerging from the river Luo, and it underlies a good deal of later Chinese cosmological arithmetic. In the tarot the fifteenth trump is the Devil, a card of bondage to the material paired, by its number, with the Lovers reversed. Fifteen is also triangular in a loose sense only, but it is exactly the sum of the numbers one through five, which is why the Pythagorean tradition treated it as a natural rounding-off of the pentad. Most of its weight, though, comes from the full moon and from the Hebrew scribal caution about the Name.
The fifteenth day of the lunar month, already noted above as the full moon of Purnima, is also the date of two of the best-attended festivals in the Chinese calendar: the Lantern Festival, Yuanxiao, closes the fifteen days of Lunar New Year on the fifteenth of the first month, and the Mid-Autumn Festival falls on the fifteenth of the eighth month, both timed to the full moon on exactly the logic that gives fifteen its place in the Hindu calendar, an independent instance of the same lunar arithmetic doing similar cultural work in an unrelated tradition.