Fatto matematico: 6820 is the number of regions formed when all diagonals are drawn in a regular 23-gon.
Trivia: 6820 is a number for which we're missing a fact (submit one to numbersapi at google mail!).
Data storica: September 2nd is the day in 1649 that the Italian city of Castro is completely destroyed by the forces of Pope Innocent X, ending the Wars of Castro.