While the cathedral does not have an outdoor labyrinth, its Cathedral Crossroads labyrinth program on the last Tuesday evening of each month is always well attended. Two 11-circuit Chartres-style labyrinths are available for walking in the transept to the accompaniment of harp and Native American flute, while a smaller labyrinth is often laid out in the Joseph of Arimathea chapel downstairs for silent walking. These evenings also include Centering Prayer (a Christian form of meditation) and a program that highlights some variety of spiritual experience. The meditation garden walks in the summer are particularly refreshing.
The cathedral itself, of course, is worth visiting, but best during the day when light streams through the stained glass windows and you can take the elevator to the tower gallery for some of the best views in Washington. Finding a silent spot for private prayer or reflection can be tough, though, with the tour groups wandering about. The quietest place is the Center for Prayer and Pilgrimage on the crypt level, and if it's not open, then try the tiny Good Shepherd chapel.