Hmm -- you're quite right that I meant the sphere, and also that the space of uninteresting dissections is larger than I was thinking. Perhaps the good question again is to ask for one with not all pieces touching the center?
Nah, ... the right question is to ask for a classification of congruent dissections. I'll do this for the 2d case of a unit disc, but of course it only encompasses *known* congruent dissections. And, to be explicit, I'm only considering pieces that have connected interior. First, there are the obvious, uninteresting dissections: use n equally spaced segments (possibly curved) from the center to the boundary. In this case, I think it's OK to describe the "uninteresting" cases as fundamental regions of a finite subgroup of SO(2) ; we don't even need all of O(2) . Second, start by dissecting into six congruent pieces with arcs of unit circles. These pieces have mirror symmetry, so can be bisected with straight line segments. This is the figure in UPIG, and the first that Michael sent last month. Third, start with the same six-piece dissection as before. Now further dissect each into n congruent pieces using arcs of unit circles from the center. Since n of these make a shape with reflective symmetry, patches of n can be reflected (they need not have started in the same piece of the six-piece dissection). The n = 2 case, with all six pairs of pieces flipped is the second figure that Michael sent earlier; perhaps he had intended this more general situation. If this list is not complete, I'd be very interested in seeing more! On the other hand, if the list is complete, then it solves questions such as "can the center be in the interior of one of the pieces?" and "can the diameter of the pieces be less than 1 ?". Michael Reid