1) Start from the centre and reach any of the sides following a curve, such that as you traverse the curve your distance from the centre should increase, e.g.,

1) Start from the centre and reach any of the sides following a curve, such that as you traverse the curve your distance from the centre should increase, e.g., a circular arc that is less than a semi-circle. (The construction of this curve gives [at least] an uncountably infinite cubed number of degeneracies.) 2) Now take the reflection of this curve wrt the centre, and attach it to the original curve. This will give two-congruent pieces. 3) You can take the resulting curve and rotate it around the centre by pi/2. The two curves will give 4 congruent pieces. The restriction on the original curve, ie. its monotonic behaviour, guarantees that the curves in steps 2 and 3 will not intersect.