This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
Can a disk be cut into similar pieces without point symmetry about the
midpoint? Can it be done with a finite number of pieces?
Yes. Draw a circle inside the circumference of the disk, sharing a
common point on the right. Now draw another circle inside the second,
sharing a point at the left. Now draw another inside the third,
sharing a point at the right. Continue in this way, coloring in every
other region thus generated. Now, all the colored regions touch, so
count this as one piece and the uncolored regions as a second piece.
So the circle has been divided into two similar pieces and there is no
point symmetry about the midpoint. Maybe it is cheating to call these
single pieces, though.