The shape can fit on to itself 3 times before returning to its original position.
Next time try this:
1. Trace and cut out the shape 2. Label the top 'A' 3. Trace the shape one more time. DO NOT CUT THIS ONE OUT 4. Put the cut out shape on the 2nd traced one 5. Put a pen in on the point, like a spinner, and rotate it 90degrees 6. See how many it can fit into itself before 'A' returns to the top.