I recently found this article on the
blog of Alexandre Muņiz:
Flexible polyrhombs.

Here is the image he posted of a
figure with 5-fold symmetry.

This can be viewed as a construction of
pentominoes on 5 angled planes, with the pieces allowed to
"flex" across the joints. I coined the name "Flexominoes"
for puzzles of this type. Almost immediately, I wondered if I
could complete the star shape above using the 5 tetromino
pieces, and found that that I could. I also found several
more patterns with rotational symmetry before discovering
other possibilities using this new topology.

Hover over an image to see a
solution.