½ + ½ Dominoes

The elements of this puzzle are combinations of two diagonal halves of a domino tile.  The complete set of such elements consists of 11 pieces.  At first glance, it may seem that this is a set of some kind of "thorns" that are of little use for constructing beautiful geometric shapes.  However, we offer you the following tasks:

1.  Fill in the additional element into the frame.
2.  Lay out all the elements on the table and make one of the eight shapes which are illustrated in the instructions.
3.  Is it possible to tile a plane using an infinite number of this set?

Happy puzzling!

Made in acrylic by Russian designer Vladimir Krasnoukhov.