We show one solution without gaps between pieces.
This is the unique solution without gaps modulo rotation of the box.
1.
First put a H.
We place it so that it makes space at the center of the box.
2.
Place T next to it.
Arrange it so that it touches H with the same isosceles traiangle face as H has.
3.
Place another T also on the other side.
4.
Put another T in the last space on the lower side.
This time, the T touches two other T pieces with
edges of the big regular triangle.
There is a hollow in the center.
It exactly has the shape of sharp vertex of T.
5.
Place a T at the center.
6.
Put pieces on the upper side so that pieces tough through faces with the
same shape.
7.
8.
9.
Complete.
This (and its rotations) are the only solution of the puzzle
without gaps between pieces.
There are many other ways to put nine pieces if we allow gaps between
pieces.