This is a very stable polyhedron and quite lovely. Notice that there is a yellow pentagon with 5 yellow, protruding, pentagram points in the same plane. Behind that yellow plane is a second, parallel yellow plane that forms a decagram. In the same plane as the decagram but facing in the opposite direction is another yellow decagram. In front of that second decagram is another parallel pentagram.
You can make this polyhedron using 6 colors. it is easiest to begin by making one of the cups that have a pentagon base and 5 crown-shaped side pieces. The crown shaped pieces are the shape on the right in the template below.
In his book, Polyhedron Models, Father Wenninger has listed the colors of each cup piece as follows. The center pentagram of each cup is the first color listed. The second through sixth crown-shaped pieces are added to the base pentagon in a clockwise direction.
(1) W Y B O R G
(2) Y W G O R B
(3) B W Y R G O
(4) R W O Y B G
(5) W R G O Y B
(6) G W R B O Y
To see the full-sized template click on this image:
Cup 1 will be at the center of your hemisphere surrounded by cups 2 through 6. Each of the cups will connect in the manner shown in the picture below.
An assembled cup will have side pieces that are glued together. The places where the side pieces are connected will be folded in opposite directions and glued to the back of the face of the other cup. See the picture below to help you visualize the technique.
A bobby pin placed connecting the two cups after gluing will easily hold the pieces together until dry.
At any time during your construction you can begin making the star-armed dimples that fit between the cups. Notice that the star arms finish the pentagram that can be seen exuding from every pentagonal face. Using this observation, you will be able to choose the appropriate colored kites to align into the star shaped dimple. Father Wenninger suggests that you can also use the color table given for the Great Dodecahedron to arrange these dimples.
The list of cup pieces is, as usual, only one-half of the polyhedron. The second half of the polyhedron is enantiomorphic to the first half. You can use the same color chart above but reverse your application of pieces 2 through 6 to be counter-clockwise.