Lots of cardboard
Box cutter/Hobby knife
Step 1: Cut Out the Pieces
This step takes a long time. I did it over the course of a week because I went through several blades and it tires out your shoulder. The cardboard gets bent and damaged sometimes but that is not a big deal.
When spray painting mine, I put a primer down first to help stiffen the cardboard up a bit. I then used indoor chrome spray paint because it sticks best.
Step 2: Start Assembly
A frabjous has a total of 20 points on the surface connecting 3 "S" shapes together. In order to figure out where each piece goes, it is best to number the points (which I have done in one of the pictures) and divide them into four layers. The Upper Layer contains points 1-5, the Upper Middle Layer points 6-10, the Lower Middle Layer points 11-15, and the Lower Layer points 16-20.
When assembling, it is best to always work with the Upper Layer and renumber the points to put the pieces in place. As you can see in the diagram, point 1 connects across to point 13. What's not show is where the other pieces that are connected to point 1 go. Those other two pieces are connected to the Lower Layer at points 17 and 19. When looking at this two dimensionally, it looks a little like a "W" - all the pyramids within the structure follow this pattern.
Point 2 connects to 14, 18, and 20
Point 6 connects to 8, 9, and 18
Point 11 connects to 4, 13, and 14
Point 16 connects to 3, 5, and 9
In my second picture, the two pyramids are held at points 1 and 3, and they are connected at point 19. In the 3rd picture with three pyramids, the intersections are points 1, 3, and 4 connected at points 19 and 17.
Only the intersections at points 1, 3, and 4 are glued. All other connections from on out should be with tape in order to correct mistakes.
Step 3: Continue the Pattern
On each side of the dodecahedron we start with points 1,3, and 4 because it helped stabilize the frabjous and made the construct pattern the same everywhere. When you add in points 2 and 5, the top blossom should be finished (Pic 2).
Picture 3 shows the underneath of the first blossom, which has now become the Upper Layer. Re-number and insert the next five piece to create the second blossom. Make sure the spiral pattern is correct on all the blossoms. Each piece should go over 2 pieces, and under the other 2 pieces.
Once that is finished, flip over to another unfinished blossom, re-number, and finish the blossom. The hardest part here is figuring out if a piece should go under or over one already established. Keep the blossom spiral in mind and you should be able to figure it out. We had to undo and reconnect at least 10 shapes because they were out of place. Don't be afraid to be forceful with the cardboard, it can handle it. An accidental fold is barely noticeable when it is finished.
Step 4: Glue It Together
Edit: Added some pictures with a light in the center due to popular demand. Unfortunately my low quality camera does a poor job capturing the true look, but I hope it gives everyone a good idea of what it looks like.