Based on other designs I saw over the internet, I designed an Iris-box made completely from 4mm wood and glue, without the need for screws. I added the ornaments in Lightburn, but any other tool should work. Hope you enjoy it, and the following instructions are understandable.

Supplies

The box consists of the bottom plate, 20 wall segments, 5 petals, 4 wooden rings and 10 wooden rods of 12mm and 20mm length.

What you need:

4mm thick Wood sheets
4mm diameter Wood rod
Wood glue

I also cut the wooden rods using my laser, but a saw will also do just fine. Only be careful that they are as precise as possible.

Step 1: Combining Bottom and Wall

At first glue the wall segments into the bottom plate and let it sit aside to dry. Try to keep them as straight up as possible, such that they do not dry in the wrong orientation.

Step 2: Prepare Top Rings

Glue the 12mm rods flush into Ring 2 and the 20mm rods into Ring 4. Let it dry.

Step 3: Add Petals Onto Ring 2 and Glue Ring 1 On

Put the Petals onto Ring 2 by putting the rods through the normal hole in the petals. Then put Ring 1 onto the petals and glue it there. Be careful not to get any glue on the petals or the middle of the rod! The petals need to be able to rotate on that rod in the end. Also, don't squeeze the two rings too hard together. I put a thick piece of paper between the petals and Ring 1 as a placeholder, such that when removing the paper later, there was a small gap, and the petals had space to move. Let it dry.

Step 4: Add the Assembly From Step 3 to Ring 4 and Ring 3

Put the assembly from Step 3 onto the assembly of Ring 4 with the 20mm rods. The rods need to go through the long holes in the petals, and in Rings 1 and 2. Then put Ring 3 on top and glue it there. Again, be careful not to get any glue on the wrong point, and don't squeeze too hard. Let it dry.

Step 5: Combine Top and Bottom Assembly

Finally, combine the top and bottom assemblies. It's a little tricky to get the wall segments into place but it is possible with some tactile sensitivity. Let it dry and you are done!