Helicone is an interactive sculpture by John Edmark (more info). You can buy one on Amazon. By rotating the shaft, the arms can settle into a helix, or a pinecone shape that uses (half of) the Golden Angle to fill out the space completely.
I saw a few videos of it in action and decided I wanted to try to make one. I found a few models online, but they all relied on externally visible notches for stops, which the Helicone does not exhibit. After I came up with my solution, I saw a "making of" video with some screen shots of the laser pattern which indicates that it uses a very similar mechanism.
A video of it working.
You can use the Customizer to change various dimensions and petal shape. If you download the OpenSCAD file, you can also adjust the number of slices and experiment with different functions to change the helix angle and 3D outline.
The color change in the photos/video is due to running out of black, and having to replace two slices due to breakage.
Print out two of each slice. The slice number is embossed on the underside of one petal to help keep them organized when you drop them. To assemble, thread all the slices and top and bottom caps onto a 3/16" dowel, working them around until they spin freely. Glue the bottom cap to the rod (I used a dot of superglue; be careful not to glue any other parts to the rod). You can also glue to the top cap into position if you want to prevent the parts flying off the top.