I made an animated gif! It is the 6th image.
I made a mini version of this as a test to see if the design that I reversed engineered worked(Look here for the full story: https://www.instructables.com/id/Diamond-Puzzle-Sketchbook/
) and it did, but for the mini version I used cardboard for the bottom and that caused it to be hard to move the segments all at once, I fixed this problem by having the top and bottom made of card stock. I am able to move it very smoothly, I might make a video in the future, or if someone asks.
And to explain for how it works go to the last image, that explains most of it. I dont know if the creator's book's design is like that but it works for me, and it is a very cool notebook design.
Unfortuantly I wont be able to make a step by step instructable for this exact one because I only took pictures of the first half and I was too busy making the rest that I forgot about taking pictures. I am going to make another one out of wood, and you could use that same instructable for a paper one, you would just have to modify the hinge a little bit.