【Research Paper on Pop-Up Books】 Geometry and Mathematics in Making Pop-up Books

My own pop-up book was inspired by works from Peter Dahmen

The gradual changes in shapes and angles are really aesthetic. While trying to make something like his work, I encountered some problems with designing. So I decide to do some research in the math rules that we can use while designing our own pop-up books.

For beginners like us, it’s pretty hard to construct some 3D structure just based on our imagination. Even though sometimes we got the design,


In Algorithms for Designing Pop-Up Cards, the authors have categorized several models that appeared very often in pop up cards. The first one is this linkage structure. They are like the cross section of the pop-up structure. However, compared to the 3D version of the real popped up thing, this is much easier to understand and design. Software like Linkage can help us to design this kind of structure digitally and simulate the popping movement.

Box structure is the one that’s most widely used by beginners like us. However, with this model introduced here.


Integrating box structures together became possible. We can actually create stuff that looks really complex but actually pretty simple to construct. The black dots stand for common joints, which is like a folding line on the paper. The white points are places that one is attached to another but able of folding like a hinge. and the crosses stand for slice forms where paper are intersected with each other. We can also change the basic structure that we are using from boxes into polygons that vary in shape. With further development, we can get something as beautiful as the one in Figure 1.

Apart from Linkage, there is more software that can help us with using digital computing in making pop-up books. Popup workshop is a software that can help us to simulate the actual folding in 3D. QQ20160515-2

Useres can design the pattern as well as viewign the output at the same time. Though there are only simple cuttings and foldings that we can use, it’s still helpfull in reducing the time that are wasted in calculating the length of each cut.


Work Cite:

Abel, Zachary, et al. “Algorithms for designing pop-up cards.” LIPIcs-Leibniz International Proceedings in Informatics. Vol. 20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2013. http://drops.dagstuhl.de/opus/volltexte/2013/3940/

Popup Workshop http://l3d.cs.colorado.edu/~ctg/projects/popups/

Leave a Reply