A Geometric Study of V-style Pop-ups: Theories and Algorithms


Xian-Ying Li1   Tao Ju2   Yan Gu1   Shi-Min Hu1


1Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing

2Department of Computer Science and Engineering, Washington University in St. Louis








Pop-up books are a fascinating form of paper art with intriguing geometric properties. In this paper, we present a systematic study of a simple but common class of pop-ups consisting of patches falling into four parallel groups, which we call v-style pop-ups. We give sufficient conditions for a v-style paper structure to be pop-uppable. That is, it can be closed flat while maintaining the rigidity of the patches, the closing and opening do not need extra force besides holding two patches and are free of intersections, and the closed paper is contained within the page border. These conditions allow us to identify novel mechanisms for making pop-ups. Based on the theory and mechanisms, we developed an interactive tool for designing v-style pop-ups and an automated construction algorithm from a given geometry, both of which guaranteeing the popuppability of the results.





A Geometric Study of V-style Pop-ups: Theories and Algorithms [34.3M Paper] [1.8M Paper] [8.9M Video] [13.9M Slides]

Xian-Ying Li, Tao Ju, Yan Gu, and Shi-Min Hu.

ACM Transactions on Graphics (Proceedings of SIGGRAPH 2011), 30(4): article 98.





  author = {Li, Xian-Ying and Ju, Tao and Gu, Yan and Hu, Shi-Min},
  title = {A Geometric Study of V-style Pop-ups: Theories and Algorithms},
  journal = {ACM Transactions on Graphics},
  year = {2011},
  volume = {30},
  number = {4},
  pages = {98:1--10},




About Pop-up Books


"Design and creation of such books is known as paper engineering, a term not to be confused with the term for the science of paper making. It is akin to origami in so far as the two arts both employ folded paper. However, origami in its simplest form doesn't use scissors or glue and tends to be made with very bendy paper, pop-ups rely on glue, scissors and stiff card. What they have in common is folding. ...

The audience for early movable books were adults, not children. It is believed that the first use of movable mechanics appeared in a manuscript for an astrological book in 1306. The Catalan mystic and poet Ramon Llull, of Majorca, used a revolving disc or volvelle to illustrate his theories. "

--- from Wikipedia


"We cut, fold and glue thousands of pieces of card stock to begin developing the pop-up mechanisms themselves. This is the most difficult part of making a pop-up book. Many times the pop-ups designed don't work and we have to start all over again. But this is also one of the most exciting parts of a project, especially when the pop-ups work! All of the pop-ups are white since we don't need to waste any time adding color at this early stage. Designing the rough prototype pop-ups can take between three and six months. "

--- by Robert Sabuda, the paper engineer of "Alice's Adventure in Wonderland".





Popup: Automated Paper Architectures from 3D Models (Our Previous Work)


Robert Sabuda's website for Pop-up Books


"Pop-up Book" in Wikipedia






V-stylePopupMaker (14.5M for Windows Vista/7)