Paper Logic

Paper logic is the model this site uses to explain paper structures: each piece of paper is treated as a rigid sheet with no thickness that turns only along its creases.

In a pop-up

In a v-fold, a folded piece glued to both pages across the gutter, paper logic gives a clean result: four creases meet at one point on the spine, so the opening angle alone sets where the piece stands. The same model also shows where it stops matching paper. As the book closes, the gap between each folded piece and the facing page shrinks with the opening angle. Measured 5 mm out from the spine corner where the sheets meet, the narrower of the two pieces’ gaps in the model is 0.07 mm at a 2° opening and 0.36 mm at 10°. Those numbers hold only because the model gives paper no thickness, so in those last few degrees the drawing shows the model, not a sheet of card. How pop-up books work follows the v-fold through the whole opening and names the angles that make it jam.

Limits

Paper logic is a model, not a rule. Geometric studies of pop-ups make the same simplification, and they also note that real paper bends and has thickness. Pages that use the model say what it leaves out, and they say where the model and real paper part company, such as thick card, many stacked layers or curved forms. See Which layer was checked for how paper mechanisms are checked.