A Study on the Boundary Conditions of 90° Paper Pop-up Structures

Abstract

The design of a pop-up book or card has hitherto been labour intensive with tasks of trials and errors. The constructions of collapsible pop-up structures can be demanding and inefficient without adequate knowledge of their geometric properties. This paper examines the properties of creases in 90° pop-up structures. A 90° pop-up structure is one that erects fully when two adjacent base pages, on which it sits, are opened to a right angle. In particular, we define a boundary region for creating 90° pop-ups. Similarly, paper folds are able to achieve pop-up effects and can be integrated with 90° pop-up constructions. The development of these pop-up structures can be represented graphically. Through this study, a fundamental foundation for pop-up topology and geometry is built. This foundation would be vital for understanding the applications of pop-up making techniques. The mathematical relationships devised would be useful for developing computer-enhanced pop-up design.