Pop-Up Geometry
The Mathematics Behind Pop-Up Cards

Explores the beautifully intricate dynamics of pop-up cards using high school mathematics, making tangible what is often dry and abstract.

Joseph O'Rourke (Author)

9781009096263, Cambridge University Press

Paperback / softback, published 24 March 2022

144 pages
22.8 x 15.2 x 0.8 cm, 0.24 kg

'Using an engaging and conversational tone, O'Rourke strikes a nice balance between rigor and informality. It's easy to follow the details in the elementary material, yet also get the big picture for the more advanced topics. The book includes ample exercises, ranging from mathematical computations to card constructions, along with solutions. Whether you are a casual reader or a serious student of pop-ups, you will come away with new insights. This book will open a world of possibilities. Get it and get to work. Happy folding!' Richard H. Hammack, Math Horizons

Anyone browsing at the stationery store will see an incredible array of pop-up cards available for any occasion. The workings of pop-up cards and pop-up books can be remarkably intricate. Behind such designs lies beautiful geometry involving the intersection of circles, cones, and spheres, the movements of linkages, and other constructions. The geometry can be modelled by algebraic equations, whose solutions explain the dynamics. For example, several pop-up motions rely on the intersection of three spheres, a computation made every second for GPS location. Connecting the motions of the card structures with the algebra and geometry reveals abstract mathematics performing tangible calculations. Beginning with the nephroid in the 19th-century, the mathematics of pop-up design is now at the frontiers of rigid origami and algorithmic computational complexity. All topics are accessible to those familiar with high-school mathematics; no calculus required. Explanations are supplemented by 140+ figures and 20 animations.

Preface
1. Parallel Folds
2. V-Folds and Rotary Motion
3. The Knight's Visor
4. Pop-up Spinner
5. Polyhedra: Rigid Origami and Flattening
6. Algorithms for Pop-Up Design
7. Pop-Up Design is Hard
8. Solutions to Exercises.

Subject Areas: Origami & paper engineering [WFTM], Mathematical theory of computation [UYA], Algorithms & data structures [UMB], Geometry [PBM]