Pop-Up Geometry

We have all received birthday, holiday, or congratulatory pop-up greeting cards.  The motions and actions of these pop-up cards provided us with an additional amusement. And, many of us have watched children learn with delight when they page through a Pop-Up book.  How many of us have ever wondered about or considered the complexity of making these cards and books?  The actions and mechanisms in these pop-ups are elaborate and complex.

 

Joseph O’Rourke has reduced much of this complexity to geometry, algebra, and trigonometry which is accessible to those with a basic understanding of high school mathematics. The book begins by providing the reader with card notation.  These notations and definitions are then used throughout the remainder of the book to explain the numerous illustrations and to develop theorems that describe the motions and connections among the pop-up elements.

 

The depth of geometry needed to understand the motions of a card include such things as dihedral angles, the intersection of two circles, and the intersection of three spheres, which the author points out is also used in the global positioning networks that we use to determine our bearings here on earth.  

 

Over 120 figures, many with multiple illustrations, provide comfortable references and direction as one follows the derived theorems or attempts the exercises included in the text.  This book is stocked with exercises to help one ‘test for understanding’ as they proceed through the text.  Complete solutions are provided for all of the exercises.  The exercises range from the readily solvable to the complex.  For example, the first exercise asks the reader to determine the dimensions of a pop-up so that it does not ‘stick out’ when the card is fully closed.  While another exercise asks the reader to determine the conditions under which three unit-radius spheres have just one point of intersection. 

 

The motions of the card structures in this book are readily seen in the internet animations from the author’s website.  O’Rourke’s website also provides templates for each illustration in his book.  The reader can readily construct a physical model of each of the illustrations in the book.  Once I viewed the website, I was unable to withstand the temptation to build my own pop-up card making a template from the site.  And, with help from my wife, we constructed the Knight’s Helmet Visor pop-up.   

 

This text can readily be used as a supplement to a geometry course.  I also see this book serving as a foundation for a multi-disciplined extracurricular activity called “Pop-Up Card Design”.  This activity would encourage students interested in enhancing their skills in English, Mathematics, and Art as they work together in a cooperative effort to produce Pop-Up cards or Pop-Up books. 

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Tom French has a B.S. and a M.S. degree in Mathematics from Minnesota State University, Mankato.  He has 35 years of engineering and business experience with UNIVAC and its successor companies.  He was part of the design team that first implemented medium-scale and large-scale integrated circuits into computers.  Tom was the program manager for several large computer innovations and was one of the leaders who implemented the technology revolution into the banking system in the Russian Federation.  He has lectured on mathematics and computer systems throughout the world and has taught mathematics at a number of US colleges and universities.