Contents

• Introduction: The story of the Flexagon
• Building and Operating the Flexagon
• The Covering Space Representation of Flexagon Operation
• Maps and Plans
• The Pat Structure
• New Angles
• G-Flexagons
• Proper Flexagons
• The Flexing Operation and Tubulating Proper Flexagons
  • Flexing Characteristics
  • Tubulating Proper Flexagons
  
  
• Improper Flexagons
• Class Distinctions
• Zero Angles
• The Heterocyclic Flexagons
• Compound Faces
• Playing Cards
  • DUALS
  
  
• Requirements
• The Number of Flexagons
• The Distant Relations
  • A. Hinged Tetrahedra
  • B. The Mushrooms and Bregdoids
  • C. The Fleptagon
  
  
• Bibliography
• About this document ...

Acknowledgements

In 1956, when the authors first learned of flexagons, nothing like the present work was conceived of. Even when in the following summer, on the basis of encouraging theoretical and practical developments, a paper was begun on the basis of encouraging and practical developments, nothing like the final size was envisioned. However, as work progressed, things rapidly mushroomed; what seemed like minor difficulties often developed into major extensions of the subject. This continued during several summers. Finally, the last few years have seen the reigning in of all the loose ends and coordination of the finished text. Drawings were prepared by Anthony Conrad, Bari Thompson, and artists of the Martin Company.

RIAS in Baltimore has contributed in several practical and organizational ways to the completion of the paper.

Mike Schlesinger has constructed many models of flexagons which proved useful in the research.

The authors wish especially to thank the many flexagon enthusiasts with whom they have conferred or communicated. These include Harold V. McIntosh, C. O. Oakley and R. J. Wisner, L. B. Tuckermann, Arthur Stone, and John Tukey.

Finally, the authors would appreciate news of new developments which may reach the ears of the reader; correspondence may be addressed simply Baldwin, Maryland.

Anthony S. Conrad

Daniel K. Hartline

Cambridge, Massachusetts

May 5, 1962.

Flexagons are ostensibly a mathematical recreation, and as is frequently the case turn out not only to have a substantial underlying theory, but a close relation to other seemingly unrelated topics as well. At first, they have been studied simply because they were interesting, and that along is sufficient justification. It is accordingly gratifying to find that the theory is closely related to certain types of programming languages, because in this way those who feel that mathematical research must have an application can take the same delight in a beautiful theory brought to an elegant conclusion, that I have found in supporting this study on its own merits.

Harold V. McIntosh

Baltimore, 14 May 1962.