hexaflexagons
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| hexaflexagons | |
|---|---|
| Name | Hexaflexagons |
| Invented | 1939 |
| Inventor | Arthur H. Stone, Bryant Tuckerman (credit disputes) |
| Type | Paper toy |
| Related | Tetraflexagon, Octaflexagon, Flexagon |
hexaflexagons are folded paper constructions that reveal hidden faces when flexed, notable for their surprising topology and recreational mathematics connections. Originating in the late 1930s, they link amateur puzzle culture with formal work in Dudley and Princeton mathematics, and later influenced mathematical exposition and popular culture. The objects have been studied by both recreational mathematicians and researchers in graph theory and topology, inspiring collections, exhibitions, and educational activities.
History
Hexaflexagons were first noticed by students at Princeton after Arthur H. Stone folded strips of paper while a graduate student; credit and formal description involved Bryant Tuckerman and Richard Feynman among peers at Princeton, with early demonstrations at campus gatherings. The phenomenon entered print through the circuit of amateur problem solvers connected to Scientific American's Martin Gardner column and through the informal distribution of the "Flexagon Committee" notes in the 1950s and 1960s. Subsequent clarifications and generalizations were pursued by mathematicians at institutions including University of Michigan, Massachusetts Institute of Technology, and Cambridge University, with modern historical treatments referencing archives from Princeton University Library and oral histories involving figures from Bell Labs and Bell Telephone Laboratories where some early researchers worked.
Construction and types
Constructing a common form involves a strip of paper divided into equilateral triangles; folding patterns produce different species such as the tri-hexaflexagon, hexahexaflexagon, and higher-order variants. The tri-hexaflexagon (often called the "trihex") uses a strip length associated with 10 or 9 triangles depending on crease pattern and was demonstrated by early aficionados at Princeton. The hexahexaflexagon (six-faced) requires a longer strip and more complex branching; enthusiasts at institutions like MIT and Harvard University have produced catalogs of templates. Other constructions extend to octaflexagons and decaflexagons, with practical examples assembled by contributors at Smithsonian Institution exhibits and makerspaces at Cooper Hewitt-affiliated workshops.
Materials and tools historically include plain paper, colored surfaces, and adhesive; published templates have circulated through the American Mathematical Society outreach, Mathematical Association of America chapters, and recreational groups tied to Dudley, Massachusetts alumni. Physical variants have been constructed for demonstrations at Museum of Mathematics events and in classrooms affiliated with St. Olaf College and University of Chicago outreach programs.
Mathematical properties and topology
Hexaflexagons exhibit combinatorial and topological structure that links to graph theory, group theory, and surface topology. The set of faces and flex operations can be modeled as a state graph whose nodes represent visible faces and whose edges correspond to elementary flexes; research appearing in conferences at Institute for Advanced Study and papers by contributors associated with Bell Labs explored these transition graphs. The underlying paper strip corresponds to a planar embedding whose identification of edges produces a nontrivial surface; analyzing this leads to connections with orientable and nonorientable surfaces studied at Princeton University topology seminars. Algebraic invariants such as permutation groups generated by flex operations have been classified in works linked to researchers at University of Cambridge and Imperial College London.
Counting problems—number of distinct flexagons for given strip lengths—relate to enumerative combinatorics problems encountered in seminars at University of California, Berkeley and Stanford University. The fold patterns correspond to tilings by equilateral triangles and can be encoded using words in generators studied in group theory courses at Harvard University.
Folding mechanisms and flexes
A flex is an operation that transforms one visible face into another by reconfiguring internal layers; canonical flexes include pinwheel and helical moves demonstrated in workshops at Museum of Mathematics and in video lectures by educators at Massachusetts Institute of Technology. These moves are reversible and generate subgroups of the full symmetry group of the flexagon, with sequences of flexes producing cycles studied by participants in Mathematical Association of America meetings. Experimental analysis of mechanical stability and fatigue has been conducted by hobbyists associated with Bell Labs and by students at engineering departments of University of Michigan.
Instructional descriptions often reference creasing sequences named after early popularizers at Princeton and techniques for identifying hidden faces using colored patterns were popularized in demonstrations at Smithsonian Institution and Cooper Hewitt events.
Notable examples and variants
Famous exemplars include the tri-hexaflexagon demonstrated by Arthur H. Stone contemporaries, the hexahexaflexagon featured in early popular accounts circulated by Martin Gardner-era contributors, and museum specimens curated by Smithsonian Institution and Museum of Mathematics. Variants include the tetraflexagon and octaflexagon forms displayed in collections at Princeton University and Harvard University recreational mathematics clubs. Modern reinterpretations have appeared in installations by artists affiliated with Cooper Hewitt and in educational kits marketed by organizations tied to Mathematical Association of America outreach.
Academic variants explore generalized polyflexagons and planar foldings studied in seminars at Institute for Advanced Study and at colloquia hosted by University of Cambridge mathematics departments.
Cultural impact and applications
Hexaflexagons influenced recreational mathematics culture propagated by figures around Princeton University and through columns in Scientific American associated with Martin Gardner. They appear in popular science demonstrations at Smithsonian Institution and in pedagogical materials used by educators at St. Olaf College, University of Chicago, and Museum of Mathematics. Artists and designers in residencies at Cooper Hewitt and galleries linked to Cambridge have used flexagon principles in kinetic sculpture. Educational applications include hands-on activities in outreach programs of Mathematical Association of America and American Mathematical Society, where flexagons illustrate permutation concepts, topology intuition, and constructive problem solving.