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A hexaflexagon, shown with the same face in two configurations
A hexaflexagon, shown with the same face in two configurations

In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front.

Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon.

In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.[1][2]

Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation.[1]

History

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Discovery and introduction of the hexaflexagon

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The discovery of the first flexagon, a trihexaflexagon, is credited to the British mathematician Arthur H. Stone, while a student at Princeton University in the United States in 1939. His new American paper would not fit in his English binder so he cut off the ends of the paper and began folding them into different shapes.[3] One of these formed a trihexaflexagon. Stone's colleagues Bryant Tuckerman, Richard Feynman, and John Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon.[4] Tuckerman traverses are shown as a diagram that maps each face of the flexagon to each other face. In doing so, he realized that each face does not always appear in the same state.

Flexagons were introduced to the general public by Martin Gardner in the December 1956 issue of Scientific American in an article so well-received that it launched Gardner's "Mathematical Games" column which then ran in that magazine for the next twenty-five years.[3][5] In 1974, the magician Doug Henning included a construct-your-own hexaflexagon with the original cast recording of his Broadway show The Magic Show.

Attempted commercial development

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In 1955, Russell Rogers and Leonard D'Andrea of Homestead Park, Pennsylvania applied for a patent, and in 1959 they were granted U.S. Patent number 2,883,195 for the hexahexaflexagon, under the title "Changeable Amusement Devices and the Like."

Their patent imagined possible applications of the device "as a toy, as an advertising display device, or as an educational geometric device."[6] A few such novelties were produced by the Herbick & Held Printing Company, the printing company in Pittsburgh where Rogers worked, but the device, marketed as the "Hexmo", failed to catch on.

Varieties

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Tetraflexagons

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Tritetraflexagon

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Diagram for folding a tritetraflexagon
A tritetraflexagon can be folded from a strip of paper as shown.
Sides of a tritetraflexagon
This figure has two faces visible, built of squares marked with As and Bs. The face of Cs is hidden inside the flexagon.

The tritetraflexagon is the simplest tetraflexagon (flexagon with square sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat. The construction of the tritetraflexagon is similar to the mechanism used in the traditional Jacob's Ladder children's toy, in Rubik's Magic and in the magic wallet trick or the Himber wallet.

The tritetraflexagon has two dead ends, where you cannot flex forward. To get to another face you must either flex backwards or flip the flexagon over.

Tritetraflexagon traverse

Hexatetraflexagon

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A more complicated cyclic hexatetraflexagon requires no gluing. A cyclic hexatetraflexagon does not have any "dead ends", but the person making it can keep folding it until they reach the starting position. If the sides are colored in the process, the states can be seen more clearly.

Hexatetraflexagon traverse

Contrary to the tritetraflexagon, the hexatetraflexagon has no dead ends, and does not ever need to be flexed backwards.

Hexaflexagons

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Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure. (Note that the word hexaflexagons [with no prefixes] can sometimes refer to an ordinary hexahexaflexagon, with six sides instead of other numbers.)

Trihexaflexagon

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This trihexaflexagon template shows 3 colors of 9 triangles, printed on one side, and folded to be colored on both sides. The two yellow triangles on the ends will end up taped together. The red and blue arcs are seen as full circles on the inside of one side or the other when folded.

A hexaflexagon with three faces is the simplest of the hexaflexagons to make and to manage, and is made from a single strip of paper, divided into nine equilateral triangles. (Some patterns provide ten triangles, two of which are glued together in the final assembly.)

To assemble, the strip is folded every third triangle, connecting back to itself after three inversions in the manner of the international recycling symbol. This makes a Möbius strip whose single edge forms a trefoil knot.

Hexahexaflexagon

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This hexaflexagon has six faces. It is made up of nineteen triangles folded from a strip of paper.

A strip of paper, divided into triangles, which can be folded into a hexaflexagon.
A series of photos detailing construction and "flexing" of a hexaflexagon
Figures 1-6 show the construction of a hexaflexagon made out of cardboard triangles on a backing made from a strip of cloth. It has been decorated in six colours; orange, blue, and red in figure 1 correspond to 1, 2, and 3 in the diagram above. The opposite side, figure 2, is decorated with purple, gray, and yellow. Note the different patterns used for the colors on the two sides. Figure 3 shows the first fold, and figure 4 the result of the first nine folds, which form a spiral. Figures 5-6 show the final folding of the spiral to make a hexagon; in 5, two red faces have been hidden by a valley fold, and in 6, two red faces on the bottom side have been hidden by a mountain fold. After figure 6, the final loose triangle is folded over and attached to the other end of the original strip so that one side is all blue, and the other all orange. Photos 7 and 8 show the process of everting the hexaflexagon to show the formerly hidden red triangles. By further manipulations, all six colors can be exposed.

Once folded, faces 1, 2, and 3 are easier to find than faces 4, 5, and 6.

An easy way to expose all six faces is using the Tuckerman traverse, named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from exactly the same corner every time. If the corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:

1 → 3 → 6 → 1 → 3 → 2 → 4 → 3 → 2 → 1 → 5 → 2

And then back to 1 again.

Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 can be flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)

Hexahexaflexagons can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.

Other hexaflexagons

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While the most commonly seen hexaflexagons have either three or six faces, variations exist with any number of faces. Straight strips produce hexaflexagons with a multiple of three number of faces. Other numbers are obtained from nonstraight strips, that are just straight strips with some joints folded, eliminating some faces. Many strips can be folded in different ways, producing different hexaflexagons, with different folding maps.

Higher order flexagons

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Right octaflexagon and right dodecaflexagon

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In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes.[7] The division of the square faces of tetraflexagons into right isosceles triangles yields the octaflexagons,[8] and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons.[9]

Pentaflexagon and right decaflexagon

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In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72–54–54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides one and two on the front and back, to displaying its previously hidden sides three and four.[10]

By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon.[11]

Generalized isosceles n-flexagon

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The pentaflexagon is one of an infinite sequence of flexagons based on dividing a regular n-gon into n isosceles triangles. Other flexagons include the heptaflexagon,[12] the isosceles octaflexagon,[13] the enneaflexagon,[14] and others.

Nonplanar pentaflexagon and nonplanar heptaflexagon

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Harold V. McIntosh also describes "nonplanar" flexagons (i.e., ones which cannot be flexed so they lie flat); ones folded from pentagons called pentaflexagons,[15] and from heptagons called heptaflexagons.[16] These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of isosceles triangles, and they can be made to lie flat.

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Flexagons are also a popular book structure used by artist's book creators such as Julie Chen (Life Cycle) and Edward H. Hutchins (Album and Voces de México). Instructions for making tetra-tetra-flexagon and cross-flexagons are included in Making Handmade Books: 100+ Bindings, Structures and Forms by Alisa Golden.[17]

A high-order hexaflexagon was used as a plot element in Piers Anthony's novel 0X, in which a flex was analogous to the travel between alternate universes.[18]

Vi Hart, a well-known recreational mathematician and public educator, gained attention for their video on hexaflexagons.

See also

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References

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Bibliography

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Flexagon

View on Grokipedia
from Grokipedia
A flexagon is a flat, polygonal paper model constructed by folding a strip or sheet into a compact form, such as a hexagon, that can be manipulated or "flexed" along its creases to reveal and conceal different faces, creating an illusion of multiple hidden surfaces.[1][2] Flexagons were invented in 1939 by British mathematician Arthur H. Stone, then a graduate student at Princeton University, who discovered the concept while idly folding strips of paper trimmed from the margins of his notes.[1][2] Stone's initial creation, the trihexaflexagon—a hexagon with three distinct faces—sparked interest among peers, including Richard P. Feynman, Bryant Tuckerman, and John W. Tukey, who formed an informal "Flexagon Committee" to explore their properties and develop folding techniques.[1] Though early work by Tukey and others laid foundational mathematical theory in the 1940s, flexagons remained obscure until popularized in 1956 by Martin Gardner in his Scientific American column on recreational mathematics, which introduced constructions like the hexahexaflexagon to a wider audience.[1] The most common flexagons are hexaflexagons, based on equilateral triangular units folded into hexagonal rings, with varieties distinguished by the number of visible faces they can display through flexing motions such as "pinches" and "V-folds."[1][2] Notable examples include the trihexaflexagon (order 3, revealing 3 faces from a 10-triangle strip), the tetrahexaflexagon (4 faces), pentahexaflexagon (5 faces), and hexahexaflexagon (order 6, up to 3,420 distinct mathematical faces from a 19-triangle strip).[1][2] Higher-order flexagons, such as the dodecahexaflexagon (order 12), can exhibit billions of faces, analyzed through combinatorial algorithms that count configurations via recursive formulas.[2] Mathematically, flexagons connect to topology—modeling surfaces like Möbius bands—and geometry, with their folding patterns demonstrating principles of polyhedral nets and cyclic permutations, while inspiring educational applications in visualization and pattern recognition.[1][2]

Fundamentals

Definition and Properties

A flexagon is a flat, folded paper model formed from a strip of polygons, typically equilateral triangles or other regular polygons, arranged in a chain and overlapped to create a multi-layered cyclic structure. These structures possess the intriguing property of revealing different hidden faces through a process known as flexing, where vertices are pinched and manipulated to cycle through visible surfaces.[1][3] The first notable example, the trihexaflexagon, demonstrates this by displaying three distinct faces in a repeating sequence.[1] Key properties of flexagons include their multi-layered construction, which enables a cyclic permutation of the visible faces as the model is flexed, creating the illusion of new surfaces emerging without altering the overall form. When flexed, flexagons exhibit non-convex geometry, temporarily deforming into irregular shapes that allow inner layers to become outer ones. Additionally, the total surface area is conserved throughout these transitions, as the folding merely rearranges the existing polygonal faces without expansion or contraction.[4][3][4] The net of a flexagon begins as a linear chain of identical polygons connected edge-to-edge, which, when folded, results in overlapping layers that obscure some faces until flexing exposes them—for instance, envision a sequence of equilateral triangles folding into a compact hexagonal envelope with concealed interior patterns.[4][1] This layered arrangement underpins the mathematical elegance of flexagons, blending geometry and topology in a deceptively simple form.[5]

Construction Basics

To construct a basic flexagon, begin with appropriate materials: a strip of fairly stiff paper at least 1.5 inches wide, scissors for cutting, and glue or tape for assembly.[1] Stiff paper, such as cardstock or adding machine tape, ensures durability during repeated flexing, while still allowing the necessary folds.[6] For a generic net, such as that of a trihexaflexagon, first draw or cut a straight chain of 10 equilateral triangles along the paper strip.[1] Label the triangles sequentially if desired for tracking faces. Score all fold lines lightly with a ruler or fingernail to create creases, folding each line both forward and backward multiple times before proceeding—this prevents tearing and facilitates smooth flexing later.[6] Begin the folding sequence by folding the strip backward along the line between the second and third triangles (denoted as line ab in standard diagrams), then turn the entire strip over. Next, fold backward again along the line between the eighth and ninth triangles (line cd), tucking the ninth triangle over the first. Finally, fold the tenth triangle backward and glue or tape it to the underside of the first triangle, overlapping the ends slightly to form a closed hexagonal ring.[1] The resulting structure should lie flat with six visible triangular faces, ready for flexing to reveal hidden layers. Common pitfalls include incomplete creasing, which can cause the paper to resist flexing or tear during use, and uneven chain lengths that prevent proper closure into a ring.[7] Over-tightening folds during initial assembly can lock layers together, hindering the multi-layered reveals that define flexagons.[6] While the basic process uses triangular strips, it can be adapted briefly for other polygons, such as squares for tetraflexagons: start with a net of squares in a specific pattern, such as 6 squares for the tri-tetraflexagon, fold into a diamond pattern, and join ends similarly with tape, though the creasing requires diagonal lines for tetrahedral symmetry.[8] For pentagons, wavy strips of regular pentagons follow a comparable chaining and overlapping method, but the non-planar result demands extra care in creasing to maintain flexibility.[9]

History

Discovery of the Hexaflexagon

The hexaflexagon was invented in 1939 by Arthur H. Stone, a 23-year-old English graduate student in mathematics at Princeton University. Arriving from Britain, Stone encountered the challenge of adapting to larger American notebook paper, which required trimming an inch from the edges to fit his accustomed size. During these adjustments, he experimented with the resulting strips of leftover paper, folding them into equilateral triangles in the course of tiling explorations, inadvertently creating the first known flexagon: a trihexaflexagon constructed from a strip of 10 equilateral triangles that could be manipulated to reveal different faces.[10][1] In early 1940, Stone shared his discovery with fellow Princeton graduate students John W. Tukey, Bryant Tuckerman, and Richard P. Feynman, sparking collaborative interest among the group. They formed an informal "Flexagon Committee" to systematically investigate the structures, exploring variations by adjusting the length and folding patterns of paper strips, which led to the development of additional hexaflexagon models such as the hexahexaflexagon. This teamwork produced an early mathematical theory of flexagons, including methods for construction and analysis of their surfaces, though it remained unpublished at the time and circulated only within academic circles.[11][1] The term "flexagon" originated from this committee, derived from the distinctive flexing action that allowed the paper models to pivot along folds and expose hidden faces, distinguishing them from static geometric forms. The hexaflexagon, particularly the trihexaflexagon, emerged as the simplest non-trivial example in their studies, serving as the foundational model for further inquiry. While Stone's tiling experiments echoed contemporary interests in tessellations, such as those inspiring M.C. Escher's work in the late 1930s, no documented prior inventions of flexagon-like manipulable polygons exist, marking the 1939 creation as a serendipitous innovation in recreational mathematics.[12][10]

Early Development and Commercial Attempts

Following the initial discovery, collaborative efforts in the early 1940s advanced flexagon variations through the Princeton Flexagon Committee, comprising Arthur H. Stone, Bryant Tuckerman, John Tukey, and Richard P. Feynman, who developed theoretical frameworks and new constructions despite their work remaining unpublished at the time.[1] These explorations expanded beyond basic hexaflexagons to include multifaceted designs, laying foundational insights into flexing mechanisms and surface mappings that influenced later recreational mathematics.[10] The popularization of flexagons occurred in the mid-1950s via Martin Gardner's "Mathematical Games" column in Scientific American, with his December 1956 article introducing hexaflexagons to a broad audience and sparking widespread interest among mathematicians and hobbyists.[1] Gardner's exposition, later compiled in his 1959 book Hexaflexagons and Other Mathematical Diversions, detailed construction methods and encouraged experimentation, significantly boosting engagement in paper-folding puzzles as a form of accessible mathematical recreation.[13] Commercialization efforts emerged in the late 1950s, exemplified by the 1955 patent application from Russell Rogers and Leonard D'Andrea, granted as U.S. Patent 2,883,195 in 1959 for a "hexahexaflexagon" device titled "Changeable Amusement Devices and the Like," aimed at producing interactive toys with interchangeable faces.[14] Despite such initiatives, marketing attempts faced challenges from the intricate assembly required for paper-based models, resulting in limited adoption by toy manufacturers and only niche availability in educational outlets by the mid-1960s. By the 1970s, interest waned as plastic-based puzzles and construction toys gained prominence for their durability and ease of use, overshadowing the delicate, hands-on nature of flexagons.[15] Nonetheless, these early developments fostered enduring hobbyist communities, with Gardner's writings inspiring ongoing amateur explorations and contributions to recreational mathematics literature.[16]

Mathematics

Geometric Structure

Flexagons are modeled topologically as quotients of polygonal strips folded cyclically, yielding a non-orientable surface akin to a Möbius band due to half-twists in the folding process. This framework accounts for the invariant topological band underlying all positions of the flexagon, distinguishing it from its apparent ring-like geometry during flexing. The non-orientable nature arises from the identification of edges in the strip, creating multiple sheets that overlap in a covering space. The net geometry of a flexagon consists of a linear strip of equilateral triangular sectors (pats) arranged in a cycle, where vertices represent hinge points and edges denote folds between adjacent pats. These pats form the fundamental units, with the overall net closing upon itself through the folding process. For example, the trihexaflexagon is constructed from a strip of 10 triangles, while the hexahexaflexagon uses 19 triangles, folding into compact hexagonal forms with layered overlaps.[1] During flexing, the surface undergoes a continuous transition that interchanges hidden layers with exposed ones, effectively permuting the visible configuration without tearing or altering the underlying connectivity. This deformation preserves the topological invariants, notably the Euler characteristic χ = 0 for the Möbius band-like structure. Such transitions highlight the flexagon's role in illustrating homotopy equivalence in low-dimensional surfaces.[2] For flexagons constructed from non-triangular bases, such as quadrilateral or higher polygonal forms, adaptations using isosceles trapezoids in the net ensure foldability by aligning angle bisectors at hinge lines, preventing angular mismatches that could distort the cyclic symmetry. This modification maintains the required dihedral angles for smooth folding while preserving the overall topological structure.

Enumeration Methods

Enumeration of flexagons involves combinatorial techniques to classify and count distinct configurations based on their folding patterns and symmetries. Approaches apply group theory to account for equivalences under flexing operations, such as rotations and reflections induced by pinching. Classification schemes organize flexagons by their order n, defined as the number of distinct colorable faces, where pats are the equilateral triangular regions forming the net. In frameworks like those of Oakley and Wisner, flexagons are categorized into orders such as tetra- (order 4) and hexa- (order 6), with regular flexagons requiring n ≡ 0 (mod 3) for closed cycles. Parity distinguishes right-handed and left-handed twists in the folding sequence, where each pinch introduces a half-twist of ±180°. The total number of surfaces S accounts for overlapping configurations during flexing; for example, the trihexaflexagon has 3 mathematical surfaces.[17] Advanced counting employs algorithms to handle irregular nets and asymmetric sequences, particularly for hexaflexagons with varying numbers of faces. By modeling sequences as necklaces under cyclic and dihedral group actions, the number of distinct hexaflexagons with n faces can be computed recursively. Early enumerations by the Flexagon Committee, including Bryant Tuckerman, identified 18 distinct configurations for the hexahexaflexagon in the 1940s, including variations from straight, bent, and clover-shaped nets.[1] Limitations arise from parity mismatches in twist sequences, preventing closure for certain nets. For example, bases with odd-sided polygons often fail to form closed flexagons due to incompatible half-twist parities, restricting constructible cases to even-order or specific modular conditions. Non-regular pats allow broader constructions but increase computational complexity in enumeration, as symmetries no longer align neatly with modular arithmetic.[17]

Varieties

Tetraflexagons

Tetraflexagons are flexagons constructed from nets of squares, forming structures based on 4-sided polygons that reveal multiple surfaces upon flexing, typically displaying two surfaces at a time while hiding others in layered overlaps. These differ from triangular-based flexagons by their 90-degree angular geometry, enabling simpler mechanics with flex points aligned to square edges. A basic tetraflexagon net often consists of a linear or cyclic strip that closes after folding, though more complex variants extend this for additional layers.[18][19] The tritetraflexagon represents the smallest tetraflexagon, featuring 3 visible sides and folded from a net of 6 squares (2 rows of 3) arranged in a strip, numbered on both sides to guide assembly. Construction involves creasing the strip into a compact square form with double thickness, where flexing cycles through the minimal layers by opening and refolding along hinged edges, exposing one new surface per cycle without dead ends in optimal paths. This results in a structure with 6 paper layers overall, providing basic topological transitions between the three faces.[20][18] The hexatetraflexagon extends this to a 6-sided variant, utilizing a 16-square net (4x4 grid) that allows for more intricate overlaps and multi-axis flexing, distinct from hexaflexagons due to its right-angled flex points rather than 60-degree ones. Folded into a square or hexagonal profile, it supports complex layer interactions, revealing all six surfaces through vertical and horizontal manipulations. Enumeration methods indicate dozens of distinct hexatetraflexagon configurations, though only a subset are cyclic without traps.[18][20] Construction of tetraflexagons emphasizes precise 90-degree folds along scored lines to maintain uniform thickness, typically doubling the paper for stability.[19][20]

Hexaflexagons

Hexaflexagons represent the most accessible and varied family of flexagons, built from nets composed of equilateral triangles that assemble into a hexagonal form, enabling the revelation of hidden surfaces through targeted folding and pinching actions. Their triangular basis provides rotational symmetry and straightforward construction, distinguishing them from other flexagon types while emphasizing the diversity of visible patterns achievable with minimal materials.[6] The trihexaflexagon serves as the foundational example, utilizing a classic 10-triangle net that folds into a compact hexagon exhibiting 3 distinct surfaces. Discovered by Arthur H. Stone in 1939 during his graduate studies at Princeton University, this model reveals one new face per complete flex cycle when a corner is pinched and the structure is gently opened.[6][21] The tetrahexaflexagon uses a 13-triangle net to display 4 faces, bridging the tri- and penta- varieties with moderate complexity in flexing sequences. The pentahexaflexagon employs a 16-triangle net for 5 faces, requiring more precise folding to access all surfaces without trapping.[22] Building on this, the hexahexaflexagon employs a 19-triangle net to produce 6 surfaces, incorporating intricate overlaps of layered triangles that support bidirectional flexing for exploring faces in forward or reverse sequences.[21][6] Additional hexaflexagons expand this framework, such as the isohexaflexagon with its symmetric traps that promote balanced folding paths, and tetraxial variants featuring 4 axes for multifaceted rotations.[6][21] A hallmark of hexaflexagons is their reliance on 60-degree angles inherent to equilateral triangular components, which facilitate smoother transitions between surfaces during flexing. Their net designs further ensure durability, preventing tears through precise overlapping and gluing that withstands repeated manipulations.[6][21]

Higher-Order Flexagons

Higher-order flexagons extend the principles of basic varieties to polygons with more than six sides, introducing greater structural complexity and often requiring non-standard folding patterns to achieve viable models. These include right-angled variants like the octaflexagon and dodecaflexagon, which utilize specific triangular sectors to form regular polygonal bases while maintaining flexing capabilities through specialized twists. The right octaflexagon, an 8-sided model constructed from 45-45-90 right triangles, features nets composed of 16 sectors where eight triangles meet at the center to form the octagonal shape. Flexing occurs via pyramidal twists, such as 4-fold pinch flexes that manipulate flaps in two distinct positions, allowing the model to reveal multiple faces despite its higher symmetry. Similarly, the right dodecaflexagon is a 12-sided variant built from 30-60-90 right triangles, with nets containing 24 sectors that enable 12 triangles per face and support symmetries including 2-fold, 3-fold, 4-fold, 6-fold, and 12-fold rotations. Its flexing relies on pyramidal twists, such as pyramid shuffles and pinch flexes with 3-fold or 6-fold symmetry, which transition the model through hybrid faces and non-hexagonal configurations like propeller shapes.[23][24][25] In contrast, flexagons based on pentagonal and decagonal polygons introduce irregularities due to their odd or higher even side counts, demanding asymmetric folds that can lead to incomplete or partial closures during assembly. The pentaflexagon, with a 5-gon base, assembles from strips where five triangles converge at the center, necessitating irregular folds like slot flexes—where adjacent pairs are folded and slid through openings—and pyramid shuffles that bend leaves into tetrahedral intermediates before reopening to pentagonal states. The odd number of sides poses challenges, often resulting in partial closures where final flaps must be tucked face-to-face and taped, limiting full planar traversal and requiring puzzle-like sequences to access all faces, such as transitioning from one set of faces to another via repeated slot and shuffle operations. The decaflexagon, based on a 10-gon, uses variants like 36-72-72 isosceles triangles or 36-54-90 right triangles across 10 per face, with irregular folds including pinch flexes exhibiting 5-fold symmetry and tuck flexes that fold halves differently to reveal star or ring patterns. These models highlight the difficulties of even higher orders, where the increased sector count amplifies folding inconsistencies, though they remain viable through pre-creasing and precise taping of numbered pairs.[9][26] A generalized framework for isosceles n-flexagons accommodates arbitrary n greater than 6 by dividing regular n-gons into isosceles triangular sectors, typically trapezoidal in net form, to create flexible polygons with n leaves per face. This approach defines universal flex operations, such as pinch flexes that rotate and pair leaves according to patterns like (1,2)(3)…(n) transforming to (^1)(5,^3)…(^n-2)(2,^n), ensuring consistent handedness for viability. Viability requires n to be even for dual-sided strips and the base angle to divide evenly into 360 degrees, allowing complete closure without distortions; odd n variants, like the pentaflexagon, succeed only with single-sided constructions and adapted flexes. Nonplanar examples, such as the pentaflexagon and heptaflexagon, function as 3D hybrids by folding out-of-plane to expose extra surfaces beyond the primary two faces, with the heptaflexagon forming configurations like a 6-triangle base plus an protruding "tent" via pinch flexes that mix faces across dimensions. These nonplanar forms were first described in 1970s literature on advanced flexagon geometries, expanding beyond planar constraints to incorporate twisted linkages and 3D shuffles.[27][28]

Applications and Culture

Flexing Techniques

Flexing a flexagon involves manipulating its layered structure to reveal hidden surfaces through controlled folding motions. The basic process begins by pinching two adjacent triangles at one corner of the flexagon between the thumb and forefinger, while simultaneously pushing the opposite corner inward with the other hand to form a temporary three-dimensional shape. This action pops a hidden layer forward, allowing the flexagon to open like a flower and expose a new face. For a standard trihexaflexagon, repeating this sequence three times—each followed by a 120-degree rotation—cycles through all three faces and returns the model to its starting configuration.[29][30] Advanced maneuvers expand on the basic pinch to access multiple surfaces more efficiently. Rocking involves holding one diagonal of the hexagon horizontally and alternately flexing the left and right sides, which reveals additional faces through a swinging motion. Twisting, often part of a Tuckerman traverse, entails flexing repeatedly at the same corner until the motion becomes obstructed, then rotating the flexagon slightly to an adjacent vertex and continuing; this systematic approach, developed by Bryant Tuckerman, systematically uncovers all available faces without redundancy.[30][31][29] If a flexagon becomes stuck during manipulation, troubleshooting typically requires gently re-flattening the model by pressing it evenly against a surface to realign the layers, ensuring no unintended creases have formed that could lock the structure. To prevent such issues, users should avoid excessive force and verify that all folds maintain a single paper thickness at hinge points during initial assembly. Tetraflexagons, constructed from strips of 45-45-90 degree triangles, demand sharper angles and more precise pinching compared to hexaflexagons, which use equilateral triangles and allow smoother, 60-degree rotations.[29][19] During transitions, flexing produces striking visual effects, such as sudden color shifts or pattern inversions as hidden layers emerge, creating optical illusions of impossible geometries that momentarily distort the apparent shape before settling into the new flat state.[32][30]

Popular Culture and Education

Flexagons gained prominence in popular culture through the writings of mathematician and author Martin Gardner, who introduced them to a broad audience in his December 1956 Scientific American column "Mathematical Games: Flexagons," describing their construction and surprising properties as recreational mathematics.[1] This exposure led to their inclusion in Gardner's 1956 book Mathematics, Magic and Mystery, where they were presented alongside other geometric curiosities as engaging puzzles. In media, flexagons appeared in a 1996 episode of the Canadian documentary series The Nature of Things hosted by David Suzuki, which profiled Gardner and demonstrated flexagon folding as an example of accessible mathematical play.[33] More recently, online videos have revitalized interest; Vi Hart's 2012 animated tutorial on hexaflexagons popularized their history and assembly, amassing millions of views and inspiring DIY enthusiasts.[34] Similarly, a 2019 Numberphile video featuring mathematician Matt Parker explored the "forgotten" tetraflexagon, highlighting its lesser-known variants and sparking discussions in mathematical communities.[35] In education, flexagons serve as hands-on tools for teaching geometric concepts and introductory topology, allowing students to explore folding symmetries and hidden surfaces without advanced prerequisites.[36] They appear in geometry curricula to illustrate non-Euclidean transformations and problem-solving, often integrated into lessons on polyhedra and paper folding.[37] Post-2010 digital adaptations have extended their classroom use; online simulators like the Flexagon Playground enable virtual construction and experimentation with various flexagon types, facilitating remote learning and customization.[38] While VR-specific flexagon simulations remain limited, interactive apps and web tools support topology visualizations, bridging traditional paper models with computational geometry. As toys and hobbies, flexagons have seen a revival in maker communities through 3D printing, with designs shared on platforms like Thingiverse for durable, hinged models that mimic paper flexing without folding wear.[39] These printable variants, popular in the 2020s, appeal to hobbyists for their fidget-like qualities and customization potential using multi-color filaments.[40] Flexagons also feature in puzzle challenges, such as mazes embedded in their surfaces, enhancing their role in recreational problem-solving.[41] Recent online tutorials on sites like Instructables and YouTube provide step-by-step guides, while STEM kits distributed by libraries and educational organizations include pre-cut templates for assembly, promoting accessible math exploration in homes and schools.[42][43]

References

  1. Mathematical Games: Flexagons | Scientific American
  2. The combinatorics of all regular flexagons - ScienceDirect.com
  3. Introduction | SpringerLink
  4. [PDF] flexagon.pdf
  5. [PDF] Flexagons Author(s): C. O. Oakley and R. J. Wisner Source
  6. [PDF] Hexaflexagons - Assets - Cambridge University Press
  7. None
  8. How to Fold a Hexa-tetraflexagon - National Museum of Mathematics
  9. Pentaflexagon - Loki3 Home Page
  10. Introduction: The story of the Flexagon - Erik Demaine
  11. Flexagon History - Eighth Square
  12. [PDF] Down the Rabbit Hole with Flexagons - Gathering 4 Gardner
  13. Hexaflexagons and Other Mathematical Diversions: The First ...
  14. Milestones — Bassetti Architects
  15. [PDF] Flexagons Inside Out - Assets - Cambridge University Press
  16. Martin Gardner and Scientific American
  17. https://www.jstor.org/stable/2310544
  18. Tetraflexagon -- from Wolfram MathWorld
  19. Tetraflexagon - Mathematische Basteleien
  20. [PDF] The Second Book of Mathematical Puzzles and Diversions
  21. (PDF) Constructing and Counting Hexaflexagons - ResearchGate
  22. Octaflexagon
  23. Dodecaflexagon
  24. Dodecaflexagons - The Flexagon Portal
  25. Decaflexagon
  26. [PDF] Flex Theory - Loki3 Home Page
  27. Flexagons Inside Out - Cambridge University Press & Assessment
  28. Flexagon theory
  29. Flexagons - Mathematische Basteleien
  30. Cycles and Traverses - Loki3 Home Page
  31. Explorable Flexagons
  32. Martin Gardner and Flexagons - YouTube
  33. Hexaflexagons - Vimeo
  34. The Forgotten Flexagon - Numberphile - YouTube
  35. Flexagon Flexomania - Base Camp Math
  36. [PDF] ASF03 - Flexagon: Designing and Making a Geometric Toy
  37. Flexagon Playground
  38. Hexaflexagon REDO v5 by ThreeHamsWillKillHim - Thingiverse
  39. Hexa-hexa-flexagon by jetpad MakerWorld: Download Free 3D ...
  40. Flexagon Puzzles - Loki3 Home Page
  41. How to Fold an Origami Flexagon - Instructables
  42. 3D Flexagons - Bronson Brookside Library - Facebook
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