THE MATHEMATICS OF MARTIN GARDNER

HexaflexagonsMartin Gardner

A prolific writer on mathematics and science, MartinGardner (1914–2010), is best known for the immenselyinfluential series of “Mathematical Games” columns that hewrote for Scientific American from December of 1956through 1981. This special issue of The CollegeMathematics Journal is dedicated to the mathematics thatMartin Gardner promoted and nourished for over 54 years.Through his writings, Gardner substantially broadened thefield of mathematics by making recreational mathematicsaccessible, popular, and enjoyable for a very wide audienceof professional and amateur mathematicians. The issuebegins with his very first “Mathematical Games” column.

Flexagons are paper polygons, folded from straight or crooked strips of paper, whichhave the fascinating property of changing their faces when they are “flexed.” Had itnot been for the trivial circumstance that British and American notebook paper are notthe same size, flexagons might still be undiscovered, and a number of top-flight math-ematicians would have been denied the pleasure of analyzing their curious properties.

It all began in the fall of 1939. Arthur H. Stone, a 23-year-old graduate studentfrom England, in residence at Princeton University on a mathematics fellowship, hadjust trimmed an inch from his American notebook sheets to make them fit his Englishbinder. For amusement he began to fold the trimmed-off strips of paper in variousways, and one of the figures he made turned out to be particularly intriguing. He hadfolded the strip diagonally at three places and joined the end so that it made a hexagon(see Figure 1). When he pinched two adjacent triangles together and pushed the oppo-site corner of the hexagon toward the center, the hexagon would open out again, like abudding flower, and show a completely new face. If, for instance, the top and bottomfaces of the original hexagon were painted different colors, the new face would comeup blank and one of the colored faces would disappear!

This structure, the first flexagon to be discovered, has three faces. Stone did somethinking about it overnight, and on the following day confirmed his belief (arrived atby pure cerebration) that a more complicated hexagonal model could be folded withsix faces instead of only three. At this point Stone found the structure so interestingthat he showed his paper models to friends in the graduate school. Soon “flexagons”were appearing in profusion at the lunch and dinner tables. A “Flexagon Committee”was organized to probe further into the mysteries of flexigation. The other membersbesides Stone were Bryant Tuckerman, a graduate student of mathematics; Richard

http://dx.doi.org/10.4169/college.math.j.43.1.002MSC: 00A08, 97A20

2 „ THE MATHEMATICAL ASSOCIATION OF AMERICA

D

C

B

A

Figure 1. Trihexaflexagon construction. A strip of 10 triangles (A) is folded backward alongab and turned over (B), then folded again along cd and the next to last triangle placed on topof the first (C). The last triangle is now folded backward and glued to the other side of the firsttriangle (D).

Feynman, a graduate student in physics; and John W. Tukey, a young mathematicsinstructor.

The models were named hexaflexagons—“hexa” for their hexagonal form, and“flexagon” for their ability to flex. Stone’s first model is a trihexaflexagon (“tri” forthe three different faces that can be brought into view); his elegant second structure isa hexahexaflexagon (for its six faces).

To make a hexahexaflexagon you start with a strip of paper (the tape used in addingmachines serves admirably) which is divided into 19 equilateral triangles (see Figure2). You number the triangles on one side of the strip 1, 2 and 3, leaving the 19th

Figure 2. Hexahexaflexagon construction. A strip of 19 triangles (A) is folded in a spiral toform a strip of 10 triangles (B), which is then assembled as in (C) and (D) to complete theconstruction.

VOL. 43, NO. 1, JANUARY 2012 THE COLLEGE MATHEMATICS JOURNAL 3

Figure 3. The pinch flex.

triangle blank. On the opposite side the triangles are numbered 4, 5 and 6, accordingto the scheme shown. Now you fold the strip so that the same underside numbers faceeach other—4 on 4, 5 on 5, 6 on 6 and so on. The resulting folded strip (Figure 2B)is then folded on the lines ab and cd (Figure 2C), forming the hexagon (Figure 2D);finally the blank triangle is turned under and pasted to the corresponding blank triangleon the other side of the strip. All this is easier to carry out with a marked strip of paperthan it is to describe.

If you have made the folds properly, the triangles on one visible face of the hexagonare all numbered 1, and on the other face are all numbered 2. Your hexahexaflexagonis now ready for flexing. You pinch two adjacent triangles together (see Figure 3),bending the paper along the line between them, and push in the opposite corner; thefigure may then open up to face 3 or 5. By random flexing you should be able to findthe other faces without much difficulty. Faces 4, 5 and 6 are a bit harder to uncoverthan 1, 2 and 3. At times you may find yourself trapped in an annoying cycle that keepsreturning the same three faces over and over again.

Tuckerman quickly discovered that the simplest way to bring out all the faces of anyflexagon was to keep flexing it at the same corner until it refused to open, then shift toan adjacent corner. This procedure, known as the “Tuckerman traverse,” will bring upthe six faces of a hexahexa in a cycle of 12 flexes, but 1, 2 and 3 turn up three timesas often as 4, 5 and 6. A convenient way to diagram a Tuckerman traverse is shown inFigure 4. When the model is turned over, a Tuckerman traverse runs the same cycle inreverse order.

Figure 4. Tuckerman traverse of a hexahexaflexagon.

4 „ THE MATHEMATICAL ASSOCIATION OF AMERICA

By lengthening the chain of triangles, the committee discovered, one can makeflexagons with 9, 12, 15 or more faces; Tuckerman managed to make a workable modelwith 48! He also found that with a strip of paper cut in a zigzag pattern (i.e., a stripwith sawtooth rather than straight edges) it was possible to produce a tetrahexaflexagon(four faces) or a pentahexaflexagon. There are three different hexahexaflexagons—onefolded from a straight strip, one from a chain bent into a hexagon and one from a formthat somewhat resembles a three-leaf clover. The decahexaflexagon (10 faces) has 82different variations, all folded from weirdly bent strips. Flexagons can be formed withany desired number of faces, but beyond 10 the number of different species for eachincreases at an alarming rate. All even-numbered flexagons, by the way, are made ofstrips with two distinct sides, but those with an odd number of faces have only a singleside, like a Mobius surface.

Pearl Harbor called a halt to the committee’s flexigation program, and war worksoon scattered the four charter members to the winds. Stone became a lecturer in math-ematics at the University of Manchester in England. Feynman was a famous theoreticalphysicist at the California Institute of Technology. Tukey, as a professor of mathemat-ics at Princeton, made brilliant contributions to topology and to statistical theory whichbrought him world-wide recognition. Tuckerman became a mathematician at IBM’sresearch center in Yorktown Heights, New York.

Summary. This is a reprint of Martin Gardner’s very first “Mathematical Games” column(which appeared December 1956 in Scientific American). Here Gardner introduces flexagons(paper polygons folded from straight or crooked strips of paper which have the fascinatingproperty of changing their faces when they are “flexed”), discusses their history prior to WorldWar II, and explains how to make several hexaflexagons. The reprint, part of a special issueof The College Mathematics Journal devoted to “Martin Gardner’s Mathematics,” is followedby two contemporary papers that describe some of the ways that the study of flexagons hasdeveloped since 1956. Susan Goldstine and Ethan Berkhove provided vital assistance editingall three flexagon papers.

VOL. 43, NO. 1, JANUARY 2012 THE COLLEGE MATHEMATICS JOURNAL 5