The Faces of the Tri-Hexaflexagon
Transcription
The Faces of the Tri-Hexaflexagon
The Faces of the Tri-Hexaflexagon Author(s): Peter Hilton, Jean Pedersen and Hans Walser Source: Mathematics Magazine, Vol. 70, No. 4 (Oct., 1997), pp. 243-251 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690860 . Accessed: 18/02/2014 15:54 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine. http://www.jstor.org This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions ARTICLES The Faces oftheTri-Hexaflexagon PETER HILTON of New York StateUniversity NY 13902-6000 Binghamton, and University ofCentralFlorida Orlando, FL 3281 6 JEAN PEDERSEN Santa Clara University Santa Clara, CA 95053 HANS WALSER ETH Zurich CH-0892 Zurich Switzerland and Santa Clara University Santa Clara,CA 95053 Introduction Hexaflexagonswere inventedat Princetonin 1939 by ArthurH. Stone, then a graduate student,now ProfessorEmeritus of Mathematicsat the Universityof account(see [1]) of Stone'sworkand Rochester.MartinGardnergivesan interesting his collaboration withBryantTuckerman,thena graduatestudentand now a retired fromIBM (YorktownHeights,NY), the late RichardP. researchmathematician Feynman,thena graduatestudentin physicsand latera Nobel Laureate,and JohnW. Tukey,then a young mathematicsinstructorand now an EmeritusProfessorat to remarkthatthediagramsFeynmandevisedforanalyzing Princeton.It is interesting in modern were forerunners ofthe famousFeynmanDiagramns 6-facedhexaflexagons atomicphysics.A descriptionof how to constructa 3-facedhexaflexagon may be is given foundin anyofthe references [1] through[4]. Further,a detaileddescription in [3, pp. 63-74] of how to construct with3n faceswithoutthe use of hexaflexagons or compass. straightedge we will considerin thisarticleis the tri-hexaflexagon,' The particularhexaflexagon so named because it has 3 faces; thatis, in anygivenstateof the flexagon,one face (consistingof 6 equilateraltriangles)willbe up, one facewillbe down,and one face of the faceswillvaryfromstateto state,the willbe hidden.Althoughthe orientation same 6 triangleswillalwaysappeartogetheron a face. We will show in thisarticlehow,by drawinga humanvisageon each face of the color for each face, we can keep trackof all the flexagon,and using a different possiblepositionsofthe flexagonas it lies in a plane.We are therebyable to discover We mayreferto the tri-hexaflexagon as simply"the flexagon"ifno confusionwould result. 243 This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 244 MATHEMATICS MAGAZINE thatthe set of motionsof this flexagonwhichbringit into coincidencewithitself the dihedralgroupD18. constitutes 1. How to BuildtheTri-hexaflexagon The tri-hexaflexagon is constructedfroma stripof paper containing10 equilateral triangles2as shownin FIGURE 1. In orderthatthe finalmodelwill flexeasilythe fold lines between the trianglesshould be creased firmlyin both directions.Now we / /J\ \ / / \J / / \ \ / \/ / \ / \ FIGURE 1 Thestrip decoratethe stripas shownin FIGURE 2, wherewe make the bottomsurfaceof the the entirepatternpiece overa horizontalaxisas indicatedby stripvisibleby flipping the figure(wherethe verticesA, B, C, D shouldcorrespondwith A', B', C', D', Top surfaceofthestrip B C_ Bottomsurfaceofthestrip B ~~~~~~~~~~~~ A' A A\ Color Color2 Color3 FIGURE 2 Decoratingthestrip afteryouhaveflippedthepiece over).Caution:Be carefulhere!Flipping respectively, the patternpiece over a verticalaxis, and then decoratingit as shown does not producethe desiredflexagon. 2Notice that,fromthe pointof view of decoratingthispiece, we have availablea totalof 20 triangles (because the stripof paper has two surfaces,the top surfaceand the bottomsurface).When two of the trianglesare glued to each otherthereremain18 triangleswithwhichto formthe 3 faces. This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 245 VOL. 70, NO. 4, OCTOBER 1997 Now we suggestthatyouviewtheconstruction oftheflexagonas a puzzle. Here are some hintsforconstructing the flexagonwithsmiling(and frowning) faces. (1) The firsttriangleon the upperportionof the stripis ultimately glued to the last triangleon the bottomportion(and it doesn't matterwhich one is on top of which).We suggestthatyou attachthesetriangleswitha paper clip at first,and save the actualgluinguntilyou are certainaboutthe correctness of the construction. (2) The completedflexagonshouldshowthevisageof a smilingface,entirely of color 1, as youlayit downas shownin FIGURE3(a). Andwhenyoufliptheflexagonover, about a horizontalaxis,it shouldshow the visageof a frowning face,entirelyof color3, orientedas displayedin FIGURE3(b). (b) (a) FIGURE 3 Thetri-hexaflexagon (3) The stripthatcreatedthehexagoncontainsthreehalf-twists; thus,likethe Mo5bius it has one surface this band, only (or side). Geometrically means therewill be three slitson any face of thisflexagon,symmetrically located at 1200 intervals about its center. These slits are created by edges of the stripthat go from alternateverticesof the hexagonto its centeras shownin bothpartsof FIGURE 3. Fromthe lasthintabovewe knowthatthe flexagonnowhas onlyone surface.After at manipulating you become proficient withyour yourflexagon you maywishto verify own modelthatthe repetitive patternofthreemouths,threerighteyes,and threeleft occursas shownin FIGURE4. eyes,in the colors1, 2, 3, respectively, _n~~~~~~O These triangles are glued together FIGURE 4 Theentiresurface ofthestrip This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 246 MATHEMATICS MAGAZINE 2. The HappyGroup Firstwe willalwaysneed to startwiththe flexagonin a standardinitialposition,that is, withthe smilingface of color 1 up and orientedpreciselyas shownin FIGURE 5(a). E F9 I ~~~~~flexiing up (b) (a) FIGURE 5 Themotion f, flexing up Now we assume n 2 0 and definethe following motions: motion1, whichmeanswe retaintheinitialposition, (the identity f= themotion from offlexing theinitial up,starting position, offlexing fromtheinitialposition. fn = the motion up n times,starting More precisely, the motionf consistsof lifting theverticesof the hexagonlabelled E, F, and G (in FIGURE 5(a)) above the flexagonuntiltheymeet,when the flexagon will come apart at the bottomand fall into the shape of a new hexagonwiththe verticesE, F, and G at itscenter.If thisis done correctly (it is important notto rotate the flexagonin eitherdirection), you willsee the upside-downsmilingface of color3 shownin FIGURE5(b). Noticethatthe slitsin the flexagonhave revolved' of a turn. Thus, whenyou flexup the second timeyou will have to bringthe verticesmarked withasterisks(*) togetherabove the flexagon.A simplewayto rememberwhatto do is that,in each case, thevertexat the foreheadof the humanvisagegetsliftedto the center(and it disappearsas the motionis completed). a motionwithits effect We now followthe usual,obviousprocedureof identifying motion. on the initialposition.Whenwe do thiswe see thatf'8 = 1, the identity Once you have masteredthe motionsfn, you mayverifythe sequence of motions which produce the Happy Group shown in FIGURE 6; here we have adopted the identification indicatedabove. Since f18 is the identity, we see thatthe Happy Group is the cyclicgroup C18, generated byf. Nextwe defineflexingdown.To describethismotionf, we begin,as before,with the flexagonin the standardinitialpositionshown3in FIGURE 7(a). Then f meansthat we pushtheverticesofthehexagonlabelled H, J, and K downwardsuntiltheymeet; at thatstagethe flexagonwillcome apartat the top and fallintothe shape of a new hexagonwiththeverticesH, J, and K at itscenter,but underneaththehexagon(this is indicatedby puttingH, J, and K in parenthesesin FIGURE 7(b)). If thisis done we willobtainthesmilingpiratefaceofcolor2 as shownin FIGURE7(b). Just correctly, 3Thisis, of course,the same initialpositionas thatin FIGURE 5(a), but the labelinghas changed. This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 9 247 VOL. 70, NO. 4, OCTOBER 1997 1:. .f3 f4 ,f6: f:f87 f'2: f' f f 0 3* 0 f=l016: * f5:* f14 f7 FIGURE 6 The HappyGroup as withthe up-motions, it is important notto rotatethe flexagonin eitherdirectionas we flexit. To obtainfn, we simplyrepeatthe processof flexingdown n times(notice thatwhenwe flexdownthe secondtimeit is theverticeslabelledwiththe asterisk(*) thatcome togetherbeneaththe flexagon).It is interesting that,in flexingdown,the vertexat the foreheadof the humanvisagemovesup (as whenflexingup), but in this case the flexagonvisiblysplitsacross the foreheadbeforeit fallsflat,revealingthe pirate. This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions MATHEMATICS MAGAZINE 248 K @ I flexing down KI H (b) (a) FIGURE 7 Themotion f,flexing down position,you mayverify thatf Beginningwiththe flexagonin the standardstarting yieldsthesamefaceas f17ofFIGURE 6. Thusf =f'7 =f-'. Thismeansthat,ifyou startwiththe positionindicatedon the rightof FIGURE 7 and flexdown,you get the initialposition.In otherwords,flexingup is the inverseof flexingdown (and vice versa),as you mightexpect. all of the steps If you'reenjoyingthisyou maycheckyourflexingskillby reversing of the Happy Groupin FIGURE 6. 3. The Entire Group We realizethatthe fullgroupforthisflexagonmustbe largerthan C18 because no faceseverappearedunderthe motionsfn. Cheerfulas thissituationis, it is frowning in thisworldthisflexagonhas good (happy)and plainlynotcomplete.Like everything need to have a bad (unhappy)features.In orderto get the entiregroupwe certainly motionthat makes the unhappyfaces visible.Tro achieve thiswe introducea new motion, t = turnover(so therotationis abouta horizontalaxis). Thus, if we beginwiththe flexagonin the standardinitialpositionand performthe face of color3 (see FIGURE 8). motiont we will see a frowning FIGURE 8 Themotion over t,turning Obviouslyt is an involution,that is, t2 = 1. FIGURE 9 shows that the motionft (meaningfirstdo t, thendo f ) is not the same as tf(meaningfirstdo f, thendo t). thatthe flexagonshouldbe in the standardinitialposition, Check this(remembering in bothcases,whenyou start).Thus we see thatour new motiont does not comtnute This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 249 VOL. 70, NO. 4, OCTOBER 1997 FIGURE 9 fti=#tf his patchcovershis lefteye, withf. We also noticethatwhenthe piratefrowns one!4 insteadoftheright f t: f3: tf3 :Q O* P rf9 ti.e tf9 tf f4 O * fllo f12:0tfl2: f13:*tf13* fls:0tf15: fl6 f2*tf2 tf4 f5tf5 tf7 o :*ttf89 tfl'1 * f14*tf14 *t 46* fIl*7tf170 FIGURE 10 The entiregroup forfn and tf n. Noticethatthe first, FIGURE10 showsall the possibilities third,and fifth columnsare just the smilingfacesfromFIGURE 6. This observationmaygiveyou an idea of an easywayto confirmthatthe visagesin FIGURE 10 are correct. itis clearthatthis thisparticular we coulddo without flexagon, 4Although anyeyepatchesin analyzing we trackofthefaceson morecomplicated feature flexagons-and mayprovidea betterwayofkeeping addition tothisgroup(visually andsocially!). thepiratemadean interesting thought This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions 250 MATHEMATICS MAGAZINE We have alreadyseen that tfnif't. However,since flexingup, as viewed from abovetheflexagon, is thesameas flexing down,as viewedfrombelowtheflexagon, we have, fnt=tfn. Thus we see thatthe groupgeneratedbyf and t has 36 elementsand is therefore ofourflexagon. thefullgroupofmotions Sincethegenerators thedefining f, t satisfy set of relationsfl8 = 1, t2 = 1, ft= t-', the groupis the dihedralgroup D18, the groupofsymmetries oftheregular18-gon(shownin FIGURE 11). The figure on the frontcoverof thisissue showsthe effectsof the groupelements,wherethe arrows denotethef actionandthedouble-headed single-headed arrows denotethet action. 5 4 o36 9Ith o r 5~~~~~~~~ 13 14 FIGURE 11 Theregular 18-gon 4. A NormalSubgroup If we are onlyinterested in thedifferent on thefacesof our flexagon, expressions without to orientation, we haveonly6 cases(as seenin FIGURE 12), instead respect of36. FIGURE 10 motivates thefollowing argument. We obtainthe groupof motions of theunoriented facesby addingthe relation 3 = I to our The of f groupD18. resulting quotient group DI, bythenormal subgroup itself generated byf3 is thengenerated by F and t, subjectto F3 = 1, t2 = 1, _ _ _ ~~~~~~~~~e~ ,~~~~~~~ _~~~~~~~~FGR_12 ~xrsin of th flxao This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions VOL. 70, NO. 4, OCTOBER 1997 251 Ft = tF-. Here, F is, of course,the image in the quotientgroup of f; and the quotientgroupis just the symmetric groupS3. 5. A Challengeto theReader In [4] the tri-hexaflexagon was discussedand the group S3 was obtainedby usinga flexagonwhereeach of the 3 facessimplyhad different colors.In [2] the group Dg was obtainedbya systematic labelingoftheverticesofthe6 triangleson each ofthe3 facesofthetri-hexaflexagon. However,in orderto obtainthe entiregroupD18,it was betweenthe different orientanecessaryto introducea finermethodof distinguishing tionsof the faces;distinguishing betweensmilingand frowning visagesdid the trick. The obviousnextquestionto exploreis whetheror notthis,or some refinement of it, will help to identifythe mathematicalstructureof the hexa-hexa-flexagon (with 6 faces). If we had a quick,or easy,answerto thisquestionwe wouldn'tbe stopping here! REFERENCES 1. MartinGardner,The ScientificAmericanBook of MathematicalPuzzles and Diversions,Simon and Schuster,New York,NY, 1959. 2. MichaelGilpin,Symmetries thisMAGAZINE 49, No. 4 (1976), 189-192. of the trihexaflexagon, 3. PeterHiltonand JeanPedersen,Build Your Own Polyhedra,Addison-Wesley, Menlo Park,CA, 1994. 4. JeanPedersen,Sneakingup on a group,Two-yearCollegeMathematics Journal3 (1972), 9-12. This content downloaded from 98.150.67.43 on Tue, 18 Feb 2014 15:54:07 PM All use subject to JSTOR Terms and Conditions