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
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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
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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.
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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
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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.
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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.
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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
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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
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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
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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.
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