Exercise 1.2.1. Findangles x and y asshowninFigure 1.2.11. Thelines m and n areparallel.
60°
45°
Figure 1.2.11
x
y
m
n
Exercise 1.2.2. Findangles α, β and γ asshowninFigure 1.2.12. Thelines p and q areparallel.
115°
80°
p
α
β
γ
Figure 1.2.12
q
Exercise 1.2.3. Westatedabovethatifwearegivenaline m andapoint A, wecanconstructa
linethrough A thatisperpendicularto m. Showthatthereisonlyonesuchline. Thedemonstration
hastwosubcases, dependinguponwhether A ison m ornot.
Exercise 2.1.1. Supposethataparallelogramhasatleastone 90◦ angle. Showthatallfourangles
mustbe 90◦ , andhencetheparallelogramisarectangle. (Useonlyresultsfromthissectionand
previoussections.)
Exercise 2.1.2. Supposethatahexagonhasthepropertythateachpairofoppositeedgesisparallel.
(1) Showthatoppositeanglesinthishexagonareequal.
(2) Mustitbethecasethatoppositeedgeshaveequallengthsinthishexagon? Ifyes, showwhy.
Ifnot, giveanexample.
Exercise 2.2.1. UseProposition 2.2.1(1)todemonstrateProposition 1.2.4.
Exercise 2.2.2. WeknowfromProposition 2.2.1thattheanglesinatrianglearerelatedtoeach
other; in particular, we could not take three arbitrary angles, and expect there to be a triangle
withthosethreeangles. Thisexerciseconcernsrelationshipsbetweenthelengthsoftheedgesofa
triangle.
(1) Isthereatrianglewithedgesoflengths 2, 3 and 4? Explainyouranswer.
(2) Isthereatrianglewithedgesoflengths 2, 3 and 6? Explainyouranswer.
(3) Whatcanyousayabouttherelationshipbetweenthelengthsoftheedgesofatriangle. In
particular, trytocomeupwithcriteriaonthreenumbers a, b and c thatwouldguarantee
thatthereexistsatrianglewithedgesoflength a, b and c. Explainyouranswer.
Exercise 2.2.3. Findexamplestoshowthatthereisno“Angle-Side-SideTheorem.” Thatis, find
twotriangles △ABC and △A ′ B ′ C ′ suchthat ∡A = ∡A ′ , that |A B| = |A ′ B ′ |, andthat |B C| =
|B ′ C ′ |, andyetthetwotrianglesarenotcongruent. Exactmeasurementsofsuchtrianglesarenot
needed; asketchofthetriangles, togetherwithadescriptionofwhatyoumean, wouldsuffice.
Exercise 2.2.4. Istherean“Angle-Angle-SideTheorem”? Eitherdemonstratewhysuchatheorem
istrue, orgiveanexampletoshowthatitisnottrue.
Exercise 2.2.5. Supposethatinaquadrilateral, bothpairsofoppositeedgeshaveequallengths.
Showthatthequadrilateralisaparallelogram. (Becarefulwithnotconfusingatheoremandits
converse—thisfactcannotbeprovedbysimplyquotingProposition 2.2.5(1).)
Exercise 2.2.6. Supposethatinaquadrilateral, apairofoppositeedgesareparallelandhaveequal
lengths. Showthatthequadrilateralisaparallelogram.
Exercise 2.2.7. Showthatthetwodiagonalsinaparallelogrambisecteachother.
Exercise 2.2.8. Supposewearegivenatriangle △ABC. Showthatif ∡B = ∡C, then |A B| =
|A C| (sothatthetriangleisisosceles).
Exercise 2.2.9. UsecongruenttrianglestodemonstrateProposition 1.3.1.
Exercise 2.2.10. Supposewehaveacircle, andsupposethat A, B and C arepointsonthecirclesuch
thatthelinesegment A B isadiameterofthecircle. Formthetriangle △ABC. SeeFigure 2.2.8.
Showthattheangleat C is 90◦ .
C
A
B
Figure 2.2.8
Exercise 2.2.11. A treecastsashadowthatis 20 ft.long. Atthesametimeofday, a 3 ft.stickcasts
a 5 ft.shadow. Howtallisthetree?
Exercise 2.3.1. Showthataquadrilateralhasatmostoneinterioranglethatisgreaterthan 180◦ .
Exercise 2.3.2. Recall, statedinSection 2.1, thatarectangle isdefinedtobeaquadrilateralin
whichallfouranglesareequal.
(1) Showthatallfouranglesinarectangleare 90◦ .
(2) Showthateveryrectangleisaparallelogram.
(3) Showthatoppositeedgesinarectanglehaveequallengths.
Exercise 2.3.3. Supposethataquadrilateralhastwopairsofequaladjacentangles. Showthatthe
quadrilateralisatrapezoid.
Exercise 2.3.4. Showthatanyparallelogramisconvex.
Exercise 2.3.5. Supposethataquadrilateralhastwooppositeedgesthathaveequallengths, and
thatthesetwoedgesarebothperpendiculartooneoftheedgesthatisinbetweenthem. Thegoal
ofthisexerciseis toshow thatthequadrilateralisa rectangle. Weoutlinetwodemonstrations,
oneusingcongruenttriangles(insteps(1)–(3)below), andtheotherusingsimilartriangles(insteps
(4)–(8)below); thereaderisaskedtofillinthedetailsofeachstep.
Supposethat ABCD isaquadrilateral. Supposethat |A D| = |B C|, andthatboth A D and B C
areperpendicularto A B. SeeinFigure 2.3.10. Wenowproceedasfollows.
(1) Showthattriangles △ABC and △ABD arecongruent. Deducethat |A C| = |B D|.
(2) Showthattriangles △ACD and △BCD arecongruent. Deducethat ∡D = ∡C.
(3) ByProposition 2.3.3(1)weknowthatthesumoftheanglein ABCD is 360◦ . Deducethat
∡C = 90◦ and ∡D = 90◦ . Itfollowsthatallfouranglesinthequadrilateralareequal, and
hencethequadrilateralisarectangle.
(4) Wenowgiveanotherdemonstrationofthefactthatthequadrilateralisarectangle. Asa
firststep, wewanttoshowthatthequadrilateralisaparallelogram. ItfollowsfromProposition 1.2.5that A D and B C areparallel. Nowsupposethat A B and C D arenotparallel.
Thenextendthemuntiltheymeet, sayinpoint P . Wewillarriveatalogicalcontradiction.
(5) Showthatthetriangles △ADP and △BCP aresimilar.
(6) Deducethat
|A D|
|A P|
=
.
|B C|
|B P|
(7) Usethefactthat |A P| > |B P| todeducethat |A D| > |B C|. Explainwhythisisalogical
impossibility, giventhehypothesesofthisexercise. Deducethat A B isparallelto C D, and
hencethequadrilateralisaparallelogram.
(8) NowuseExercise 2.1.1toshowthatthequadrilateralisarectangle.
A
B
D
C
Figure 2.3.10
Exercise 2.4.1. DemonstrateProposition 2.4.2(2).
Exercise 2.4.2. Supposethattriangles △ABC and △A ′ B ′ C ′ aresimilar. Let Q and Q ′ respectivelydenotetheareasof △ABC and △A ′ B ′ C ′ . Showthat
2
Q
|A B|
=
2.
′
Q
|A ′ B ′ |
Exercise 2.4.3. FindtheareasofthepolygonsshowninFigure 2.4.8.
2
2
2
1
3
2
2
2
2
1
(i)
2
(ii)
Figure 2.4.8
(iii)
Exercise 2.4.4. Showthatiftworectangleshavethesameareaandthesameperimeter, thenthey
havethesamedimensions. (Thisoneusessomealgebra.)
Exercise 2.4.5.
(1) Findtwopolygonsallofwhoseedgeshaveintegerlengths, whichhavethesameareasand
sameperimeters, butthatarenotcongruent.
(2) IfthetwoconvexpolygonsyoufoundinPart (1)werenotconvex, findtwoconvexpolygons
allofwhoseedgeshaveintegerlengths, whichhavethesameareasandsameperimeters,
butthatarenotcongruent.
Exercise 2.4.6. A kite isaquadrilateralthathastwopairsofadjacentedgeswithequallengths.
Wecallthediagonalthathasonepairofadjacentedgeswithequallengthsononesideandthe
otherpairontheothersidethe crossdiagonal; theotherdiagonaliscalledthe maindiagonal. (The
nomenclatureinthisexerciseisnotstandardized.) Forthesakeofthisexercise, assumethatallkites
underdiscussionareconvex(thatobviatestheneedconsideringdifferentsubcases), thoughthere
arealsonon-convexkites.
(1) Showthatthemaindiagonalinakitebreaksthekiteintotwocongruenttriangles.
(2) Showthatthemaindiagonalinakitebisectseachoftheanglesatitsendpoints.
(3) Showthatthemaindiagonalinakitebisectsthecrossdiagonal.
(4) Showthatthecrossdiagonalinakiteisperpendiculartothemaindiagonal.
(5) Showthattheareaofakiteistheonehalftheproductofthelengthsofthediagonals.
Exercise 2.4.7. Findtheareaforeachofthefollowingregularpolygons.
(1) Anequilateraltrianglewithedgesoflength 1.
(2) A regularhexagonwithedgesoflength 1.
(3) A regularoctagonwithedgesoflength 1. (Hint: Donottrytosolvethisproblembydividingtheoctagonintoeightcongruenttriangleswithacommonvertexinthecenterofthe
octagon—thatmethodisquitetricky.)
Exercise 2.4.8.
(1) Supposewehaveacircle, andsupposethat A and B arepointsonthecirclethatarenot
diametricallyoppositeeachother. Supposefurtherthat C isanotherpointonthecirclethat
isbetween A and B. Considertheareaofthetriangle △ABC. Explainwhyofallpossible
choicesofpoints C, theonewhere △ABC hasthemaximalareaiswhere C ismidway
between A and B.
(2) Giveaninformalexplanationthat, amongallpolygonswith n vertices, andwithitsvertices
allonagivencircle, theregular n-gonwillhavethemaximalarea.
Exercise 2.5.1. Thetwosidesofarighttriangleare 6 and 11 inchesrespectively. Howlongisthe
hypotenuse?
Exercise 2.5.2. A 40 ft.wireisstretchedfromthetopofapoletotheground. Thewirereachesthe
ground 25 ft.fromthebaseofthepole. Howhighisthepole?
Exercise 2.5.3. ProvethePythagoreanTheoremusingFigure 2.5.3insteadofFigure 2.5.1.
c
a
b
c
b
a
a
b
a
c
Figure 2.5.3
b
c
Exercise 3.1.1. Foreachofthefollowingquestions, iftheanswerisyes, giveanexample, andifthe
answerisno, explainwhynot. (Toexplainwhyapolyhedroncannotbeintwodifferentcategories,
itdoesnotsufficesimplytostatethatthetwocategoriesareconstructeddifferently, becausesometimestwodifferentconstructionscanyieldthesameresult, forexampletheoctahedron, whichis
bothabipyramidandanantiprism.)
(1) Canapolyhedronbebothabipyramidandaprism?
(2) Canapolyhedronbebothaprismandanantiprism?
(3) Canapolyhedronbebothapyramidandeitheraprismoranantiprism?
(4) Canapolyhedronbebothapyramidandabipyramid? (Ifyouthinkthattheanswerisno, it
isnotsufficientsimplytosaythatapyramidhasoneconepoint, andabipyramidhastwo
conepoints. Perhapsifyoulookatacertainpolyhedroninonewaythereisoneconepoint,
andviewedanotherwaytherearetwoconepoints; perhapsnot.)
Exercise 3.1.2. Findallpyramidsthathaveallregularfaces.
Exercise 3.1.3.
(1) Whatisthedualtoabipyramidoveran n-gon?
(2) Whatisthedualtoapyramidoveran n-gon?
Exercise 3.1.4. Suppose P isaconvexpolyhedron. Whatistherelationbetween P andthedual
ofthedualof P ?
Exercise 3.2.1.
(1) Whichoftheregularpolyhedraarepyramids?
(2) Whichoftheregularpolyhedraarebipyramids?
(3) Whichoftheregularpolyhedraareprisms?
(4) Whichoftheregularpolyhedraareantiprisms?
Exercise 3.2.2. Findallconvexpolyhedrathatarebothbipyramidsandantiprisms.
Exercise 3.2.3. Supposethatyouhaveacubemadeoutofclay; supposefurtherthattheclayisred,
buttheoutsideofthecubeispaintedblue. Youthenslicethecubeinastraightlinewithaknife,
causingthecubetobreakintotwopieces. Eachpiecehasanexposedredpolygon, wherethecube
wassliced. Dependinguponhowyouslicethecube, youmightgetdifferentexposedpolygons; all
theexposedpolygonswillbeconvex. Forexample, ifyousliceparalleltooneofthefacesofthe
cube, yourexposedpolygonwillbeasquare; ifyousliceoffacornerofthecuberightnexttoa
vertex, yourexposedpolygonwillbeatriangle. Whatareallthepossibleexposedpolygonsthat
couldbeobtainedbyslicingthecube?
Exercise 3.3.1. WhatcanbesaidaboutthefacesofthedualofanArchimedeansolid?
Exercise 3.3.2. Weknowthatthedualofeachregularpolyhedronisitselfaregularpolyhedron. Can
ithappenthatthedualofanon-regularsemi-regularpolyhedronisitselfasemi-regularpolyhedron?
Iftheanswerisyes, giveanexample, andiftheanswerisno, explainwhynot.
Exercise 3.4.1.
deltahedra.
Findallconvexface-regularpolyhedrathathaveidenticalfaces, otherthanthe
Exercise 3.4.2. FindatleasttwoJohnsonsolidsthatarenotpyramids, bipyramidsordeltahedra,
andaredifferentfromtheoneshowninFigure 3.4.2.
Figure 3.4.2
Exercise 3.4.3. A face-regularpolyhedroniscalled elementary ifitcannotbebrokenupintotwo
ormoreface-regularpolyhedrathatarejoinedalongacommonface. Forexample, theoctahedronisnotelementary, becauseitcanbebrokenupintotwopyramidswithsquarebases. Findat
leastoneothernon-elementaryface-regularpolyhedron, andatleastoneelementaryface-regular
polyhedron.
Exercise 3.4.4. Suppose we have two face-regular polyhedra, and one of the faces in the first
polyhedronisidenticaltooneofthefacesinthesecondpolyhedron. Wecanthengluethetwo
polyhedraalongtheiridenticalfaces, yieldingonelargerpolyhedra. Forexample, startingwitha
cubeandapyramidwithasquarebaseandequilateralsides, andgluingthetwoalongasquare
faceineach, resultsinthepolyhedronshowninFigure 3.4.2.
(1) Is the polyhedron that results from the gluing two face-regular polyhedra by this process
necessarilyface-regular? Explainyouranswer.
(2) Supposetheoriginaltwoface-regularpolyhedrawerebothconvex. Isthepolyhedronthat
results from the gluing necessarily convex? If the answer is yes, explain why, and if the
answerisno, giveanexampletoshowwhynot.
Figure 3.4.2
Exercise 3.4.5. Showthatthereareinfinitelymanynon-convexface-regularpolyhedra.
Exercise 3.5.1.
(1) Whatisthefacevectorofeachoftheregularpolyhedra?
(2) Whatisthefacevectorofeachofthesemi-regularpolyhedra?
Exercise 3.5.2. Findaconvexpolyhedronsuchthatneither E nor F isdivisibleby 3.
Exercise 3.5.3.
(1) Whatisthefacevectorofapyramidoveran n-gon?
(2) Whatisthefacevectorofabipyramidoveran n-gon?
(3) Whatisthefacevectorofaprismoveran n-gon?
(Note: Youranswerforeachpartofthisexercisewillinvolve“n.”)
Exercise 3.5.4. Findabipyramidthathasthesamefacevectorastheicosahedron.
Exercise 3.5.5. Supposethataconvexpolyhedron P hasfacevector (V, E, F). Whatistheface
vectorofthedualof P ?
Exercise 3.5.6. Supposethat P isapolyhedron.
(1) Supposethatallthefacesof P are n-gons. Showthat nF = 2E.
(2) Supposethatallthefacesof P aretriangles. Showthat F isdivisibleby 2, and E isdivisible
by 3.
(3) Supposethateveryvertexof P iscontainedin q edges. Showthat qV = 2E.
Exercise 3.5.7. ThisexerciseusesExercise 3.5.6.
Supposethat P isaconvexpolyhedron.
(1) Supposethatallthefacesof P aretriangles. Findaformulaforeachof E and F intermsof
V.
(2) Supposethatallthefacesof P arequadrilaterals. Findaformulaforeachof E and F interms
of V .
(3) Supposethatallthefacesof P arepentagons. Findaformulaforeachof E and F intermsof
V.
Exercise 3.5.8. Supposethat P isaconvexpolyhedron, andthatallthefacesof P aretriangles.
Assumefurtherthat P hasatleastfivevertices. Istherealwaysabipyramidoversome n-gonthat
hasthesamefacevectoras P ? Ifthereis, find n intermsofthenumberofverticesof P .
Exercise 3.5.9. Supposethataconvexpolyhedron P isself-dual(thatis, thepolyhedronisit’sown
dual). Findaformulaforeachof E and F intermsof V .
Exercise 3.5.10. Findallpossibleconvexpolyhedra P thatareself-dual(asinExercise 3.5.9), and
allthefacesofwhicharetriangles.
Exercise 3.5.11. Supposethat P isaconvexpolyhedron. Showthat E ≤ 3F − 6.
Exercise 3.5.12. Supposethat P isaconvexpolyhedron. Showthat P mustcontainatleastone
facethathaseither 3, 4 or 5 edges. (Inotherwords, thisexerciseshowsthattherecannotbea
convexpolyhedronwithallfaceshaving 6 ormoreedges.)
Exercise 3.5.13. Foreachofthesetsofthreenumbersgivenbelow, statewhetherornotitisthe
facevectorofaconvexpolyhedron. Ifitisthefacevectorofaconvexpolyhedron, findsucha
polyhedron; ifnot, explainwhynot.
(1) (5, 10, 6).
(2) (12, 18, 8).
(3) (23, 33, 12).
(4) (10, 20, 12).
Exercise 3.6.1. Findtheangledefectateachoftheverticesofthefollowingpolyhedra.
(1) A cube.
(2) A regularicosahedron.
(3) ThepolyhedronshowninFigure 3.4.2(assumingthatallthetrianglesareequilateral).
Figure 3.4.2
Exercise 4.2.1. Drawtheeffectontheletter R showninFigure 4.2.4astheresultofrotatingthe
planeby 60◦ clockwiseabouteachofthepointsshown. (Therewillbefouranswers, oneforeach
point.)
A
R
D
B
C
Figure 4.2.4
Exercise 4.2.2. Drawtheeffectontheletter R showninFigure 4.2.7astheresultofreflectingthe
planeineachofthelinesshown. (Therewillbethreeanswers, oneforeachline.)
n
R
m
p
Figure 4.2.7
Exercise 4.2.3. Suppose m isalineintheplane. Describeallfixedlines ofthereflection Mm .
Exercise 4.2.4. Supposethat A and B aredistinctpointsintheplane. Let m betheperpendicular
bisectorof A B.
(1) Showthat Mm takes A to B andtakes B to A, andthat Mm istheonlyreflectionofthe
planetodoso.
(2) Showthat Mm fixesanypointthatisequidistantto A and B.
Exercise 4.3.1. IneachofthethreepartsofFigure 4.3.8areshowninitialandterminalletters F,
obtainedbyusinganisometry. Foreachofthethreecases, statewhattypeofisometrywasused.
Moreover, iftheisometryisarotation, indicateitscenterofrotation; iftheisometryisatranslation,
indicatethetranslationbyanarrow; iftheisometryisareflection, indicatethelineofreflection.
terminal
initial
F
initial
F
(i)
(ii)
initial
F
F
F
terminal
terminal
F
(iii)
Figure 4.3.8
Exercise 4.4.1. ReferringtoFigure 4.4.1, computethefollowingcompositions; thatis, foreachof
thefollowingexpressions, findasingleisometrythatisequaltoit.
A
(1) RA
1/4 ◦ R1/3 .
(2) RA
1/4 ◦ Mn .
(3) Mn ◦ Mm .
(4) Ml ◦ RA
3/4 ◦ Mn .
A
(5) RA
1/4 ◦ Mm ◦ R1/2 .
n
m
A
l
k
Figure 4.4.1
Exercise 4.5.1. Drawtheeffectontheletter R showninFigure 4.5.3astheresultofglidereflecting
theplaneusingeachofthepairsoflineofreflectionandtranslationvectorshown.
u
n
R
v
m
p
Figure 4.5.3
w
Exercise 4.5.2. Supposethat G isaglidereflection. Whatkindofisometryis G ◦ G?
Exercise 4.5.3. Suppose m isalineintheplane, and v isavectorthatisnotparallelto m. Isit
alwaysthecasethat Mm ◦ Tv ̸= Tv ◦ Mm ? Explainyouranswer.
Exercise 4.5.4. IneachofthethreepartsofFigure 4.5.5areshowninitialandterminalletters F,
obtainedbyusinganisometry. Foreachofthethreecases, statewhattypeofisometrywasused.
Moreover, iftheisometryisarotation, indicateitscenterofrotation; iftheisometryisatranslation,
indicatethetranslationbyanarrow; iftheisometryisareflection, indicatethelineofreflection; if
theisometryisaglidereflection, indicatethelineofreflectionusedintheglidereflection.
F
terminal
initial
F F
terminal
initial
F
(i)
(ii)
initial
F
terminal
F
(iii)
Figure 4.5.5
Exercise 4.6.1. InthedemonstrationofProposition 4.6.3(3), weassertedthattheanglefromthe
←→
line A Y totheline n isequaltotheanglefromtheline m totheline n; seeFigure 4.6.4. Use
congruenttrianglestodemonstratethisclaim.
Z
W
A
n
m
Figure 4.6.4
Exercise 4.6.2. Supposearotationisfollowedbyaglidereflection, whichisthenfollowedbya
reflection. Whattypeofisometry(orisometries)couldbeobtainedasaresultofthiscomposition?
Exercise 4.6.3. Describetheisometrythatresultsfromahalfturnfollowedbyanotherhalfturn?
(Theresultdependsuponwhetherthetwohalfturnshavethesamecenterofrotationornot.)
Exercise 4.6.4. A halfturnisfollowedbyareflection. Supposethatthecenterofrotationofthe
halfturnisonthelineofreflection. Showthattheresultingisometryisareflectioninthelinethrough
thecenterofrotationandperpendiculartotheoriginallineofreflection.
Exercise 4.6.5. A halfturnisfollowedbyareflection. Supposethatthecenterofrotationofthe
halfturnisnotonthelineofreflection. Showthattheresultingisometryisaglidereflection, which
haslineofglidereflectionthroughthecenterofrotation, andperpendiculartotheoriginallineof
reflection.
Exercise 4.6.6. A halfturnisfollowedbyaglidereflection. Supposethatthecenterofrotationof
thehalfturnisonthelineofglidereflection. Showthattheresultingisometryisareflection, which
haslineofreflectionperpendiculartothelineofglidereflection, andatadistancefromthecenter
ofrotationhalfthedistanceofthetranslationintheglidereflection.
Exercise 4.6.7. A reflectionisfollowedbyaglidereflection. Supposethatthelineofreflectionis
perpendiculartothelineofglidereflection. Showthattheresultingisometryisahalfturn, withthe
centerofrotationonthelineofglidereflection.
Exercise 5.1.1. ForeachoftheobjectsshowninFigure 5.1.4, listallsymmetries.
(i)
(ii)
(iv)
(v)
Figure 5.1.4
(iii)
Exercise 5.1.2. ForeachoftheobjectsshowninFigure 5.1.6, findandindicatethecentersofrotation
andthelinesofreflection.
(i)
(ii)
(iv)
(v)
Figure 5.1.6
(iii)
Exercise 5.1.3. ThefollowingquestionsinvolvewordsinEnglishwrittenincapitalletters. Assume
thatalllettersareassymmetricaspossible, andthatW isobtainedfromM by 180◦ rotation.
(1) Findfourwordsthathaveahorizontallineofreflection. Findthelongestsuchwordyoucan
thinkof.
(2) Findoneormorewordsthathaveaverticallineofreflection. Findthelongestsuchword
youcanthinkof.
(3) Findoneormorewordsthathavea 180◦ rotationsymmetry. Findthelongestsuchwordyou
canthinkof.
Exercise 5.2.1. Listallthesymmetriesofaregularpentagon, andofaregularhexagon.
Exercise 5.2.2. Constructthecompositiontableforthesymmetrygroupofthesquare. Uselines
ofreflectionlabeledasinFigure 5.2.6. Calculateeachentryinthetabledirectly; donotsimply
copythepatternofTable 5.2.1. (Thepointofthisexerciseistoverifybyactualcalculationthatthe
compositiontableforthesquarehasthesamepatternasfortheequilateraltriangle.)
L4
L3
L2
L1
Figure 5.2.6
Exercise 5.2.3. Fortheregularoctagon, computethefollowingsymmetries(thatis, expresseach
asasinglesymmetry).
(1) R1/8 ◦ R3/8 ;
(2) R1/4 ◦ R5/8 ;
(3) R1/8 ◦ M3 ;
(4) M1 ◦ M5 ;
(5) (M6 )−1 ;
(6) (R3/8 )−1 .
Exercise 5.3.1. Foraregular n-gon, showthat rn−a = r−a foranyinteger a. Inparticular, deduce
that rn−1 = r−1 .
Exercise 5.3.2. Constructthecompositiontableforthesymmetrygroupofthesquare, analogously
toTable 5.3.3. Useonlyouralgebraicrules, withoutactuallydrawingasquare. (Calculateeach
entryinthetabledirectly; donotsimplycopythepatternofTable 5.3.3.)
Exercise 5.3.3. Simplifyeachofthefollowingexpressionsforarbritraryregularpolygons. Ineach
case, theanswershouldbeoftheform 1 or ra or mra forsomeinteger a.
(1) mrmr2 .
(2) r5 mmrm.
(3) r7 m3 r2 mr.
(4) mr3 mr3 mr4 mr4 .
(5) m4 rm3 rm2 rm.
Exercise 5.3.4. Simplifyeachofthefollowingexpressionsfortheregularpolygonindicated. In
eachcase, theanswershouldbeoftheform 1 or ra or mra forsomenon-negativeinteger a, where
a islessthanthenumberofedgesofthepolygon.
(1) rmr2 m fortheequilateraltriangle.
(2) m3 r6 mrm fortheequilateraltriangle.
(3) mr9 m2 r forthesquare.
(4) mr4 mr3 mr2 mr fortheregularpentagon.
(5) mr4 mr3 mr2 mr fortheregularhexagon.
Exercise 5.3.5. Simplifyeachofthefollowingexpressionsfortheregularpolygonindicated. In
eachcase, theanswershouldbeinthegeometricnotation.
(1) R1/3 ◦ M1 ◦ R2/3 ◦ M3 fortheequilateraltriangle.
(2) M2 ◦ R1/3 ◦ M3 ◦ M1 fortheequilateraltriangle.
(3) R1/2 ◦ M1 ◦ M3 ◦ R3/4 forthesquare.
(4) R3/5 ◦ M1 ◦ M5 ◦ R2/5 ◦ M3 fortheregularpentagon.
(5) R1/2 ◦ M1 ◦ M3 ◦ R3/4 fortheregular 60-gon.
(6) R1/2 ◦ M1 ◦ M50 ◦ M3 ◦ R3/4 fortheregular 80-gon.
Exercise 5.4.1. ForeachoftherosettepatternsshowninFigure 5.4.6, listthesymmetries, andstate
whattypeofsymmetrygroupithas.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Figure 5.4.6
Exercise 5.4.2. Foreachofthefollowingcollectionsofsymmetries, statewhetherornotitisthe
symmetrygroupofsomeplanarobject. Ifyes, giveanexampleofanobjectwiththatsymmetry
group; ifno, explainwhynot.
(1) {I, R1/3 , R2/3 , M1 , M2 }.
(2) {I, R1/3 , R2/3 }.
(3) {I, R1/2 , R3/4 }.
(4) {I, M1 , M2 }.
(5) {I, M1 }.
(6) {I, R1/2 , M1 , M2 }.
Exercise 5.5.1. Suppose that a frieze pattern has glide reflection symmetry, and has reflection
symmetryinahorizontalline. Showthattheglidereflectionsymmetrymustbetrivial.
Exercise 5.5.2. Listthesymmetriesinthefriezegroup f1m, similarlytothewaywelistedthe
symmetriesin f11. Onceagainlet t denotethesmallestpossibletranslationsymmetrytotheright
ofthisfriezepattern, andlet h denotereflectioninahorizontalline.
Exercise 5.5.3. ForeachofthefriezepatternsshowninFigure 5.5.7, statetheanswerstoQuestions
A–D,andstatewhatsymmetrygroupithas.
Figure 5.5.7
Exercise 5.5.4. Findandphotocopy 7 friezepatterns, allwithdifferentsymmetrygroups. Foreach
ofthefriezepatternsyoufind, statetheanswerstoQuestionsA–E,andstatewhatsymmetrygroup
ithas.
Exercise 5.5.5. ThemathematicianJohnH.Conwayhascomeupwiththefollowingdescriptive
namesforthesevenfriezegroups, eachonebasedonaformofbodilymotion: hop, jump, step,
sidle, spinninghop, spinningjumpandspinningsidle. Performeachtypeofmotion(partofthe
problem is figuring out what each motion is), and look at your footprints. The footprints from
eachmotionformafriezepattern. Matchupthesefootprintfriezepatternswiththesevenlistedin
Table 5.5.2.
Exercise 5.6.1. ForeachwallpaperpatternsshowninFigure 5.6.4, findandlabelonecenterof
rotationperequivalenceclass(ifthereareany).
(i)
(ii)
(iii)
(iv)
Figure 5.6.4
Exercise 5.6.2. FindtheorderofeachwallpaperpatternshowninFigure 5.6.4.
(i)
(ii)
(iii)
(iv)
Figure 5.6.4
Exercise 5.6.3. Supposewearegivenawallpaperpattern. Supposefurtherthat, amongitscenters
ofrotation, thewallpaperpatternhasanorder 2 centerofrotationandanorder 3 centerofrotation.
Fromthispartialinformation, canyoudeterminetheorderofthewallpaperpattern? Ifyes, whatis
theorder, andwhy?
Exercise 5.6.4. ForeachwallpaperpatternshowninFigure 5.6.4, findandlabelonelineofreflection
perequivalenceclass(ifthereareany).
(i)
(ii)
(iii)
(iv)
Figure 5.6.4
Exercise 5.6.5. Supposethatawallpaperpatternhasalineofreflection. Showthatthislineof
reflectionmustalsobealineofglidereflection(foratrivialglidereflectionsymmetry). Thisexercise
usesideasfromAppendix C.
Exercise 5.6.6. Supposethatawallpaperpatternhasanon-triviallineofglidereflectionthatis
paralleltolinesofreflection. Showthatthelineofglidereflectionishalfwaybetweenthelinesof
reflection. ThisexerciseusesideasfromAppendix C.
Exercise 5.6.7. Supposethatawallpaperpatternshasanon-triviallineofglidereflection, andit
hasacenterofrotationthatisnotonthelineofglidereflection. Showthatthereisanothercenter
ofrotationatthesamedistancefromthelineofglidereflection, butontheotherside(thoughnot
directlyacrossfromtheoriginalcenterofrotation). ThisexerciseusesideasfromAppendix C.
Exercise 5.6.8. Findanexampleofawallpaperpatternthathasnon-triviallinesofglidereflection
andhaslinesofreflection, andsuchthatthenon-triviallinesofreflectionintersectsomelinesof
reflection. Thereissuchanexampleamongthewallpaperpatternsshownsofarinthissection,
thoughitslinesofreflectionandlinesofglidereflectionarenotshown.
Exercise 5.6.9. ForeachwallpaperpatternshowninFigure 5.6.4, findandlabelonenon-trivialline
ofglidereflectionperequivalenceclass(ifthereareany).
(i)
(ii)
(iii)
(iv)
Figure 5.6.4
Exercise 5.6.10. Showthatnowallpaperpatterncanhaveanswers 1, yesandyestoQuestions
A–C,regardlessofwhattheanswerstoQuestionsD andE are. (Wecanthereforeeliminatethe
combinationsofanswers 1YYNNN, 1YYNNY, 1YYNYN, 1YYNYY, 1YYYNN, 1YYYNY, 1YYYYN,
1YYYYY.)
Exercise 5.6.11. Showthatnowallpaperpatterncanhaveanswers 3, yesandnotoQuestions
A–C,regardlessofwhattheanswerstoQuestionsD andE are. Similarly, showthatnowallpaper
patterncanhaveanswers 4, yesandno, oranswers 6, yesandno, toQuestionsA–C.Listallthe
combinationsofanswerstoQuestionsA–E thatcanthereforebeeliminated.
Exercise 5.6.12. ForeachofthewallpaperpatternsshowninFigure 5.6.14, statetheanswersto
QuestionsA–E,andstatewhatsymmetrygroupithas.
Figure 5.6.14
Exercise 5.6.13. Findandphotocopy 4 wallpaperpatterns, allwithdifferentsymmetrygroups.
Foreachofthewallpaperpatternsyoufind, statetheanswerstoQuestionsA–E,andstatewhat
symmetrygroupithas.
Exercise 5.6.14. Draw 4 wallpaperpatterns, allwithdifferentsymmetrygroups. (Ifyouarealso
doingExercise 5.6.13, thenmakesurethewallpaperpatternsyoudrawhavedifferentsymmetry
groupsthantheonesyoufoundandphotocopied.) Foreachofthewallpaperpatternsyoudraw,
statetheanswerstoQuestionsA–E,andstatewhatsymmetrygroupithas.
Exercise 5.7.1. Forthecube, computethefollowingsymmetries(thatis, expresseachasasingle
symmetry).
j
(1) R1/3 ◦ Ra
1/4 .
(2) Mp ◦ Rc1/4 .
(3) Ms ◦ Mp .
(4) Mq ◦ Ca
1/4,p .
Exercise 5.7.2. Foreachofthefollowingpolyhedra, listallofitssymmetries. Usepicturesorwords
todescribetheaxesofrotationandplanesofsymmetry.
(1) A pyramidoverasquare.
(2) A prismoveranequilateraltriangle, wherethesidesarerectangles, butnotsquares.
(3) A regulartetrahedron.
(4) A regularoctahedron.
Exercise 5.7.3. Howmanysymmetriesdoestheprismoveraregular n-gonhave? Assumethat
thesidesoftheprismarerectangles, butnotsquares. (Youdonotneedtolistthesymmetries, just
countthem.)
Exercise 5.7.4. Whatistherelationbetweenthesymmetriesofaconvexpolygonandthesymmetriesofitsdual?
Exercise 5.7.5.
Our goal is to show that for any finite symmetry group for an object in three
dimensionalspace, preciselyoneofthefollowingsituationsholds: eitherallthesymmetriesare
orientationpreserving, orhalfthesymmetriesareorientationpreservingandhalfareorientation
reversing. Wewillmakeuseofthefollowingtwofactsaboutisometriesthatwehaveseenforthe
plane, andwhichinfactholdtrueinthree(orhigher)dimensionalspace; wewillnotbeableto
demonstratethesetwofacts—thatwouldrequiremoretechnicalitiesthanweareusing. First, the
analogofProposition 4.4.3holdsinthreedimensions. Second, everyisometryhasaninverse.
Suppose G isafinitesymmetrygroupforanobjectinthreedimensionalspace. Theargumenthas
anumberofsteps, mostofwhichhavesomethingforthereadertodo.
(1) Supposethat A, B and C aresymmetriesin G, andthat A ̸= B. Showthat C ◦ A ̸= C ◦ B.
(2) If G hasallorientationpreservingsymmetries, thenthereisnothingtodemonstrate, soassumefromnowonthatnotallsymmetriesin G areorientationpreserving. Showthat G has
bothorientationpreservingandorientationreversingsymmetries.
(3) Let {A1 , A2 , . . . , An } denote the orientation preserving symmetries in G, and let
{B1 , B2 , . . . , Bm } denotetheorientationreversingsymmetriesin G, where n and m are
somepositiveintegers. Ourgoalistoshowthat n = m, whichwillimplythat G hasthe
samenumberoforientationpreservingsymmetriesandorientationreversingsymmetries.
(4) Considerthecollectionofsymmetries {B1 ◦ A1 , B1 ◦ A2 , . . . , B1 ◦ An }. Showthatthese
symmetriesarealldistinct.
(5) Showthatallthesymmetries {B1 ◦ A1 , B1 ◦ A2 , . . . , B1 ◦ An } areorientationreversing.
(6) Deduce that every one of {B1
{B1 , B2 , . . . , Bm }.
◦ A1 , B1 ◦ A2 , . . . , B1 ◦ An } is contained in
(7) Deducethat n ≤ m.
(8) Usesimilarideastoshowthatallthesymmetries {B1 ◦ B1 , B1 ◦ B2 , . . . , B1 ◦ Bm } are
distinct, andallarecontainedin {A1 , A2 , . . . , An }. Deducethat m ≤ n.
(9) Becausewehaveseenthat n ≤ m andthat m ≤ n, itfollowsthat n = m, whichiswhat
weneededtoshow.
Exercise 6.1.1. Determinewhichofthefivepropertiesofclosure, associativity, identity, inverses
andcommutativityaresatisfiedbyeachofthefollowingsystems.
(1) Theevenintegerswithaddition.
(2) Theoddintegerswithaddition.
(3) Allrealnumberswithaddition.
(4) Allrealnumberswithmultiplication.
Exercise 6.2.1. Ifyoustartat 7 o’clock, andgoanother 20 hours, whattimewillitbe?
Exercise 6.2.2. Whichofthefollowingpairsofnumbersarecongruentmod 12?
(1) 15 and 3;
(2) 9 and 57;
(3) 7 and −5;
(4) 11 and 1;
(5) 0 and 12.
Exercise 6.2.3.
(mod 12).
(1) a = 18;
(2) a = 41;
(3) a = −17;
(4) a = 3.
Foreachinteger a givenbelow, findtheinteger x from 0 to 11 sothat x ≡ a
Exercise 6.2.4. Calculatethefollowing.
(1) b
4+b
5;
(2) b
7+b
8;
b;
(3) b
5 + 11
(4) b
0+b
3.
Exercise 6.2.5. Findtheinverseswithrespecttoadditionof b
3, b
6, b
8 and b
0 in Z12 .
Exercise 6.2.6. Whichofthefiveproperties(closure, associativity, identity, inverses, commutativity)
holdsfor (Z12 , ·)? Forthoseelementsof Z12 thathaveinverseswithrespecttomultiplication, state
whattheirinversesare.
Exercise 6.3.1. Whichofthefollowingaretrue, andwhicharefalse?
(1) 3 ≡ 9 (mod 2);
(2) 7 ≡ (−1) (mod 8);
(3) 4 ≡ 11 (mod 3);
(4) 0 ≡ 24 (mod 6).
(5) 9 ≡ 9 (mod 5).
Exercise 6.3.2.
(mod 8).
For each integer a given below, find the integer x from 0 to 7 so that x ≡ a
(1) a = 15;
(2) a = 54;
(3) a = 1381;
(4) a = −2;
(5) a = 3;
(6) a = 8.
Exercise 6.3.3. Calculatethefollowingin Z8 .
(1) b
4+b
1;
(2) b
3+b
7;
(3) b
0+b
3.
Exercise 6.3.4. Findtheinverseswithrespecttoadditionof b
2, b
4, b
5 and b
0 in Z8 .
Exercise 6.3.5. Considerthesystem (Z6 , +).
(1) Listtheelementsofthissystem.
(2) In (Z6 , +), whatare b
5+b
2 and b
4+b
1?
(3) Constructtheadditiontablefor (Z6 , +).
(4) Findtheinverseswithrespecttoadditionof b
2, b
4, b
5 and b
0 in Z6 .
Exercise 6.3.6. Observethatintheadditiontablefor (Z8 , +), showninTable 6.3.1, alltheentries
onthedownwardsslopingdiagonalareevennumbers. Willthesamefactholdintheadditiontable
forany (Zn , +)? Ifyes, explainwhy. Ifnot, describewhatdoeshappenonthedownwardssloping
diagonalfor (Zn , +) ingeneral, andexplainyouranswer.
Exercise 6.3.7. Constructthemultiplicationtablefor (Z8 , ·).
Exercise 6.4.1. Whichofthefollowingsystemsisagroup? Whichisanabeliangroup?
(1) Theevenintegerswithaddition.
(2) Theoddintegerswithaddition.
(3) Allrealnumberswithaddition.
(4) Allrealnumberswithmultiplication.
Exercise 6.4.2. DemonstrateProposition 6.4.1(2).
Exercise 6.4.3. Foreachcollectionofobjectsandoperationtableindicatedbelow, answerthe
followingquestion:
(a) Istheclosurepropertysatisfied?
(b) Isthereanidentityelement? Ifso, whatisit?
(c) Whichelementshaveinverses? Forthosethathaveinverses, statetheirinverses? (Ifthereis
noidentityelement, thisquestionismoot.)
(d) Isthecommutativepropertysatisfied?
(e) Assumingthattheassociativepropertyholds, dothecollectionofobjectsandgivenoperationformagroup? Iftheyareagroup, isitanabeliangroup?
(1) Theset V = {x, y, z, w} withbinaryoperation ⋄ givenbyTable 6.4.4.
(2) Theset K = {m, n, p, q, r} withbinaryoperation ⋆ givenbyTable 6.4.5.
(3) Theset M = {1, s, t, a, b, c} withbinaryoperation ⊙ givenbyTable 6.4.6.
(4) Theset W = {e, f, g, h, w, x, y, z} withbinaryoperation ∗ givenbyTable 6.4.7.
⋄
x
y
z
w
Table6.4.4
x y z w
z w y x
x y z w
y z w x
z w x z
Table6.4.5
⋆ m n p q r
m n p q m r
n p r m n p
p q m r p n
q m n p q r
r r p n r q
Table6.4.6
⊙
1
s
t
a
b
c
Table6.4.7
1
1
s
t
a
b
c
s
s
t
1
c
a
b
t
t
1
s
b
c
a
a
a
b
c
1
s
t
b
b
c
a
t
1
s
c
c
a
b
s
t
1
∗ e f g h w x y z
e e f g h w x y z
f f g h e x y z w
g g h e f y z w x
h h e f g z w x y
w w x y z e f g h
x x y z w f g h e
y y z w x g h e f
z z w x y h e f g
Exercise 6.4.4. Findanexampleofafinitesetwithabinaryoperationgivenbyanoperationtable,
suchthateachelementofthesetappearsexactlyonceineachrow, andonceineachcolumn, and
yetthesetwiththisbinaryoperationisnotagroup.
Exercise 6.4.5. Let C be the set C = {k, l, m}. Construct an operation on C, by making an
operationtable, whichturns C intoagroup.
Exercise 6.4.6.
Is the group (Z4 , +) isomorphic to either of (Z, ⋆) or (Q, ⊞)? If (Z4 , +) is
isomorphictooneofthesetwogroups, demonstratethisfactbyshowinghowtorenametheelements
of (Z4 , +) appropriately.
Exercise 6.5.1.
(1) Let T bethesetofallintegermultiplesof 3, thatis, theset
T = {. . . , −9, −6, −3, 0, 3, 6, 9, . . .}.
Is (T, +) subgroupof (Z, +)?
(2) Let V bethesetofallperfectsquareintegersandtheirnegatives, thatis, theset
V = {. . . , −16, −9, −4, −1, 0, 1, 4, 9, 16 . . .}.
Is (V, +) subgroupof (Z, +)?
Exercise 6.5.2. Thegroup (Z36 , +) has 36 elements. Howmanyelementscouldasubgroupof
(Z36 , +) possiblyhave?
Exercise 6.5.3. Which, ifany, ofthefollowingsubcollectionsof Z6 aresubgroupsof (Z6 , +)? Use
Lagrange’sTheorem, andconstructoperationtablesaswedidforsubgroupsof (Z8 , +).
(1) A = {b
0, b
3};
(2) B = {b
0, b
2};
(3) C = {b
0, b
1, b
4};
(4) D = {b
0, b
2, b
4}.
(5) E = {b
0, b
1, b
2, b
3}.
Exercise 6.5.4. Let (M, ⊙) beasinExercise 6.4.3 (3). Which, ifany, ofthefollowingsubcollections
of M aresubgroupsof (M, ⊙)?
(1) E = {1, s};
(2) F = {1, a};
(3) C = {1, s, t};
(4) D = {1, a, b, c}.
Exercise 6.5.5. Findasmanypropersubgroupsasyoucanof (Z12 , +). Theoperationtablefor
(Z12 , +) isgiveninTable 6.2.2.
Table 6.2.2
b 11
b
+ b
0 b
1 b
2 b
3 b
4 b
5 b
6 b
7 b
8 b
9 10
b
b
b
b
b
b
b
b
b
b
b
b
b
0 0 1 2 3 4 5 6 7 8 9 10 11
b
b 11
b b
1 b
1 b
2 b
3 b
4 b
5 b
6 b
7 b
8 b
9 10
0
b
b 11
b b
2 b
2 b
3 b
4 b
5 b
6 b
7 b
8 b
9 10
0 b
1
b
b
b
b
b
b
b
b
b
b
b
b
b
3 3 4 5 6 7 8 9 10 11 0 1 2
b
b 11
b b
4 b
4 b
5 b
6 b
7 b
8 b
9 10
0 b
1 b
2 b
3
b
b
b
b
b
b
b
b
b
b
b
b
b
5 5 6 7 8 9 10 11 0 1 2 3 4
b
b 11
b b
6 b
6 b
7 b
8 b
9 10
0 b
1 b
2 b
3 b
4 b
5
b
b
b
b
b
b
b
b
b
b
b
b
b
7 7 8 9 10 11 0 1 2 3 4 5 6
b
b 11
b b
8 b
8 b
9 10
0 b
1 b
2 b
3 b
4 b
5 b
6 b
7
b 11
b b
b
9 10
0 b
1 b
2 b
3 b
4 b
5 b
6 b
7 b
8
9 b
b
b
b
b
b
b
b
b
b
b
b
b
b
10 10 11 0 1 2 3 4 5 6 7 8 9
b 11
b b
b
11
0 b
1 b
2 b
3 b
4 b
5 b
6 b
7 b
8 b
9 10
Exercise 6.6.1. Showthatthegroup (W, ∗) giveninExercise 6.3.4 (4)isnotthesymmetrygroup
ofanyplanarobject. Theideaisasfollows. Giventhat W isfinite, ifitwerethesymmetrygroupof
aplanarobject, itwouldhavetobethesymmetrygroupofarosettepattern(becausethosearepreciselytheplanarobjectswithfinitesymmetrygroups). ByLeonardo’sTheorem(Proposition 5.4.5),
weknowthatanyrosettepatternhassymmetrygroupeither Cn or Dn forsomepositiveinteger n.
Findreasonstoshowwhy (W, ∗) isnotisomorphictoanyofthe Cn or Dn groups.
Exercise 6.6.2. Foreachofthefollowingobjects, findallpropersubgroupsofitssymmetrygroup.
(1) Theequilateraltriangle.
(2) Theregularpentagon.
Exercise 6.6.3. Iseachofthefollowingcollectionofsymmetriesalwaysasubgroupofthesymmetry
groupofaplanarobject. Explainyouranswers.
(1) Thecollectionofallreflectionsymmetries.
(2) Thecollectionofalltranslationandallhalfturnrotationsymmetries.
(3) Thecollectionofallrotationandallreflectionsymmetries.
Exercise 6.6.4. Showthatforarosettepattern, thecollectionofallrotationsymmetriesisasubgroup
ofthesymmetrygroup.