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LaurenceLEVESQUE
1
TessellationsTessellations
Task1:WriteapresentationaboutTessellationsintheAlhambrainGranada.
Tessellationsmeandesignsfeaturinganimals,birds,andvariousshapes,whichcan……….thesurface,without…………………………:
IslamicStarpattern(AlAlhambra) EscherDogs1938
Theword'tessera'inLatinmeansasmallstonecube.Theywere used to make up 'tessellata' ‐ the mosaic picturesformingfloorsandtilingsinRomanbuildings.
Tessellations are made from a pattern, which is repeated using geometrictransformations.
LaurenceLEVESQUE
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Task2:HowtobuildanAlhambrapattern
StartwiththesquareIJKLtoreproducethefollowingdrawing.Then,writeitsprotocolofconstruction.
Likewise,reproducethesetwodrawings:
Task3:UsethethreepreviousdrawingstobuildthisIslamicpattern:
Colouritaccordingtoyourpersonaltastes.
LaurenceLEVESQUE
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Task4:ProtocolofconstructionofanAlhambrapattern
1. Takeablanksheet.
2. DrawanequilateraltriangleABCofside8cm.
3. DrawI,JandK,respectivelymidpointsofsegments[AB],[BC]and[AC].
4. DrawM,N,O, P,Q andR, respectivelymidpoints of segments [JB], [JC], [AK],
[KC],[AI]and[IB].
5. Drawtheperpendicular line to (BC) throughNandplace thepointC1on this
line,1cmfromNoutsidethetriangleABC.
6. Drawtheperpendicular line to(BC) throughMandplace thepointC2onthis
line,1cmfromMinsidethetriangleABC.
7. Drawtheperpendicular line to (AC) throughOandplace thepointC3on this
line,1cmfromOoutsidethetriangleABC.
8. Draw theperpendicular line to (AC) throughPandplace thepointC4on this
line,1cmfromPinsidethetriangleABC.
9. Drawtheperpendicular line to (AB) throughQandplace thepointC5on this
line,1cmfromQinsidethetriangleABC.
10. Drawtheperpendicular line to (AB) throughRandplace thepointC6on this
line,1cmfromRoutsidethetriangleABC.
11. DrawthecirclewithcentreC1throughJ,andthenerasethelongarcofcircleCJ.
12. DrawthecirclewithcentreC2throughJ,andthenerasethelongarcofcircleBJ.
13. DrawthecirclewithcentreC5throughI,andthenerasethelongarcofcircleAI.
14. DrawthecirclewithcentreC6throughI,andthenerasethelongarcofcircleBI.
15. DrawthecirclewithcentreC3throughK,andthenerasethelongarcofcircle
CK.
16. DrawthecirclewithcentreC4throughK,andthenerasethelongarcofcircle
AK.
17. Colourtheremainingarcsanderasetherestofthefigure.
LaurenceLEVESQUE
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Task5:Watchcarefullytheanimation(p1):howwouldyoudescribethewaythistilingismoving?
DEFINITION:TRANSLATION
Totranslateashape:everypointoftheshapemustmove:
• Thesame………………..
• Inthesame…………...
PremièreS:
The translation below shows a vector
€
u translating the top left triangle. Such vectormovementmaybewrittenasthetranslationofvector
€
u .
A’B’C’istheimageofABCunderthe…………………….ofvector…….
Inthecoordinatessystembelow,thetranslationbelowshowsavector
€
u translatingthetoptriangle……unitstotherightand……unitsdownward.Inthatcase,thevectorofthetranslationis
€
u(−4;−9)
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Task6:
ThisIslamictessellationisfromAl‐Alhambra:
Findthevectoroftranslationanddrawitdirectlyonthefigure.
Task7:WriteapresentationaboutEscher’sworkconcerningtessellations.
Lookatthefoxes.
You can see that every part needed tomake up thefull foxespicture is contained in theparallelogram.All you need to do is to repeat them by translationandyougetthefulltessellation.
EscherFoxes
Task8:
Usethegridtodrawatessellationbytranslationofaparallelogramofyourchoice.
LaurenceLEVESQUE
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Task9:Watchcarefullytheanimation(p6):trytocharacterisethistransformation.
DEFINITION:ROTATIONABOUTAPOINTTHROUGHANANGLE
The"rotation"transformationiswhereyouturnafigureaboutagiven…………….The
amount of rotation is called the…………… of rotation and ismeasured in degrees (or
radians).Byconventionarotationcounter‐clockwiseisapositiveangle,andclockwiseis
considereda…………………angle.
ThefigureA’istheimageofthefigureAundertherotationabout…….throughtheangle
………..
PremièreS:
The rotation about centre O through the angle
€
θ is the transformation such that the
imageofO isOand foranypointM≠O, the imageM’ofM is such thatOM’=……..and
€
(OM ,OM') = ..........
LaurenceLEVESQUE
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Task10:Placethecentreofrotationandfindtheangleineachcase.
AlAlhambratiling HumansEscher
Task11:Usethefollowingnettodrawatilinginvolvingrotationsonly.
Specifytheshapeofyourpatternandtheangleofrotation.
LaurenceLEVESQUE
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Task12:Watchcarefullytheanimation(pm):givethenameandthedefinitionofthistransformation.
SYMMETRYINALINE
Areflectionisatransformationthat……………..afigureacrossaline.
Task13:FindsomelinesofsymmetryinthisSamarkandMosquetessellation:
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Task14:Drawtheimageofthisfigureunderthesymmetryinline(d)
Task15:
Belowaretheimagesofthestepsgivingthecellofatessellation.
1. Namethetransformationfromstep1tostep2.2. Namethetransformationfromstep2tostep3.3. Inthenexttessellation,outlinethepattern.
LaurenceLEVESQUE
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Task16:
ThispatternisinspiredbytheromantempleofDiana,thegoddessofthehunt.
Describethebasicshapesyoucanseeinthispicture.
Whydotheyformatessellationwithoutanygap?
Task 17 :(holiday homework): Draw a tessellation using the pattern and thetransformationsofyourchoice.Colouritaccordingtoyourownfeeling.