Tessellations - IREM de la Réunionirem.univ-reunion.fr/IMG/pdf/tessellations_doc_eleve.pdf ·...

LaurenceLEVESQUE

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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

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u translating the top left triangle. Such vectormovementmaybewrittenasthetranslationofvector

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u .

A’B’C’istheimageofABCunderthe…………………….ofvector…….

Inthecoordinatessystembelow,thetranslationbelowshowsavector

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u translatingthetoptriangle……unitstotherightand……unitsdownward.Inthatcase,thevectorofthetranslationis

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u(−4;−9)

LaurenceLEVESQUE

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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

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θ is the transformation such that the

imageofO isOand foranypointM≠O, the imageM’ofM is such thatOM’=……..and

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(OM ,OM') = ..........

LaurenceLEVESQUE

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Task10:Placethecentreofrotationandfindtheangleineachcase.

Al­Alhambratiling Humans­Escher

Task11:Usethefollowingnettodrawatilinginvolvingrotationsonly.

Specifytheshapeofyourpatternandtheangleofrotation.

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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.