polyhedra by paper folding worksheet
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AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC)
AIMING HIGH
POLYHEDRA BY PAPER FOLDING Aregularpolygonhasallitsanglesequalandalledgelengthsequal.Inaregularpolyhedronallthefacesarecongruentregularpolygonsandthesamenumberofpolygonsmeetateachvertex.
Eachregularpolyhedronhasitsowncodename.Thetetrahedronis333,theoctahedronis3333,theicosahedronis33333,thecubeis444,andthedodecahedronis555.
1. Canyoucrackthecodeandexplainwhythesecodenameshavebeengiventothepolyhedra?
2. Followtheinstructionsbelowandmakethefiveregularpolyhedral.Firstfoldpapertomakeregularpolygons,theninterlockthepolyhedratogether.Thesidesofthepaperareshownindifferentcolourstomakethedirectionseasiertofollow.
a. CubeTheratiooftheshorttolongedgesofyourrectangularpieceofpapermustbe3:4soremovea17mmwidestripfromasheetofA4papertodecreasethelengthofthelongeredge.Foldthepaperasshownmaking2foldslengthwiseandthen3foldsacrossthestrip.Dothesamewiththreeseparatesheetsofpaper,andinterlockthemtoformacube.
b. Tetrahedron
FoldanA4sheetofpaperasshowntomakeatetrahedronnet.Makeanotheridenticalnet,thenslotthetwonetstogethertoformarigidtetrahedron.
[Note:Usescrappaper.ThestandardsizeA4officepaper,usedverywidelyworldwide(butnotintheUSA)hasedgelengthsintheratio√2 to1.Othersizesareinthesameproportion,biggersizesA1,A2andA3,andsmallersizesA5,A6etc.IfyoudonothaveA4paperyoucancutyourpaperinthisproportiontomakethemodels.]
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c. OctahedronMakefourtetrahedronnetsfollowingtherecipeabove.Slotthemtogethertoformanoctahedron.Slottwonetstogethertoformwhatlookslikeasquarepyramidwithflapsattached.Dothesamefortheothertwonets.Nowslotthesetwosquarepyramid-likeobjectstogether.
d. DodecahedronIfyoufollowtheseinstructionscarefully,usingpaperwithedgelengthsintheratio√2to1,youwillbeabletomakeadodecahedronliketheoneinthepicture.
e. IcosahedronFromthefinalstageofthetetrahedronrecipe,makethefoldsshownbelowtoproduceatruncatedtetrahedron.Maketwentyofthesetruncatedtetrahedraandgluethemtogethertoformanicosahedron.
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HELP Sharetheworkinagroupandeachgroupmightmakejustoneofthepolyhedra.Theneachlearneronlyhastomakeafewofthecomponentsthatgotomakingoneofthepolyhedra.Thegroupsshouldlistallthepropertiesoftheirpolyhedron,andthenpassittoanothergroupsothatallthelearnerscanhandleall5polyhedraandcontributetothelistsofproperties.
NEXT IfyouuseA4paperfortheconstructiontomakethedodecahedron,andtrytomakeregularpentagons,thereisasmallerrorintheangleatE.Findthiserrorandfindthedimensionsofthepaperthatyouwouldneedtousetogetanaccurateregularpentagonandhenceanaccurateregulardodecahedron.HintJoinAO.WhatistheanglebetweenAOandtheredfoldline?NowfindangleREO.