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.