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How to make a Möbius strip

AMöbiusstripisanintriguing3Dshapewhichhasonlyoneside.Trytothinkofanyothersurfaceandyouwillfindthatithastwosides:apieceofpaperhasafrontandback;atabletophasatopandbottom;abeachballhasaninsideandanoutside.Howcanwepossiblymakeashapewhichhasonlyoneside?

You canmake aMöbius strip using knitting, crochet, or paper and glue! Youmay enjoy tryingmultiplemethods,aseachrevealsitsownsecretsoftheMöbiusstrip.

PaperandGlue

MakingaMöbius stripoutof paper is the fastestway toexplore thismagicalshape.Takealongstripofpaper,giveitahalf-twistasshowninthepicture,andtapeorgluetheendstogethersecurely.Canyoupredictwhatwillhappenifyoucutalongthedottedline?Whataboutifyoucutathirdofthewayinfromtheedge?Howaboutcuttingitinhalf,theninhalfagain?

Crochet

You can use any needles, yarn and gauge with this project. Take your favourite yarn and anappropriate crochet hook, and chain a number of stitchesdepending on how big you’d like your project to be. Forexample,youmayliketomakeabraceletorbroochasafirstproject, and the initial chain is the circumference of theMöbiusstrip.Whenyourchain is longenough, lay thechainon a flat surface and run your finger along it so that allstitchesarefacingthesameway.Thengivethechainahalf-twist and join the last stitch to the first with a slip stitch.(Note:ifyoudon’tdothetwistcorrectlyhere,youcanalwaysfixitafterthefirstroundofcrochet.)

Begintodoublecrochetintothebottomofallthestitchesinthechain,untilyougetbacktothejoin. At this point you should check that there is indeed a half-twist. You should be able tocontinue double crocheting, now into the tops of all the stitches. Once you get back to thebeginningagaintheprojectiseasy:continuecrochetingaroundandarounduntilyourstripisthedesiredwidth.Noticethattheprojectwillgrowoutwardsonboth‘sides’atthesametime!

A final interesting thing to do is to crochet the outermost row in a different colour so as tohighlightthatitisonecontinuousloop.

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Knitting

TherearetwowaystoknitaMöbiusstrip.Thefirstwayistoknitalongstripandthengrafttheendstogetherwithahalf-twist,asin the ‘paper’ description. This will give a vertical seam,perpendiculartothestitches.

The secondmethod,which is harder butmoremathematicallypleasing,istouseaspecialcast-onwhichintroducesthetwistintheveryfirstrow(much likethecrochetmethod).ThiscreatesanalmostinvisibleseamalongthecircleoftheMöbiusstrip.

TheMöbiuscast-onwerecommendisdesignedbyCatBordhiandisbrilliantlydescribedbyherinthis video: https://youtu.be/LVnTda7F2V4.Written instructions, with diagrams, are available athttp://bit.ly/1WsRh8F.

Aswiththecrochetpattern, itdoesnotmatterwhatyarn,gauge,orsizeofneedleyouuse.Thestitchpatternthatyouusedoesmatter:sincethereisno‘front’or‘back’tothework,youneedtouseastitchpatternwhichisreversible.Oneeasyoptionisseedstitch,whichalternatesknitandpurlstitches.Youcouldalsotryaribbedpattern,asshowninthepictureabove.TheinstructionsforthiswilldependonwhetheryouareusingthefirstorsecondmethodformakingtheMöbiusstrip–canyoufigureoutthedifference?

TheMathsofMöbiusstrips

Runyourfingeraroundthesurfaceofyourstrip.Youshouldfindthatyourfingerendsuponthe‘back’oftheknitting,crochetorpapereventhough itneverwentoveranedge.This isbecausetheMöbiusstriphasonlyoneside,andlikewiseonlyoneedge.

The one-sided nature of the Möbius strip means that, for anycreature living in it, there is no concept of ‘left’ or ‘right’. Bytravelling once around the surface, a right hand is transformedinto a left hand and vice versa. If a shape has this propertymathematicians call it non-orientable. Is it possible that ouruniversecouldbelikethis,sothatwecouldflyoffinonedirectionfromtheEarthandreturn fromtheothersidewithourhearton

theright-handsidesofourbodies?Scientiststhinkthisisunlikely,buttheyhaven’truleditout!

Ifyouattach twoMöbiusstrips togetheralong theiredges,youcreateastrangeobjectcalledaKlein bottle which has no inside or outside. It cannot bemadewithout intersecting itself in 3dimensions:itstruehomeis4-dimensionalspace.

AMöbiusstripwiththreetwistsformsthesymbolforrecyclinginNewZealand.It is also the symbol for Google Drive. Try making one out of paper andcutting in half down themiddle – the result should surprise you! Can youfigureouthowtocrochetone?