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Puzzle_ Patty Paper Trisection _ Let's Play Math!
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2/6/2015 Puzzle: Patty Paper Trisection | Let's Play Math! http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 1/4 Let's Play Math! JUN 5 2007 Puzzle: Patty Paper Trisection [Feature photo above by Michael Cory (https://www.flickr.com/photos/khouri/6260501380/) via Flickr (CC BY 2.0).] (http://letsplaymath.net/2007/06/05/puzzle‑patty‑paper‑ trisection/) One of the great unsolved problems of antiquity (http://mathforum.org/dr.math/faq/faq.impossible.construct.html) was to trisect any angle (http://www‑ groups.dcs.st‑andrews.ac.uk/~history/HistTopics/Trisecting_an_angle.html) using only the basic tools of Euclidean geometry: an unmarked straight‑edge and a compass. Like the alchemist’s dream (http://en.wikipedia.org/wiki/Alchemy) of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight‑edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator (http://chemistry.about.com/cs/generalchemistry/a/aa050601a.htm) or something. One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why. About these ads Math is a game, playing with ideas.
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Transcript of Puzzle_ Patty Paper Trisection _ Let's Play Math!
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2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 1/4
Let'sPlayMath!
JUN52007
Puzzle:PattyPaperTrisection
[FeaturephotoabovebyMichaelCory(https://www.flickr.com/photos/khouri/6260501380/)viaFlickr(CCBY2.0).]
(http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/)
Oneofthegreatunsolvedproblemsofantiquity
(http://mathforum.org/dr.math/faq/faq.impossible.construct.html)wastotrisectanyangle(http://wwwgroups.dcs.standrews.ac.uk/~history/HistTopics/Trisecting_an_angle.html)usingonlythebasictoolsofEuclideangeometry:anunmarkedstraightedgeandacompass.Likethealchemistsdream(http://en.wikipedia.org/wiki/Alchemy)ofturningleadintogold,thisprovedtobeanimpossibletask.Ifyouwanttotrisectanangle,youhavetocheat.Astraightedgeandcompasscantdoit.Youhavetousesomesortofcrutch,justasanalchemistwouldhavetouseaparticleaccelerator(http://chemistry.about.com/cs/generalchemistry/a/aa050601a.htm)orsomething.
Onecheatthatworksistofoldyourpaper.Iwillshowyouhowitworks,andyourjobistoshowwhy.
Abouttheseads
Mathisagame,playingwithideas.
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2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 2/4
GatherYourMaterials
Apieceofpaperwithsquarecorners.(Iusedpattypaper.)Arulerorotherstraightedge.Apenorpencil.
OneStepataTime
Drawanyrandomangle(lessthan90degrees)withitsbaselinealongthebottomofyourpaperanditsvertexatthecorner.CallthevertexpointP.
(http://letsplaymath.files.wordpress.com/2007/06/trisection11.jpg)
Foldthepaperhorizontally,sothatpointPtouchesthesideofthepaperatsomeotherpointQ.DrawalinethroughQ,usingthefoldedupbaselineasyourguide.Unfoldthepaper,anddrawanotherlinealongthefold.Theselinesareshowninredonmypattypaper.
(Clicktheimageforalargerview.)
Canyouprovethatthesetwolinesareparalleltothebaselineofyourpaper?Canyoushowthattheymustbeequallyspaced?
(http://letsplaymath.files.wordpress.com/2007/06/trisection21.jpg)
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2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 3/4
Foldthepaper(dottedblueline)sothatpointPtouchesthelowerofthetwoparallellinesandpointQtouchesthetoplineofyouroriginalangle.ThesetouchpointsarelabeledPandQonthediagramabove.MarkpointPonyourpaper.
Unfoldthepaper.UseyourstraightedgetoconnectcornerPwithpointP.AlsodrawalinefromcornerPthroughthepointwherethisfoldmeetsthefirstfold.Thesetwolinesaremarkedingreenintheimage.
Yourangleisnowtrisected.Thelasttwolinesyoudrewcuttheoriginalangleinto3smallerangles,allequal.
CanYouProveIt?
Foldingpaperistheeasypart,butnowyourhardworkbegins.
Howcanweknowthatthismethodtrisectstheangle?
Anymathematicalproofstartswithsomesetofgivenfacts.Lookagainatthefirstpattypaperimage.Whatdoyouknowforsureaboutthoselines?Makealist!Inordertowritedownwhatyouknow,youwillhavetogivenamestosomeofthepointsandlinesinthedrawing.Eitherprintouttheimageandwriteonit,ormakeasketchofyourown.
Thenlookatthesecondimage.Isthereanythingyouknowforsureaboutthenewlines?Howdoyouknowit?Writedownanyfactsyoucanthinkof,andbesuretoincludeareasonforeachfact.
Nowconcentrateonunderstandingwhatyouaretryingtoprove.Whatexactlydoesitmeantotrisectanangle?Canyouwriteamathematicalstatementorequationthatmeansthesamethingas,Thisangleistrisected?(Youmayneedtolabelsomemorepointsoranglesonyourdiagram.)
Finally,youneedtofindawaytogetfromthefactsyouaregiventothestatementyouwanttoprove,andyouneedtoshowavalidreasonforeachstepalongtheway.Thinkbacktothethingsyoulearnedinmathclass.Whatdoyouknowabouttriangles?Whatdoyouknowaboutparallellines?Willanyofthosefactshelpyouhere?
IfYouGetStumped
IamgoingtoborrowanideafromJonathanatJD2718(http://jd2718.wordpress.com/):thesolutionspost.Iwillwriteafollowuparticlewithafewhints(http://letsplaymath.net/2007/06/05/hintsandsolutionspattypapertrisection/),andthatwillbetheplacetoshareyourquestionsorproofs,ifyouwish.Commentsherebelowarelimitedtogeneralremarksthatwontspoilanyoneelsesfuninworkingthroughthepuzzle.
[Hattip:TrisectinganAnglewithOrigami(http://www.sciencenews.org/view/generic/id/8567/title/Math_Trek__Trisecting_an_Angle_with_Origami)atMathTrek(http://blog.sciencenews.org/view/dispatches/type/column/collection_id/2/title/Math_Trek),viatheCarnivalofMathematics(http://jd2718.wordpress.com/2007/06/02/carnivalofmathematicsix/).]
DontmissanyofLetsPlayMath!: (http://feeds.feedburner.com/LetsPlayMath)Subscribeinareader(http://feeds.feedburner.com/LetsPlayMath),orgetupdatesbyEmail(http://www.feedburner.com/fb/a/emailverifySubmit?feedId=633025&loc=en_US).
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http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 4/4
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(http://www.amazon.com/JourneythroughGeniusTheoremsMathematics/dp/014014739X/)
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ByDeniseGaskinsPostedinAlgebra&Beyond,PuzzlesTaggedEuclid,Geometry,Highschool,Mathclub,Proofs,Puzzles
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