hexagonal pattern of beehives蜂箱的六角形图案

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1、HexagonalpatternofbeehivesAlignment:MACC.912.G-MG.1.1Beehivesaremadeofwalls,eachofthesamesize,enclosingsmallhexagonalcellswherehoneyandpollenisstoredandbeesareraised.Belowisapictureofsomecells:Theonlyotherregularpolygonswhichcanbeusedtotiletheplaneinthiswayareequilateralt

2、rianglesandsquares.Thisproblemexaminessomeofthemathematicaladvantagesofthehexagonaltiling.Supposewelet s denotethelengthofthewallsinthehexagonalchambers.Belowisanenlargedpictureofasinglehexagon:a.Findtheareaofaregularhexagon H withsidelength s.b.Istheratioofareatoperimete

3、rforaregularhexagongreaterthanorsmallerthanthecorrespondingratiosforanequilateraltriangleandasquare?c.Basedonyouranswerto(b),whydoyouthinkitisadvantageousforbeehivestobebuiltusinghexagonalcellsinsteadoftriangularorsquarecells?LicensedbyIllustrativeMathematicsunderaCreativ

4、eCommonsAttribution-NonCommercial-ShareAlike3.0UnportedLicenseCommentaryThegoalofthistaskistousegeometrystudythestructureofbeehives.Beehiveshaveatremendoussimplicityastheyareconstructedentirelyofsmall,equallysizedwalls.Inordertoasusefulaspossibleforthehive,thegoalshouldbe

5、tocreatethelargestpossiblevolumeusingtheleastamountofmaterials.Inotherwords,theratioofthevolumeofeachcelltoitssurfaceareaneedstobemaximized.Thisthenreducestomaximizingtheratioofthesurfaceareaofthecellshapetoitsperimeter.Thehexagonalpatternofbeehivesisawell-documentedpheno

6、menonandmanypicturescanbefoundon-line.·HoneycombstructureOneinterestingquestionwouldbewhythecellsinthehoneycombarenotcircularasacirclehasabetterratio,forthesameperimeter,ofperimetertoareathanahexagon.Oneexplanationforthisisthatunlikeregularhexagonswhichcanbeputsidebysidet

7、ofillupspacewithnogaps,withcirclestherewouldbemuchwaste.Toputthisanotherway,inthehexagonalconfigurationeachwallisawallfor two separatechambers.Withcircles,thiswouldstillbethecasebutthespacesbetweencircleswouldbesmallandoddlyshapedandperhapsnotveryuseful.Hexagonsareonlycom

8、paredwithsquaresandequilateraltrianglesasthesearetheonlythreeshapeswhichcanbeputtogethertotileth