6_what the bees know and what they do not know(Toth_1964).pdf

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1、WHATTHEBEESKNOWANDWHATTHEYDONOTKNOWL.FEJESTÓTHInthefirstpartofthispaperweconstructamoreeconomicalhoneycombthanthatofthehivebeesforanyparametersinvolvedintheproblem.Thesecondpartgivesasurveyofsomefurthersolvedandunsolvedisoperimetricproblemsconcerningcell-aggregates.I

2、.Honeycombs.Thehoneycombofthebeesisaloosetissueofwaxformingaplanelayer.Thefirstthingsonitwhichcatchone'seyearetheregularhexagonalpatternsonbothsides.Thehexagonsaretheopeningsofprismaticvessels,calledbee-cells.Keplerde•scribedtheshapeofthebee-cellsmorefully.Itturnedou

3、tthatthebot•tomofacellconsistsofthreeequalrhombiasshowninFigurelb.Thusthetwokindsofcellshavingtheiropeningsinoppositedirec•tionsareseparatedbyazigzaggedsurfaceandnotbyaplane,asonewouldexpectatfirst.Whydothebeesbuildsuchastrangeconformation?Accordingtoawidelyspreadhyp

4、othesis,goingbacktoPappus,thebeesaimateconomy:If,bysomereason,thevolumeofacellandthewidthofthewholelayeraregiven,theytrytousetheminimumamountofwaxpercell.Althoughamongthevariouseffectswhichinteractinproducingthehoneycombtheutilitarianhumanmotiveattributedtothebeessee

5、mstoplaytheleastpart,theabovehypoth•esiswasthesourceofhighlyinterestinginvestigations.Thusweac•ceptthishypothesisandtrytopointoutwhatthebeesdowellandwhattheydonotdowellfromthepointofviewofmakingthesur•face-areaoftheircellssmall.Togivetheproblemapreciseformulationwede

6、fineahoneycombasasetofcongruentconvexpolyhedra,calledcells,fillingthespacebetweentwoparallelplaneswithoutoverlappingandwithoutinter•sticesinsuchawaythat(1)Eachcellhasaface,calledbase(oropening)ononeofthetwoplanesbutdoesnothavefacesonbothplanes.(2)Inthecongruenceoftwo

7、cellstheirbasescorrespondtoeachother.Thedistancebetweentheparallelplanesisthewidthofthehoney•comb.AnaddressdeliveredbeforetheChicagomeetingoftheSocietyonApril24,1964byinvitationoftheCommitteetoSelectHourSpeakersforWesternSectionalMeet•ings;receivedbytheeditorsFebruar

8、y14,1964.468WHATTHEBEESKNOWANDWHATTHEYDONOTKNOW469Letwbethewidthofahoneycomb,vthevolumeofoneofitscellsandatheareaofthebaseofacell.C