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Philatelic12sforG4G12MichaelTanoff,Kalamazoo,MI

diandmi@sbcglobal.netWrittenfortheoccasionofGatheringforGardner12

March31–April3,2016Atlanta,GA,USA.

InMartinGardner’sSixthBookofMathematicalGamesfromScientificAmerican

(W.H.FreemanandCompany,1971),MartinGardnerintroducedhisreaderstoPatrickO’Gara,the(fictitious)mathematicalmailman.OneofO’Gara’ssubcategoriesofhisrecreationalmathematicspassionwasmathematicalphilately(stampcollecting),andhesharedwithMr.Gardnersomeofthestampsfromhiscollection,includingaGreekpostagestampfrom1955illustratingthePythagoreantheorem,andstampsfromavarietyofcountrieslikeFrance,Russia,andIrelandhonoringmathematicianssuchasPascal,Laplace,Euler,Chebyshev,andHamilton.ThroughO’Gara,Mr.Gardnerintroducedhisreaderstotheconceptoftopicalphilately,andsuppliedseveralmathematicalpostagestampreferencesforreaderswhomayhavewantedtoexplorethisareafurther.InhonorofG4G12,thetwelfthgatheringincelebrationofthelifeofMartinGardner,andinMr.Gardner’smemory,thispaperpresentsphilatelicitems—postagestamps,postmarks,postcards,etc.—ofmathematicalcontentrelatedspecificallytothenumbertwelve.

Ofcourse,thethemeoftwelveappearsonanypostagestampwhoseillustration

containsastandardanalogclock.AfunexampleofthisappearsonaSwissstampfrom2012[Figure1],onwhichaclockis“deconstructed”intoitselementsbySwisscomedianand“deconstructionartist”UrsusWehrli.

Figure1

ClockonaSwissstampdeconstructedintoitsbasicelements,includingits12major(hour)tickmarks

ThetwelvetribesofIsraelappearonseveralstampissuesbytheIsraeligovernment.Manycountrieshaveissuedstampsfeaturingthetwelvesignsofthezodiac.ThetwelveatomsonabenzenemoleculeappearonGermanandEastGermanstampsfrom1964and1979,andaBelgianstampfrom1966[Figure2].Regrettably,however,noneoftheseexamplesof“12’s”onstampswouldbeofparticularinteresttoPatrickO’Gara.For“twelve”stampswithamoremathematicalflavor,weturntogeometricfigures,bothtwo-andthree-dimensional.

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Figure2

Threestampsillustratingthe12atomsinabenzenemolecule

TheChinesemathematicianLiuHui(c.260CE)determinedupperandlowerboundsforπbycalculatingtheareasofconcentric96-and192-gons.Apagefromhisbook,JiuzhangSuanshu,TheNineChaptersOntheMathematicalArt,appearsonaMicronesiastampfrom1999[Figure3],partofalargersheetofpostagestampscelebratingthescienceandtechnologyofancientChinaduringthefirstmillennium.ThepageillustratesLiuHui’smethodofapproximatingπusingtheregulardodecagon,the12-sidedregularpolygon,asanexample.Afewhundredyearslater,ZhuChongzhi(429–500CE),alsofromChina,bestedLiuHui’supperandlowerboundsforπbyapproximatingacirclewitha213×3-sidedregularpolygon.Hiswork,alsoatthestageofa12-sidedregularpolygon,isillustratedona2015stampfromHongKong[alsoFigure3],partofafour-stampsethonoringancientChinesescientists.

Figure3

The12-sidedregulardodecagon,usedtoapproximateπ

AustriancomposerandmusictheoristJosefMatthiasHauerdevelopedamethodofcomposingmusicusingtwelveequispacedpitchesortones.Allpossiblecombinationsofthesetonescanbeenumerated,graphically,usingadodecagon,asontheAustrianstampfrom1983,commemoratingHauer’s100thbirthday[Figure4].

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Figure4

The12-sidedregulardodecagonusedtoillustrateallpossiblecombinations

of12equispacedmusicalpitches

Whilethenumberofstampsfeaturingtwelve-sidedpolygonsislimited,theofferingsincreasewhenweturntothreedimensionsandthePlatonic(regular)solids(discussedbyMartinGardnerinhissecondcollectionof“MathematicalGames”columnsinThe2ndScientificAmericanBookofMathematicalPuzzles&Diversions,SimonandSchuster,1961),which,allexceptforthetetrahedron,arerichinthenumbertwelve! Wemaybeginwiththecube,whichhastwelveedges.Whilemanystampsfeaturecubes(orothertwelve-edgedrectangularsolids),suchasthe1970ChildWelfarestampfromtheNetherlands[Figure5],issuedaspartofasetoffivecubesofdifferentcolors,aparticularlyentertainingcubeisfeaturedontheAustrianstampfrom1981[alsoFigure5],issuedinhonorofthe10thInternationalAustrianMathematicalCongressheldthatyear.

Figure5

Cubes,whetherrealorimpossible,have12edges.AnotherspecialdepictionofthecubeappearsonaChinesepostcardfrom1990[Figure6]issuedinhonorofthe31stInternationalMathematicalOlympiad,hostedbyChinathatyear.Theillustrationonthepostcarddemonstrateshowonemaybisectacubetoyieldafacethatisaregularhexagon.AnEastGermanfirstdaycancellationfrom1981usesacubetoillustrateEuler’sformulafornon-self-intersectingpolyhedra,faces+vertices–edges=2[Figure7].(Moreonthaticosahedron,shortly.)Arecentlyissuedcube-on-stampwill

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Figure6

Ademonstrationofthe12-edgedcuberevealingtheregularhexagon

Figure7

The12edgesofthecubesupportEuler’sformula.

resonatewithrecreationalmathematiciansandmechanicalpuzzleenthusiasts,alike.“Europa”stampsareissuedannuallybyparticipatingEuropeanpostaladministrations,andhaveanagreed-uponcommontheme.Thethemefor2015was“oldtoys,”andLithuania’sentrywasasetoftwoburrpuzzles,oneofthembeingintheshapeofacube[Figure8].

Figure8

Composedoftwenty-foursticks,thisburrpuzzlecubestillhas12edges.

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Thenextregularsolidistheoctahedron,whichalsohastwelveedges!ThecityofNancyinFrancechosetheoctahedronastheirsymbolforthepostmarkadvertisingtheirhostingoftheInternationalTradeFairandExhibitionsin1968[Figure9],andStockholmusedtheoctahedronasasymbolintheircommemorativepostmarkfortheInternationalConferenceonCoordinationChemistryheldinStockholmin1962[alsoFigure9].

Figure9

12edgesonaregularsolid?Notacube,butanoctahedron.AsetoffivestampsissuedbySwedenin2012illustratethespace-tilingcapabilitiesoftheoctahedron[Figure10].

Figure10

The12-edgedregularoctahedronhasspace-tilingcapabilities.

Shiftingourfocusfromedgestofaces,thedodecahedron,thenextPlatonicsolid,hastwelvefaces.TotheancientGreeks,thetetrahedron,cube,octahedron,andicosahedronrepresentedthefourbasicelementsoffire,earth,air,andwater.Thedodecahedron,however,wasconsideredarepresentationoftheentireuniverse,whichcouldexplainitsprominentplaceintheMacaucosmologystampsfrom2004[Figure11],partofMacau’songoingannual“scienceandtechnology”series.(AllfiveofthePlatonicsolidsarepicturedonthestamponthemini-sheetofthesameissue.)Mostinterestingly,

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somemoderncosmologytheoriesarereturningtotheancientGreeks’notionthattheuniverseisdodecahedralinshape!(See,forexample,Luminet,J.P.,etal.,"Dodecahedralspacetopologyasanexplanationforweakwide-angletemperaturecorrelationsinthecosmicmicrowavebackground,"Nature425(6958):593–5,2003.)

Figure11

Doestheuniverseexhibitthe12-sidedsymmetryofthedodecahedron?

Forreasonsunknowntothisauthor,theRepublicofChinausedthedodecahedronasthevehicleforcelebratingthe40thanniversaryofitsLaborInsuranceSystem,onastampissuefrom1990[Figure12].In1964,Spainissuedaseriesoffourteenstampstocommemoratetwenty-fiveyearsofpeace.The1.50pesetastampfocusedon“modern”architecturethatcouldbeachievedduringpeacetime,andfeaturedadodecahedralbuilding[alsoFigure12].Andformuchmorepurelymathematicalreasons,theWorldMathematicalYearstampfromMonacoin2000prominentlyfeaturedadodecahedronamongothersymbolsandfiguresassociatedwiththegoldenratio[alsoFigure12].

Figure12

Whethersymbolic,applied,orpurelymathematical,dodecahedronshave12faces.

Thedodecahedronhasbeenusedasasymboltoconveythestudyoffundamentalpropertiesofsolids,suchaswhenitwasusedinaSwedishcommemorativecancellationfor

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the1952InternationalSymposiumontheReactivityofSolids[Figure13],orbyFrancetocommemoratethe3rdInternationalCongressonCrystallographyheldattheSorbonneinParisin1954[alsoFigure13].Andnotsurprisingly,thedodecahedronwasusedasthefirstdayofissuepostmarkfortheMacaucosmologysetofstamps[alsoFigure13],mentionedearlier.

Figure13

12-faceddodecahedronsusedinavarietyofphilatelicapplications Lastly,weturntotheicosahedron,whichwithitstwentyfacesandthirtyedgeshasonlytwelvevertices.WecaughtaglimpseofanicosahedroninFigure7,partoftheillustrationforanEastGermanstampfrom1983,commemoratingthe200thdeathanniversaryofthegreatSwissmathematician,LeonardEuler[Figure14],andusedasanexamplewithwhichtodemonstratetheaforementionedEuler’sformula.Thefieldofvirologymakesclaimthatarrangementsofvirussubunitsaregovernedbyicosahedralsymmetry.Hence,theicosahedronwasusedapartofthedesignofa1984Japanesepostagestampissuedinhonorofthe6thInternationalVirologyCongress[alsoFigure14].

Figure14

Whetherinmathematicalanalysisorappliedtothefieldofvirology,theicosahedron’s12verticesaresignificant.

Wemayfindanotherphilatelicinstanceoftheicosahedronbeingusedtorepresentthefieldofvirology,butforthis,weneedtolookintheselvageofasheetofGermanstampsissuedin2010inhonorofthe100thanniversaryofthefoundingoftheFriedrichLoefflerInstitute,theNationalInstituteofAnimalHealthinGermany[Figure15].(Thestampitself

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illustratessomevirus-likemicrobe.)AnotherexampleofanicosahedronappearingonlyintheselvageofastampissueisforthecommemorativeissuedbyGermanyin2008honoringthe300thbirthdayofgoldsmithandinventorofscientificinstrumentsWenzelJamnitzer[alsoFigure15].ThebeautifulgoldicosahedrononthecornerselvagepieceofthestampsheetisjustoneofJamnitzer’smanyworksfeaturingthePlatonicsolidsandtheirstellatedcounterparts.

Figure15

Elusivephilatelicicosahedronsandtheir12verticesturnupinunexpectedplaces,includingavertex!

Another“viral”philatelicexampleoftheicosahedronisonthepostagemeterstripadvertisingtheanti-viraldrugZovirax(genericnameacyclovir,oraciclovirinDutch)[Figure16].Lastly,borrowingarolefromoneofthedodecahedronsinFigure13,theicosahedronwasusedinpostmarkscommemoratingthe11thand12thEuropeanCrystallographicMeetingsin1988and1989inViennaandMoscow[Figure17].

Figure16

The12-pointedvirusstructureinthefaceoftheanti-viraldrug

Foranyreaderinterestedinfurtherpursuitsinmathematicalphilately,theauthorrecommendsthattheyconsidermembershipintheMathematicalStudyUnitoftheAmericanTopicalAssociation.Thestudyunit’swebsiteisatwww.mathstamps.org.Theunit’sjournal,Philamath,ispublishedquarterlyinJanuary,April,July,andOctober.

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Figure17

The12-corneredicosahedron,likeitsdodecahedroncousin,hasbeenusedasasymbolforthestudyoffundamentalpropertiesofsolids.

(Note:Stamps,postmarks,andotherphilatelicitemsappearinginthefiguresarenotsizedtorelativescale.)Afar-from-comprehensivelistofreferencesonmathematicalphilately:

Gardner,M.,MartinGardner’sSixthBookofMathematicalGamesfromScientificAmerican,W.H.FreemanandCompany,SanFrancisco,(1971)[SeveralreferencesareprovidedattheendofChapter23,“O’Gara,theMathematicalMailman.”]

Miller,J.,“ImagesofMathematiciansonPostageStamps,”

http://jeff560.tripod.com/stamps.html

Schaaf,W.L.,MathematicsandScience:AnAdventureInPostageStamps,NationalCouncilofTeachersofMathematics,Virginia,(1978)

Wilson,R.J.,StampingThroughMathematics,Springer,NewYork,(2001)