Page 1
1
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.
Page 2
2
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].
Page 3
3
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
Page 4
4
Figure6
Ademonstrationofthe12-edgedcuberevealingtheregularhexagon
Figure7
The12edgesofthecubesupportEuler’sformula.
resonatewithrecreationalmathematiciansandmechanicalpuzzleenthusiasts,alike.“Europa”stampsareissuedannuallybyparticipatingEuropeanpostaladministrations,andhaveanagreed-uponcommontheme.Thethemefor2015was“oldtoys,”andLithuania’sentrywasasetoftwoburrpuzzles,oneofthembeingintheshapeofacube[Figure8].
Figure8
Composedoftwenty-foursticks,thisburrpuzzlecubestillhas12edges.
Page 5
5
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,
Page 6
6
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
Page 7
7
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
Page 8
8
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.
Page 9
9
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)