Euler\'s Gem: The Polyhedron Formula and the Birth of Topology

LeonhardEuler\'spolyhedronformuladescribesthestructureofmanyobjects-fromsoccerballsandgemstonestoBuckminsterFuller\'sbuildingsandgiantall-carbonmolecules.YetEuler\'sformulaissosimpleitcanbeexplainedtoachild.\"Euler\'sGem\"tellstheilluminatingstoryofthisindispensablemathematicalidea.FromancientGreekgeometrytotoday\'scutting-edgeresearch,\"Euler\'sGem\"celebratesthediscoveryofEuler\'sbelovedpolyhedronformulaanditsfar-reachingimpactontopology,thestudyofshapes.In1750,EulerobservedthatanypolyhedroncomposedofVvertices,Eedges,andFfacessatisfiestheequationV-E+F=2.DavidRichesontellshowtheGreeksmissedtheformulaentirely;howDescartesalmostdiscovereditbutfellshort;hownineteenth-centurymathematicianswidenedtheformula\'sscopeinwaysthatEulerneverenvisionedbyadaptingitforusewithdoughnutshapes,smoothsurfaces,andhigherdimensionalshapes;and,howtwentieth-centurymathematiciansdiscoveredthateveryshapehasitsownEuler\'sformula.Usingwonderfulexamplesandnumerousillustrations,Richesonpresentstheformula\'smanyelegantandunexpectedapplications,suchasshowingwhythereisalwayssomewindlessspotonearth,howtomeasuretheacreageofatreefarmbycountingtrees,andhowmanycrayonsareneededtocoloranymap.Filledwithawho\'swhoofbrilliantmathematicianswhoquestioned,refined,andcontributedtoaremarkabletheorem\'sdevelopment,\"Euler\'sGem\"willfascinateeverymathematicsenthusiast.,