triangle triangle shape formed intersection circular having center boundary curve constant width simplest best known curve circle constant width means separation parallel supporting lines independent orientation triangle answer question circle shape cover fall hole called spherical term properly refers curved surface sphere named 19th century german engineer pioneered study machines translating type motion designs shapes known time instance designers gothic church windows map projection study constant width shapes applications triangle include giving shape guitar picks drill bits drilling square holes graphic design shapes signs corporate constant width shapes given width triangle minimum area possible angle corners numerical measures centrally provides largest constant width shape avoiding points closely related shape ratio perimeter diameter perform complete rotation square times touching sides square smallest possible area shapes property covers square rotation process fails cover small fraction square area near corners property rotating square triangle known triangle sequence curves constant width formed regular odd number sides curves shapes coins triangle dimensions multiple ways intersection spheres centers lie regular constant width modified rounding edges form alternatively surface revolution triangle constant width construction triangle constructed directly circles rounding sides triangle circle construction performed compass needing true generally compass construction construction triangle particularly simple step mark arbitrary points plane eventually triangle use compass draw circle centered marked points marked point draws second circle radius centered marked point passing marked point finally draws circle radius center crossing points previous circles passing marked points central region resulting arrangement circles triangle alternatively triangle constructed triangle drawing arcs circles centered connecting constructed intersection centered radius equal length mathematical properties basic property triangle constant width meaning pair parallel supporting lines lines slope touch shape crossing lines distance regardless orientation lines pair parallel supporting lines lines necessarily touch triangle supporting line touch triangle point opposite arc distance width triangle equals radius arc mathematician discover existence curves constant width observe triangle constant width paper presented published entitled studied curves constant width called measures different measures triangle extreme curves constant width triangle smallest possible area curve given constant width area formula constant width method area formula partition triangle inner triangle regions inner triangle arcs forming triangle add areas sets extreme curve constant width maximum possible area circular disk area formula angles pair arcs corners triangle equal possible angle curve constant width additionally curves constant width triangle largest smallest inscribed largest triangle inscribed triangle connecting corners smallest connecting sides triangle consisting points belonging interior larger larger area set diameter points curve constant width triangle symmetry triangle central symmetry triangle curve constant width according different measures central measure ratio area largest centrally shape enclosed curve measure ratio area smallest centrally shape curve triangle centrally shapes determine measures inner curved sides triangle split area curve constant width maximum ratio areas diameter measure bigger triangle curves constant width shapes constant width avoid points largest width triangle axes symmetry parallel coordinate axes half line width approximately root possible circle surrounded circles touch possible arrange seven contact central triangle size maximum number possible curve constant width shape greatest ratio perimeter diameter inscribed triangle measures curves constant width including triangle equal perimeter particular perimeter equals perimeter circle width formula largest inscribed circle triangle width circle triangle formula formula respectively sum equals width triangle generally curve constant width largest inscribed circle smallest circle concentric sum constant width curve optimal packing density triangle plane remains formula density possible double packing shapes best proven upper bound packing density approximately proven highest packing density curve constant width rotation square curve constant width form square shape perform complete rotation staying square times touching sides square triangle minimum possible area axis stay fixed single point instead follows curve formed pieces angles rotating triangle reach points near angles square covers shape slightly rounded corners formed elliptical arcs point rotation corners triangle touch adjacent sides square corner triangle traces curve near opposite square shape traced rotating triangle covers approximately area square original motivation studying triangle showing single point contacts fix object single position existence shows diameter measurements verify object circular cross section overlooking fact played role space shuttle challenger disaster sections rocket launch tested measuring different round shapes cause unusually high stresses factors causing disaster connection inscribed square problem observed triangle provides example constant width shape regular sides inscribed regular described small modification shape preserves constant width prevents regular inscribed result dimensions cylinder shape cross section applications reaching corners types machinery shape triangle based property able rotate square square drill bit shape triangle modified form cutting surfaces mounted special allows bit having fixed centre rotation drill hole nearly square patented similar drills invented earlier drill octagonal holes robotic vacuum cleaner shape based triangle order ease cleaning dust corners rooms rolling cylinders class applications triangle involves cylindrical objects triangle cross section manufactured shape traditional round barrels usually promoted comfortable encouraging proper grip likely roll tables center gravity moves rolling triangle curves constant width roll makes poor wheel roll fixed center rotation object triangle cross sections roll smoothly axle attached triangle wheels bounce times revolution concept science fiction short story titled wheel bicycle floating frame supported rim triangle shaped wheel built demonstrated chinese inventor inspired shape mechanism design class applications triangle involves mechanical convert rotation fixed axis motion mechanisms studied assistance built approximately models mechanisms involved triangle models pioneering scientific investigations motion models lost collected including based triangle use mechanism design work instance steam engines early shape triangle application principle arises film application necessary advance film motion frame film stops fraction second lens quickly film moved frame mechanism rotation triangle square create motion pattern pulls film quickly new frame film motion frame projected engine shaped triangle cited example triangle curved sides somewhat triangle constant width architecture gothic architecture beginning late 13th century early 14th century triangle forms frequently windows window architectural decorations instance english gothic architecture shape associated decorated period geometric style continuing style context shape frequently called spherical triangle usual mathematical meaning spherical triangle triangle surface sphere shape commonly architecture use gothic church architecture shape triangle seen symbol trinity act opposition form circle triangle styles architecture instance shape plan fortification modern high rise building cologne germany built triangle cross section circular shape core gives varied depths rooms building early application triangle circa possibly followers direction world map spherical surface earth divided flattened shape triangle similar maps based triangle published objects guitar picks employ triangle shape combines sharp point provide strong wide tip produce warm points shape usable easier orient wears quickly compared pick single tip following suggestion antennae radio wave astronomical observatory hawaii arranged placing antennae curve constant width causes observatory spatial resolution directions provides circular observation beam curve constant width triangle leads uniform coverage plane transform signal array antennae moved triangle different observations according desired angular resolution observation precise placement antennae network places constructed observatory departs preferred triangle shape shape possible given site signs shield shapes signs corporate feature rounded specifically corporate logo belgian oil company major operations europe north_america africa triangle merger corporate logo framed triangle south pointing compass design company total identity won san year award triangle logo nature according plateau laws circular arcs dimensional soap bubble clusters meet angles angle found corners triangle based fact possible construct clusters bubbles form triangle shape isolated crystal form triangle basic triangle shape formed precipitation water system presence triangular curves constant width smooth sharp corners obtained points fixed distance triangle triangle include surfaces dimensions curves constant width sides sets provide extreme examples inequality width diameter dimensional version intersection spheres radius centered regular length called surface constant width surface constant width called replacing edge arcs curved surfaces surfaces rotation circular arc alternatively surface revolution triangle symmetry axes forms surface constant width minimum volume known surfaces revolution given constant width triangle regular odd number sides yielding shapes constant width boundaries formed circular arcs equal length constant width shapes allows use coins coin operated machines instance united_kingdom coins shape canadian dollar coin uses sides similar methods arbitrary simple curve constant width width equals diameter given resulting shape consists circular arcs sides constructed linear time drawn compass regular based odd number circular arc sides possible construct constant width shapes based irregular number sides sets sets defined convex hulls triangle circular arcs centered triangle spanning angle triangle equal equal length triangle radius small sets triangle radius large possible equal corresponding triangle shape width diameter radius largest possible circle contained shape inequality formula inequality equality sets showing improved related figures classical presentation set overlapping circles central region representing elements belonging sets takes shape triangle circles form standard drawings rings mutually linked rings realized geometric circles parts circles form figure overlapping form symbol triangle center circles form rings circular arcs form knot relatives triangle arise problem finding minimum perimeter shape fixed area includes specified points plane wide range choices area optimal solution problem curved triangle sides circular arcs equal particular points area triangle triangle optimal enclosure circular circular arc edges including triangle shapes curve type triangle curves replacing triangle concave convex composed circular arcs formed rolling circle times radius shapes curved sides include formed triangle triangle interpreted image spherical triangle angles spherical triangle parameters bounded great circle arcs surface sphere tile sphere reflection
Reuleaux triangle