Rational Equiangular Polygons

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1 Apped Mathematcs Pubshed Oe October 03 ( Ratoa Equaguar Poygos Marus Muteau Laura Muteau Departmet of Mathematcs Computer Scece ad Statstcs State Uversty of New Yor Oeota USA Ema: MarusMuteau@oeotaedu LauraMuteau@oeotaedu Receved August 3 03; revsed September 3 03; accepted September 0 03 Copyrght 03 Marus Muteau Laura Muteau Ths s a ope access artce dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orga wor s propery cted ABSTRACT The ma purpose of ths ote s to vestgate equaguar poygos wth ratoa edges Whe the umber of edges s the power of a prme we determe smpe ecessary ad suffcet codtos for the exstece of such poygos As speca cases of our vestgatos we sette two coectures vovg arthmetc poygos Keywords: Equaguar Poygo; Arthmetc Poygo Itroducto A smpe way of extedg the cass of reguar poygos s to mata the cogruece of vertex ages whe o oger requrg that the edges be cogruet I ths geeraty the ewy obtaed equaguar poygos are ot a that terestg gve oe ca fd pety of such (osmar) poygos wth a gve umber of edges Ideed drawg a parae e to oe of the edges of a reguar poygo through a arbtrary pot o a adacet edge yeds a trapezod ad a ew equaguar poygo wth the same umber of edges as the ta oe (see Fgure ) However f we aso requre that a edge egths be ratoa umbers ad that at east two of these umbers be dfferet (thus excudg reguar poygos) geera such equaguar poygos may ot eve exst For exampe f we start wth the reguar petago P ad draw the parae QQ to as Fgure ad f ad PQ are ratoa umbers the except for QQ a edge egths of the equaguar petago Q QP3P 4 P 5 are ratoa However QQ PQ cos 5 s rratoa Whe ths by o meas proves that equaguar petagos wth Fgure New equaguar petagos from od ratoa edges must be reguar t gves some credbty to the o-exstece cam above A terestg vestgato of equaguar poygos wth teger sdes s provded [] where the author cosders the probem of tg these poygos wth ether reguar poygos or other patter bocs of teger sdes I partcuar he pots out that every equaguar hexago wth teger sdes ca be ted by a set of cogruet equatera trages aso of teger sdes ad aso proposes a geera tg coecture wth a exteded tg set O the other had f oe o oger requres teger edges but ass that the vertces be teger attce pots the oy equaguar poygos that w do are squares ad octagos (see [3]) Further restrctg the cass of equaguar poygos wth teger sdes [4] R Dawso cosders the cass of arthmetc poygos e equaguar poygos whose edge egths form a arthmetc sequece (upo a sutabe rearragemet) ad shows that the exstece of arthmetc -gos s equvaet to that of equaguar -gos whose sde egths form a permutato of the set I addto some terestg exstece as we as o-exstece resuts are obtaed but the cassfcato probem for arthmetc poygos wth a arbtrary umber of edges s eft ope I ths ote we address the more geera probem of determg a equaguar poygos wth ratoa edges ad as a speca case we sette the cassfcato probem above Premares Frst we derve a ecessary ad suffcet codto for Copyrght 03 ScRes

2 M MUNTEANU L MUNTEAN 46 the exstece of cosed poygoa paths terms of edge egths ad age measures Proposto Let ad be postve rea umbers wth There exsts a cosed poygoa path P (wth P P oreted coutercocwse) havg edge egths ad the measures of the ages * 3 formed by wth 3 wth wth equa to respectvey f ad oy f ad e e e 0 () () for some teger Proof Assumg that such a poygo exsts et z be the compex umber assocated to P As the vector s the - mutpe of the rotato of P through (see Fgure ) we have z z e z z Based o the same type of argumet regardess of the oretato of trages P we have z z e zz z3 z e z z e zz z z e z z e z z z z e z z z z e Combg these reatos wth yeds z z z z z z 3 0 z z e z z e z z e 0 thus provg reato () from the cocuso Reato () foows easy as a cosequece of the ast reato the set of reatos above Coversey to prove the exstece of a cosed poygoa path wth gve satsfyg () observe that startg wth a arbtrary pot P we ca aways cosder the pots P P such that wth the excepto of ad the measure of the ages formed by wth ad wth a edge egths ad age measures are as eeded We w prove that the cosed poygoa path P satsfes the requremets To do so f we et ad deote the measure of the age formed by wth by ad wth by the we eed to show that ad By appyg the drect mpcato to our poygoa path (wth edge egths ad age measures ) we have e e e 0 Equvaety 3 e e e (3) By factorg out e ad appyg the moduus o both sdes of the equaty above we have 3 e e However the same type of operatos ca aso be apped to the reato our hypothess (vovg ) to obta 3 e e But the based o the two formuas above Now factorg out e ad repacg by (3) we have e e But we aso have 3 e (4) Fgure The poygo P * A ages are measured coutercocwse from the frst to the secod refereced ray Copyrght 03 ScRes

3 46 M MUNTEANU L MUNTEAN e e 3 e (5) Comparg reatos (4) ad (5) t foows that To show that et s ote that reato () apped to mpes for some teger P By hypothess Combg the two reatos above fshes the proof 3 Equaguar Poygos If we cosder a covex equaguar go the wth otatos as the prevous secto we have I addto f we et e the based o Proposto we obta Theorem Gve there exsts a covex equaguar -go wth sde egths (sted coutercocwse) ff 3 0 Defto A ratoa poygo s a poygo a of whose edge egths are ratoa umber Observato The edges of a o-covex equaguar poygo ca be rearraged to form a covex equaguar poygo so we w oy cocetrate o the atter As a cosequece of Theorem we obta Proposto Let e ad et N be the degree of the cycotomc poyoma X There exsts a covex ratoa equaguar -go wth edge egths (ordered coutercocwse) ff the foowg equates are satsfed: where a a 0 N a 0 N a 0 N N N are defed by N a 0 for a N 0 N Proof Let us frst ote that the defto of a maes N sese Ideed sce forms a bass of for a fxed N we ca defe a to be the coeffcets of ths bass For each N f we repace the N equaty from Theorem by we obta N a 0 a 0 N N N N a N 0 0 By reorgazg the terms the formua above becomes a a N N N N an 0 N But the we get a poyoma of degree N wth ratoa coeffcets havg as a root Ths s oy possbe ff a the coeffcets are zero thus provg the proposto Observato By fxg N the codtos the proposto above geerate a system of equatos a s N s s N wth N equatos ad N varabes N Comparg the umber of equatos ad the umber of varabes we obta three cases depedg o whether N N or To better uderstad the three cases above we have Lemma For ay postve teger we have the foowg ) ff p for some odd prme p ad some postve teger ) ff for some postve teger 3) ff m where the oegatve teger ad the odd teger mm 3 are such that ether or f 0 the m s the product of the powers of at east two dstct prmes Proof Sce the thrd case s the compemet of the frst two t s eough to prove the frst two cases So et p r where p p p are dstct prmes p p r r ad are oegatve tegers To prove () observe that the equaty s equvaet to r r p p pr p p p p p r or pp pr p p pr To show that has the desred form et us assume by cotradcto that r But the sce p ad p 3 Copyrght 03 ScRes

4 M MUNTEANU L MUNTEAN 463 we have p p 6 or equvaety p p p p 0 Ths mpes pp pp pp pp Together wth p p 3 r the secod equaty above yeds whch cotradcts the hypothess Thus r But the we must aso have p > sce otherwse Coversey t s easy to see that f p p the For () by cosderatos smar to the oes above we must have r Sce by () we caot have p t must be that p Aso t s cear that f the Next we cosder covex ratoa equaguar poygos each of the three cases gve by the emma For the overdetermed case we have the foowg: Proposto 3 If are the egths of the edges of a covex ratoa p -go wth p > prme the the poygo s equaguar ff Proof Let p pp p pp p p p p p p p p X X X p p p p X be the mma poyoma of X (see [5] page 3) I order to appy Proposto we eed to wrte for a N p pp p as a ear combato wth teger coeffcets of N Startg wth the equato gve by ts mma poyoma ad mutpyg by N we have N p p N p p p p 3p over pp pp pp Thus a f p p pp ad a 0 otherwse Wth these vaues of a the cocuso foows Cosequece Ay ratoa equaguar poygo wth a prme umber of edges s reguar Proof Ths foows based o Observato ad the case Proposto 3 Observato 3 The cosequece above proves coecture 6 from [4] For the fuy determed case we have the foowg characterzato: Proposto 4 Gve a covex ratoa poygo whose umber of edges s a power of two the poygo s equaguar ff opposte edges are cogruet Proof Let be the umber of edges of the poy- X X t foows that go Sce Thus the reato from Theorem becomes or 0 But the s a root of a ratoa poyoma of degree ess tha that of Ths s oy possbe f the poyoma s detcay zero whch mpes the cocuso As a cosequece of the proposto above we obta a dfferet proof of Theorem 3 from [4] Cosequece There does ot exst a equaguar -go wth teger edge egths a dstct For the uderdetermed case gve the ac of a smpe formua for X ths case we w oy cosder the foowg exampe Lemma 5 are the edge egths of a covex equaguar 5-go wth the edges ordered coutercocwse ff ad Proof I ths case X X 5 X 4 X By ettg 5 X X X 5 we have Based o these reatos ad Proposto we must have If we et c c c c c Copyrght 03 ScRes

5 464 M MUNTEANU L MUNTEAN ad d d d the reatos above become d d d c c d 0 c c d d d d c c c c d d c d d c Ceary these reatos are equvaet to c c c c c ad d d d d d thus provg the emma 4 Arthmetc Poygos Foowg the termoogy from [4] a poygo s sad to be arthmetc f t s equaguar ad ts edge egths ( some order) form a otrva arthmetc sequece As show the same paper a arthmetc -go exsts ff there exsts a equaguar poygo wth edge egths ( some order) I ths secto we fd a ecessary ad suffcet codto for the exstece of arthmetc poygos terms of the umber of edges Frst we have the foowg: Cosequece 3 There are o arthmetc poygos whose umber of edges s the power of a prme Proof Ths foows as a cosequece of Propostos 3 4 ad Observato Oe case whe arthmetc poygos do exst s provded by the exampe beow Exampe There exsts a (covex) arthmetc 5-go Proof If we seect the the codtos Exampe are satsfed sce ad Observato 4 The proposto above provdes a couterexampe to coecture 7 from [4] camg that o arthmetc -gos exst f s odd The exampe above suggests the foowg: Theorem There exsts a arthmetc -go f ad oy f s ot the power of a prme e has at east two dstct prmes factors Proof By Cosequece 3 t s eough to prove the coverse So et s cosder pq for some postve tegers pq Sce s ot the power of a prme p ad q ca be chose to be reatvey prme If deotes a prmtve pq- th root of uty the p p q p 0 (6) ad q q p q (7) Mutpyg reatos (6) by a a a p ad (7) b by b p b q we have ad p 0 q a p a 0 (8) 0 bq b p 0 (9) Let us ow observe that every teger betwee ad pq appears exacty oce as a expoet both (8) ad (9) due to the fact that p ad q are reatvey prme If we add a p equatos (8) ad a q equatos (9) we obta p q q p ap bq a b p 0 a0 b0 Wheever a p b q the sum of the correspodg coeffcets ab p s a teger betwee ad pq Moreover dfferet a ad b wth a p b q geerate dfferet vaues for ab p because p ad q are reatvey prme Sce there are exacty pq pars ab the vaues of abp w represet a permutato of the set pq 5 Cocusos I ths ote we determed a ratoa equaguar poygos whose umber of sdes a prme power Athough we aso determed a ratoa equaguar 5-gos the geera probem remas ope I addto we provded a compete characterzato of arthmetc poygos As a terestg appcato we ote that as metoed [6] there s a ce correspodece arsg from the Schwarz-Chrstoffe trasformatos betwee equaguar -gos ad certa areas determed by bary forms of degree wth compete factorzatos over It woud be terestg to vestgate the cosequeces of our resuts the aguage of bary forms REFERENCES [] D Ba Equaguar Poygos The Mathematca Gazette Vo 86 No pp [] P R Scott Equaguar Lattce Poygos ad Semreguar Lattce Poyhedra Coege Mathematcs Joura Vo 8 No pp [3] R Hosberger Mathematca Damods The Mathematca Assocato of Amerca Washgto DC 003 [4] R Dawso Arthmetc Poygos Amerca Mathe- Copyrght 03 ScRes

6 M MUNTEANU L MUNTEAN 465 matca Mothy Vo 9 No 8 0 pp [5] P Samue Agebrac Theory of Numbers Kershaw 97 [6] M A Bea Bary Forms Hypergeometrc Fuctos ad the Schwarz-Chrstoffe Mappg Formua Tras- actos of the Amerca Mathematca Socety Vo 347 No 995 pp Copyrght 03 ScRes