THIS LECTURE looks at bell ringing (the posh name is Tintinnalogia) which as. WE NORMALLY think of a bell as hanging mouth down. If we swing the bell,

Transcription

1 7 Bells THIS LECTURE looks t ell ringing (the posh nme is Tintinnlogi) whih s n orgnize tivity hs een roun for long time. Inee, n importnt ook y Stemn on the sujet ws pulishe in 1668 (two yers fter the Gret Fire of Lonon). As well s illustrting vrious sequenes of hnges we lso look t some of the mthemtil ies ehin the sequenes suh s permuttions n other simple spets of group theory. Rouns n plin hunting WE NORMALLY think of ell s hnging mouth own. If we swing the ell, then it will strike t regulr intervls etermine y its nturl frequeny of swinging. Like penulum, this oes not vry muh with the size of the swing ut vries with the size of the ell. If tower hs severl ells n they re tune to sy C, E, G, C', then however they re soune they will e hrmonious sine the mjor ommon hor CEGC' is hrmonious. Moreover, they will e repete t ifferent intervls euse the rte of swinging will e ifferent. If they re not in hrmony or we more ells n ontinue to swing them mouth own then the result will e ophoni mess. In English ell towers, the ells re soune with the mouth nerly upwrs. The ell swings nerly through 360 egrees with strikes t eh en of the swing in either iretion. As onsequene, the ringer is le to hol or hsten the ell fter eh strike n so ontrol the repetition of the strikes. Note tht English prtie is not to use ells s just nother musil instrument on whih meloies n e plye or to ply multiprt hors ut to explore the elegne of vrie sequenes of single notes. So the trition hs risen tht ells soun est when they re rung in sequene with uniform gp etween eh soun. The si sequene is solle rouns in whih the ells re rung in orer over n over gin strting with the highest n ening with the lowest. If there were four ells with notes C, D, E, F, then the orer of ringing rouns woul e FEDCFEDCFEDC... We usully numer the ells with 1 eing tht with the highest note. So the rouns with four ells re written in the strt form It is onventionl to put spe t the en of eh row thus For tehnil resons lternte rows re little ifferent. This is euse ells strike twie on eh yle of swings. One is lle the hnstroke n the other the kstroke. There is puse fter every other row equl to the time of one strike. This puse mkes it esier to unerstn the pttern of ringing. Springer Interntionl Pulishing Switzerln 2016 J. Brnes, Nie Numers, DOI / _7 137

2 138 Nie Numers Digrm for plin hunting on three ells. Just ringing rouns is it oring so the ie of mking hnge fter eh row ws introue. The simplest hnge is where two ells hnge ples. Thus 1234 might e followe y 1324 in whih the mile two ells hve hnge ples. It is importnt tht ell never move y more thn one ple sine this woul e too hr for the ringer. Moreover, the heviest ell (in this exmple 4) is often left in the sme ple eh time euse it is hrer to hnge the position of the heviest ell. This lso entutes the rhythm. With this requirement there re only 6 wys (= 3!) to ring the four ells. The gol is to ring these 6 rows without repetition n only with jent ells hnging ples. There re only two wys of oing this n one is the reverse of the other. Thus we hve Note tht with 4 ells (n keeping the lst fixe) there re only two moves tht n e me t ny point. Either we interhnge the first two ells or we interhnge the mile pir. One hnge will tke us k to the previous row so there is only one wy forwr. So hving eie on the first hnge the rest follow without hoie. Chnges re often epite y zigzg igrm s shown ove. Iniviul ells n e shown seprtely or superimpose. Different olours re use for lrity. (The pth of ell is often known s lue line.) The letters on the left ientify the hnge eing me. Chnge interhnges the first two ells n hnge interhnges the mile two ells. Note refully tht eh ell follows the sme pttern: in the sme ple for one row, then shifting left for two rows, the sme ple for one more row, n then shifting right for two rows. But they iffer in tht they strt t ifferent points in the yle rther like singing rouns. This very simple pttern is lle hunting. With more ells there re mny more omintions n more wys to ring them ll. Thus with 8 ells ( ommon numer n otve) n the lst fixe there re 5040 (= 7!) possile rows. To ring ll these typilly tkes out three hours n is known s ringing full pel. Bell towers often hve ors listing events when full pels were rung.

3 D A C B A B D C A B With 5 ells (n keeping the lst fixe) there re 24 (= 4!) possile rows. Moreover, it is possile to interhnge two pirs of ells in the sme hnge. Thus n so on The first hnge interhnges two pirs of ells. Suh hnges re known s oules. Similrly, with 7 ells (n gin keeping the lst fixe), we n interhnge three pirs of ells. Suh hnges re known s triples. In mthemtis, the ifferent rrngements of sy 123 re known s permuttions. Permuttions re either even or o oring s n even or o numer of single interhnges re require to give the permuttion strting from 123. Alternte permuttions in re thus even n o euse eh is otine from its preeessor y single interhnge. Note refully tht if we hve enough ells so tht we n o oules, then we nnot go through ll possile rows y just using oules euse tht woul only over ll the even permuttions ut not the o ones. Let us return to three ells (the fourth ell eing fixe) n onsier the mthemtis it more. There re two possile inepenent opertions (1, 2) interhnge first two (2, 3) interhnge mile pir 7 Bells 139 We write (1, 2) mening the ell in position 1 goes to position 2 n 2 goes to position 1. In generl permuttion theory (1, 2, 3) woul men tht the ell in position 1 goes to 2, tht in position 2 goes to 3, n tht in position 3 goes to 1. But we on t llow tht in ell ringing euse the ell in position 3 woul move two ples. Similrly we re not llowe (1, 3). However tht oul e hieve y oing, then, n then gin written. So it is not inepenent. Moreover, the sme oul e hieve y oing. Note refully tht (tht is o n then ) is not the sme s. The opertions o not ommute. This is not unusul, the opertions on squre of rotting it y right ngle n flipping it out the vertil xis o not ommute either the result epens upon the orer in whih they re one s shown elow. rotte D C flip A D B C rotte A B D C flip Rottion n flipping of squre o not ommute. Strting from the entre, flip then rotte gives fr left squre, rotte then flip gives fr right one.

4 140 Nie Numers Grph showing the hnges on three ells. We n o vrious other interesting its of lger with these opertions. For exmple note tht oing n then gin hs no effet. We sy tht 2 = 1. Similrly 2 = 1. Furthermore, the whole sequene of six hnges is or () 3 n this returns us to the strting point. So () 3 = 1. However, we hve seen tht =, so = () = () = Now 2 = 1 n 2 = 1, so finlly we get () 3 = = = = = 1 whih onfirms tht gets us k to the eginning. Another wy of epiting the sequene is y rwing grph with points representing eh row n lines etween them ientifying the hnges. In this se we get hexgon s shown ove. The lines for the hnge re shown soli wheres those for re she. If we hve four ells moving n fifth fixe then the possiilities re muh greter. The si moves re (1, 2)(3, 4) interhnge two pirs oule (2, 3) interhnge 2 n 3 (3, 4) interhnge 3 n 4 (1, 2) interhnge 1 n 2.

5 7 Bells Alternte oules n singles on four ells. As it hppens we will not use for the moment. In ny event it is equivlent to (or ). If we lterntely o n then we get the sitution shown ove. The ells gin o plin hunting. Note how the pths of ells 2 n 3 re mirror imges s re the pths of ells 1 n 4. But this sequene lone is no goo euse fter 8 hnges we re k to the eginning wheres we know tht we hve to o 24 hnges for four moving ells in orer to over ll possiilities. The solution is to introue nother hnge n thus voi the return to rouns. This hnge reks the plin hunting. Plin Bo Minimus THE SOLUTION is to o hnge rther thn hnge fter 7 hnges. This gets us into ifferent yle of 8 n then we o nother to prevent tht repeting n finl to finish off. The result then overs ll 24 hnges. We n write the whole sequene s (() 3 ) 3 This sequene is known s Plin Bo Minimus n is shown overlef. Minimus simply refers to the ft tht there re 4 moving ells. Other terms re Doules (5 ells), Minor (6), Triples (7), Mjor (8), Cters (9), Royl (10), Cinques (11), n Mximus (12). Note tht ell 1 (the highest or trele) ontinues to o plin hunting, wheres the pths of the three other ells hve kinks in them. These kinks ue to the introution of the hnge re known s the work n the ells re si to e oging euse they oge k n forth rther thn ontinue smooth hunting. Note lso tht the pths of the working ells 2, 3, n 4 re gin ientil ut simply shifte in time.

6 142 Nie Numers Work Work Plin Bo Minimus Furthermore, within eh group of eight hnges, the ells gin hve pths tht re mirror imges in pirs. We sw tht in the first group, the pirs were numers 1, 4 n 2, 3. In the seon group they re 1, 2 n 3, 4; n in the thir group they re 1, 3 n 2, 4. Another wy to ring the hnges on four ells (with fixe fifth ell) uses Single Chnges only. This style of ringing ws use in erlier times when the erings of the ells were less smooth n mking hnges ws onsequently somewht more iffiult. The hnges use re those lle,, n ove. The result is shown opposite. This metho is known s Doule Cnterury. Bell 1 gin oes plin hunting. The other three hve more lngui pttern wherey ell stys in the sme ple for three rows. However, they o ll hve extly the sme sequene lthough shifte in time. Altogether, it oesn t look so pretty n it lso souns it monotonous. The mthemtil sequene is not quite so ler either. Bsilly, it onsists of lternte groups n joine y n repete three times thus (()()) 3

7 7 Bells Single hnges on four ells: Doule Cnterury ut this is not very ler euse it strts prt wy through one of the sequenes. The next question we might sk is n we represent these possiilities with four moving ells s grph like the hexgon for three moving ells. The nswer is yes ut it gets omplite. The grph overlef shows the 24 possile rows (ell 5 is omitte for simpliity sine it is fixe) n the llowe hnges joining them. Different styles of lines ientify the ifferent kins of hnges. Thus soli lines represent the oules. Note refully tht the opposite sies of the grph hve to e ientifie. So the lines oming out of the point mrke A t the ottom right re the sme lines oming out of the orresponing point A t the top left. Topologilly, the figure onsists of mp of six squres, four hexgons, n three otgons rwn on rel projetive plne. Eh point is where squre, hexgon, n otgon meet. In orer to onstrut set of hnges ll we hve to o is to tre pth through ll 24 points tht visits them ll just one. This n e one in severl wys.

8 144 Nie Numers The two methos we hve enountere, Plin Bo Minimus n Doule Cnterury, re shown on the next two grphs. In eh se the pth tully tken is shown in ol re n lue in the iretion shown y the rrows. In the se of Plin Bo Minimus shown opposite every oule hnge is use. There re tully three otgons in the grph. One is in the entre n the other two re split one ross the sies n one from top to ottom. The sequene strts from 1234 (in ol) y going roun the entrl otgon (in re) with lternte oules (soli line) n mile singles (sh ot) (this is () 3 ) n then hnges on lue otte line (, whih is the work) from 1324 to 1342 to go roun the otgon ross the sies n then hnges on nother (1,2)(3,4) (2,3) A (3,4) (1,2) D B C C B D A Grph showing the hnges on four ells.

9 lue otte line () from 1432 to 1423 to go roun the thir otgon n finlly tking nother lue otte line () from 1243 to 1234 to finish. Hene we get (() 3 ) 3 7 Bells 145 By ontrst, the grph overlef shows the sequene using Doule Cnterury. It is quite ifferent. No oules () re use n so none of the soli lines is trverse. But mny of the she lines () re use inste. The si theme is tht the pth onsists of exursions roun the four points of the six squres using she n otte lines (in re). These orrespon to (1,2)(3,4) (2,3) A (3,4) (1,2) D B C C B D A The pth of Plin Bo Minimus.

10 146 Nie Numers hnges (1, 2) on the first pir n hnges (3, 4) on the lst pir. In some ses the pth uses oth igonls () n in others it only uses one (). An these squres re then joine y the lue sh-otte lines () whih represent the hnges on the mile pir (2, 3). Hene we see how we get (()()) 3 So we hve looke t this grph s either silly three otgons or s six squres joine together. In Plin Bo Minimus, we go roun the otgons n with single hnges we go roun the squres. (1,2)(3,4) A (2,3) (3,4) D (1,2) B C C B D A Using Doule Cnterury on four ells.

11 7 Bells 147 However, we n lso look upon the grph s omprising four hexgons. Eh hexgon onsists of those hnges where one of the four ells is fixe t the k. This provies nother wy of ringing ll the hnges. Go roun eh hexgon in turn n then jump etween them s neessry. The metho known s Single Court Minimus is lmost like this n is isusse s one of the Exerises (ut the trele lwys oes plin hunting). This is n interesting wy of looking t the grph euse it shows how the grph for four ells reltes to tht for three ells whih ws just hexgon. This lso hints t how the grph oul e extene to five moving ells. We just replite the grph for four ells five times n join up the points s neessry. Clerly it gets too omplex to ontemplte! Plin Bo Doules THE PLAIN BOB METHOD n lso e rung on five moving ells n is known s Plin Bo Doules. We sw how 8 hnges on four ells gve plin hunting n returne to rouns. Similrly 10 hnges on five ells o the sme. We nee two oules (1, 2)(3, 4) interhnge first two pirs (2, 3)(4, 5) interhnge lst two pirs Agin we see tht every ell oes plin hunting ut they re stggere. In orer to o the omplete extent we lso nee two single hnges (3, 4) interhnge 3 n 4 (2, 3) interhnge 2 n Plin hunting on five ells.

12 148 Nie Numers In orer to prevent the return to rouns we o hnge rther thn t the en so tht the sequene eomes () 4 whih shoul e ompre with the sequene () 3 of Plin Bo Minimus. We then rry on plin hunting for nother 9 hnges n then o gin. If we o this group of 10 hnges four times giving totl of 40 hnges thus (() 4 ) 4 then this will return us to plin hunting one more. Note the nlogy with Plin Bo Minimus whih in totl ws (() 3 ) 3. The first 20 or so re shown elow. Note how the ell in ple 5 t eh point of work (when the trele les) oes four strikes in the sme ple. This is hrteristi of the metho on n o numer of moving ells. This sequene of 40 hnges is known s Plin Course Work Work The first 24 hnges of Plin Bo Doules.

13 However, we nee 120 hnges ltogether n so further moifition is require to prevent it reverting to rouns fter 40 hnges. In ft the hnge is reple y the hnge. This interhnges 2 n 3 rther thn 3 n 4. The net effet is tht the ells in positions 2, 3, n 4 re yle roun. This mnoeuvre is known s Bo. The igrm elow shows hnges 21 to 44. The hnge keeps the ells in ples 1, 3, n 5 unmove. Agin the ell in ple 5 oes four onseutive strikes in tht ple. But exiting oging is not involve. All tht relly hppens is tht one ell reverses its hunting erly. Chnge ( Bo) is lso me fter 80 hnges n finlly t the en fter 120 hnges whih then returns immeitely to rouns. So the omplete sequene for Plin Bo Doules is ((() 4 ) 3 () 4 ) 3 7 Bells 149 The finl en of Plin Bo Doules is shown overlef. The lst Bo returns immeitely to rouns Work Bo Chnges 21 to 44 of Plin Bo Doules.

14 150 Nie Numers Bo The finl en of Plin Bo Doules. Ringers know the plin ourse ut ller will inite when Bo is to e one y lling Bo n then when it is out to return to rouns y lling Tht s ll. Note tht the full extent oes not hve to e rung extly s inite. A Bo n e lle t ny time (ut inevitly when the trele is t the front). For exmple if Bo were lle t hnge 30 n then gin t 70 n 110 then we woul fin tht we h to o 10 more hnges efore returning to rouns. There re in ft three sets of 40 hnges the si Plin Course n two vritions otine y rotting ells in positions 2, 3, n 4. We n swp etween these three sets y oing Bos t pproprite times n so n ring the full extent in mny ifferent wys. Note refully tht the term Bo refers to the hnge. The metho Plin Bo Minimus with only four moving ells oes not tully inlue ny Bos! The sme pproh n e use with more ells. An itionl kin of interhnge known s Single will e require to prevent premture returning to rouns if full extent is to e rung. Thus with 6 moving ells we hve (1, 2)(3, 4)(5, 6) interhnge three pirs triple (2, 3)(4, 5) interhnge mile two pirs (3, 4)(5, 6) interhnge lst two pirs

15 7 Bells 151 So Plin Course of Plin Bo Minor is (() 5 ) 5 whih is similr to tht for Plin Bo Doules with 4 reple y 5. Thus it hs 60 hnges. In orer to o full extent we lso nee (2, 3)(5, 6) the Bo e (5, 6) the Single n y lling these t pproprite points we n o the full extent of 720 hnges. Stemn FABIAN STEDMAN ( ) wrote two fmous ooks on ell ringing, Tintinnlogi in 1668 n Cmpnlogi in He introue numer of new methos for ringing ells n his nme is immortlize y the nmes of these methos. The title pge of the thir eition of Cmpnlogi (te 1733 n thus posthumous) sys By Plin n Methoil Rules n Diretions, wherey the Ingenious Prtitioner my, with little Prtie n Cre, ttin to the Knowlege of Ringing ll Mnner of Doule, Tripple, n Quruple Chnges. With Vriety of New Pels upon Five, Six, Seven, Eight, n Nine Bells. As lso the Metho of lling Bos for ny Pel of Tripples from 168 to 2520 (eing the Hlf Pel:) Also for ny Pel of Quruples, or Ctors from 324 to One ie of Stemn s ws to rek the ells up into groups. For exmple, with five moving ells, we rek the ells into group of three n group of two. Suppose the group of three o the hnges on three ells isusse erlier wheres the group of two just keep oging. After six hnges this woul return to rouns s shown elow

16 152 Nie Numers The first 20 hnges of (our vrition of) Stemn Doules. In orer to voi this we exhnge one ell etween the two groups. The hnges re (1, 2)(4, 5) (2, 3)(4, 5) (1, 2)(3, 4) n so silly we hve followe y n then followe y, n so on. Note tht lternte groups of hnges on the threes re reverse in iretion. If this were not one the first two ells woul oge too muh. The first few hnges re shown ove. Note refully tht this is not s Stemn is normlly strte ut simplifies the mthemtis of the permuttions. Although it might not e ovious, in ft every ell follows extly the sme pttern ut they re phse ifferently. Thus ell 2 repets the pttern of ell 1 ut 12 hnges lter n ell 5 follows 12 hnges fter tht n so on. The iniviul ptterns look omplite lthough the superimpose pttern hs n elegnt symmetry. After 60 hnges we hve one plin ourse (() 2 () 2 ) 5

17 7 Bells The proper strt of Stemn Doules. n we return to rouns. However, this mens tht we hve only one hlf of the totl extent of 120. The reson is ovious. The hnges,, n re ll oules n so we hve only elt with the even permuttions. In orer to o the other 60 we must insert single (twie). We n use (2, 3) the Single n if we use it t the en the full extent will e ((() 2 () 2 ) 4 () 2 () 2 ) 2 Note some interesting ptterns. In prtiulr, note how ell 5 goes from position 5 to 1, oes oule strike on 1, n then goes out to 5 gin. This is lle running in quik n running out quik respetively. In prtie ell ringers o not strt quite s we hve one ut with the trele running out quik. This mens tht they strt prt wy through sequene n this somewht spoils the mthemtil nlysis (though mye souns prettier). This proper strt is shown ove. (Grmmrins woul prefer to run in n run out quikly no out.) One onsequene of strting prt wy through touh of six hnges thus... is tht the Single is lso pplie prt wy through touh thus...

18 154 Nie Numers Single Chnges 19 to 63 of (our vrition of) Stemn Doules showing the Single.

19 Single 7 Bells Prt of norml Stemn Doules showing the Single. n s onsequene the full sequene s normlly rung is ((() 2 () 2 ) 4 () 2 ) 2 The orresponing prt of the norml Stemn inluing the Single is shown ove. Note tht we nnot relly o Stemn with less thn five moving ells sine we nee t lest three plus two. Inee, Stemn relly only works on n o numer of moving ells. So we n hve seven ells with one group of three n two groups of two; the ells then hnge etween the vrious groups.

20 156 Nie Numers The first few hnges of (our version of) Stemn Triples This is known s Stemn Triples n the key hnges re (1, 2)(4, 5)(6, 7) (2, 3)(4, 5)(6, 7) (1, 2)(3, 4)(5, 6) The si ourse is s in Stemn Doules nmely (() 2 () 2 ). Wheres in Doules this reverts to rouns if pplie 5 times, in Triples it reverts to rouns if pplie 7 times (fter 84 hnges). Agin, s normlly rung, it strts with the trele running out quik n so it strts () 2... In orer to o full extent we n use Bo n Single. These re (1, 2)(3, 4)(6, 7) the Bo e (1, 2)(3, 4) the Single Unlike Stemn Doules, the Bo n Single simply reple n instne of n thus we hve sequenes s follows e Curiously, the Single is in ft oule! Moreover, the norml hnges n the Bo re ll triples n so hnge the permuttion from o to even n k t eh hnge. Thus, unlike Stemn Doules, we o not get loke into even permuttions n so o not inevitly hve to use the Single to o the o permuttions. It hs een shown s reently s 1994 tht it is inee possile to ring full extent of Stemn Triples using only the hnges,,, n.

21 Grnsire NO OVERVIEW of ell ringing woul e omplete without mention of the Grnsire methos. In the plin Bo methos one ell oes plin hunting. In Stemn s, no ells o plin hunting. But in Grnsire methos two ells o plin hunting (more or less). Below is shown the strt of Grnsire Doules. It is ells 1 n 2 tht o plin hunting ut they nnot e jent sine otherwise no ell oul pss them euse it woul hve to move two ples n tht is not llowe. So we strt with hnge tht vois this. We hve (1, 2)(3, 4) (2, 3)(4, 5) (1, 2)(4, 5) n the sequene silly is simply repete. () 4 7 Bells The first 22 hnges of Grnsire Doules.

22 158 Nie Numers Note tht ells 3, 4, n 5 follow pttern lmost ientil to the working ells in Plin Bo Minimus. After 30 hnges it returns to rouns. In orer to o full extent (120) two importnt points must e notie. First, the hnges use so fr re ll oules n so we must o single to o the o s well s the even permuttions. Moreover, if ell 2 lwys follows ell 1 two ples ehin, then it nnot possily over the full extent euse tht requires ell 2 to e in ll possile ples fter ell 1. The solution is to rrnge for the other three ells to tke their turn t oing plin hunting two ples ehin ell 1. We n hieve this y introuing single, nmely (4, 5) the Single In ft we reple the just efore the hnge y the single. This hs the effet of reversing the roles of ells 2 n 3. So ell 3 now oes plin hunting n follows ell 1 s ell 2 previously i. But this is not enough sine if we o it gin t 60, ll tht hppens is tht ells 2 n 3 swp k n so we woul revert to rouns. To voi this we lso reple the previous hnge y so tht the sequene eomes... The net result is tht 3 now oes plin hunting n 2, 4, n 5 re yle roun s well s shown elow Single Chnges 20 to 42 of Grnsire Doules showing Single ().

23 7 Bells 159 So the initil plin roun is reple y At 60 it is reple y n t 90 y An then t 120 it reverts to rouns hving one the full extent. The replement of y is the opertion (2, 3, 5, 4). Sine this hs length 4, it follows tht pplying it four times returns to the originl stte s we esire. A Bo is lso efine for Grnsire Doules. However, this oes not involve n itionl permuttion. It simply onsists of repling the norml y. In other wors we simply reple the whih is two efore the y nother. This reples y This hs the sme effet s the Single exept tht 3 n 5 re interhnge. So 5 is now the ell tht oes plin hunting n follows 1. So it is the opertion (2, 5, 4) n ell 3 is unffete. It is not very useful euse it still leves us with even permuttions. Groups THE MATHEMATICS ehin ll this is lle group theory. A group is set of things whih hve opertions on them. In our se they re the vrious permuttions of N things n the opertions re hnges suh s (2, 3). Groups n e suivie into smller prts some of whih re genuine groups in their own right. Other importnt suivisions re lle osets. See Appenix E for n introution to some simple properties of groups. In the se of Plin Bo Doules, the Plin Course visits just sugroup of size 40 of the whole group of permuttions on 5 ojets (whih hs size 120). The other two susets of 40 re osets (they re not groups in their own right sine they o not hve plin roun). Whenever we o Bo we swith from one oset to nother. Further reing CHAPTER 7 OF Musi n Mthemtis eite y Fuvel, Floo n Wilson overs ell ringing from gently mthemtil perspetive. A tritionl ell ringers ook is Bell-ringing y Ron Johnston. Digrms of mny methos will e foun in Digrms in the Jsper Snowon Series. Fsimiles of Tintinnlogi n Cmpnlogi were pulishe y Kingsme Reprints (1970) n Christopher Groome (1990) respetively. The ltter ws limite eition. For eep mthemtil insight (n possily hehe) onsult vrious ppers y Arthur T White suh s Ringing the Cosets in the Amerin Mthemtil Monthly for 1987 (pp ) n Fin Stemn: The First

24 160 Nie Numers Group Theorist? in the Amerin Mthemtil Monthly for 1996 (pp ). They re ville t The topology of the projetive plne is esrie in Gems of Geometry y the uthor. Also see Appenix E on Groups. Exerises 1 The following metho is known s Cnterury Minimus Drw its igrm niely. 2 Using the nottion tht the permuttions re (1, 2)(3, 4) interhnge two pirs oule (2, 3) interhnge 2 n 3 (3, 4) interhnge 3 n 4 (1, 2) interhnge 1 n 2. give the lgeri sequene for Cnterury Minimus. Tht is the form in whih Plin Bo Minimus is (() 3 ) 3. 3* The metho for Single Court Minimus strts In the Grph for hnges on four ells shown erlier, Single Court Minimus goes roun three of the hexgons in turn lthough it strts prt wy through the hexgon for ell 4. The fourth hexgon (for ell 1) is visite in stges n is not trverse in one lump. This enles ell 1 (the trele) to o plin hunting s norml. Bells 2, 3, n 4 follow the sme pttern ut stggere; this pttern inlues sequene of six strikes. Complete the metho, rw its igrm n give the lgeri sequene.