Experiments in Generalizing Geometry Theorems Stephen B. Gray

Transcription

1 Experimens in Generalizing Geomery Theorems Sephen B. Gray. INTRODUCTION: THE PDN THEOREM Well-known advances in geomery have been made wih experimenal, or compueraided echniques. The irs was he proo o he Four-Color Conjecure, and he second was he presumed proo o he Kepler Conjecure. Less well known was Kimberling s compuerized search or new special poins, lines, and circles in riangles. I will presen anoher example o advancing geomery experimenally. Firs, I need o esablish a saring poin. bou 4 years ago I ound and proved a heorem abou consrucions on plane n-gons, bu i had been ound and proved as early as 98. Wriers have described his Per-Douglas-Neumann (PDN) heorem as beauiul and remarkable; agreeing wih ha assessmen, I ook i up again a ew years ago. On each side o a general plane n-gon - we place isosceles riangles wih apex angles π/n, making anoher n-gon,, rom hose apices. The heorem saes ha i his is done n imes wih =,,,n, a single poin or degenerae n-gon will resul. I ollows ha n-gon n is regular. I will illusrae in Fig. wih n=5. In he ollowing maerial, verices o n-gon are denoed,,,,,,n-. Verex subscrips are aken modulo n Y Z Z Y Z Z = = Z Y 3 Z 6 Z Z 88 = 3 = 4 Models: apex = π n ngles Y 4 ps ps ps 3 ps 4 ps poly poly poly 3 poly Fig. : Model riangles and arge polygon The riangles on he le are prooypes, or models, or hose used in he sages o consrucion. The yellow model riangle corresponds o =, and so on down o model = n. Fig. shows he model riangles mouned on all our sages o he arge, which is he righ hand igure. ccording o he heorem, he angles o polygon 3 and he single-poin propery o 4 are invarian wih changes in. Penulimae polygon 3 (brigh red) says regular bu may

2 ranslae, roae, and dilae. Here n=5, bu n can have any value greaer han, and here will be n sages o consrucion. I ll say a word abou proos, which have all been algebraic. We can express, or example, poin as a linear uncion o poins and, and similarly or he oher verices o. In he usual way, we represen polygons by column vecors, and each consrucion sage corresponds o a square marix wih exacly wo nonzero elemens per row. (Eq. a) This shows wo o he n marix muliplies. = 4 3 * 4 3 = 4 3 * 4 3 Equaion : Sample marices ll n- operaions are represened as shown here: = M = M = Q = M = M M = Q 3 = M 3 = M 3 M M = Q 3 4 = M 4 3 = M 4 M 3 M M = Q 4 Equaion a: Marix operaions The produc Q 4 o he our speciic marices M has idenical rows; his implies ha every verex o he inal polygon 4 is he same uncion o he verices o, so 4 is a single poin. Working backwards, he previous polygon 3 mus be regular. The consans are uncions only o he model angles. For he inal polygon o have a special shape, he consrucion mus be irreversible, which in algebraic erms means ha all he Q s and M s are singular. lso, or origin independence o he algebraic represenaion o he consrucion, any row o any M mus sum o.. COMPLETING THE MODELS I s producive o draw he ive riangles used in sage = assembled ogeher, orming a penagon (in yellow). Triangles or he oher sages are also assembled ino penagons, some relexive. The verex common o he sage model is denoed Y. I ve varied he shades o yellow

3 so he arge and model mach. The consrucion or sage can be represened as ollows, where k incremens rom o n as we go around he model and arge ogeher. Z k Z k+ Y,k+,k,k Eq. b: Typical verex correspondence For example in sage =, riangles are ranserred as ollows: Z Z Y, Z Z 3 Y, ec. Similarly, he riangles used in sages =3,4 are respecively arranged around Y 3 and Y 4. (I ll show a complee consrucion key below.) Figure illusraes he n- sages and heir polygons. The riangles come rom generally nonadjacen model verices bu always go o adjacen arge verices. For a consruced polygon o have a special shape, such as being regular or degenerae, sage o consrucion mus be irreversible, oherwise we could always ind a polygon on which he consrucion would give, so a consruced polygon could be regular, degenerae, ec. I can be shown ha he geomeric crierion or irreversibiliy o sage is ha he consrucion riangles can be assembled o orm a closed n-gon wih a coninuous perimeer, like all n models shown here. Z 3 Z Z 4 Y Z = 3 3 Z Z 3 Z Y = Z 4 Z 4 Z Z 3 Z Z 4 Y 3 Z =3 4 Z 4 Z Z 3 Z Z Y 4 Z =4 4 Regular -3 Clear Targe poly 3 poly Loci Y ps Key inerior - ps poly inerior - ps poly inerior 3-4 ps 3 ps 4 ps 3 inerior Fig. ll n compleed models and consrucion sage = on he arge. ngles 3. THEOREMS ND Inspired by his ype o model display, we can ry moving all he Y poins ogeher away rom he model cenroids. Even hough he consrucion riangles are no longer similar, 3 says regular, and 4 is sill one poin. I always rue, his is clearly a generalizaion o he PDN heorem. For a second experimen, we modiy he shape o he model penagons hemselves, all ogeher. Now 3 is no longer regular, bu is sill similar o he models. This is a second generalizaion, which wih he arbirary locaion o he Y, is called Theorem, appearing in he March 3 merican Mahemaical Monhly []. Finally, i we move he common verices separaely, he shapes o n and n are invarian. Tha s a hird generalizaion, called 3

4 Theorem, which I ound and proved aer he paper was acceped by he Monhly. (I couldn revise he paper a ha poin because I had already annoyed he edior enough.) These exensions o he original PDN heorem became possible o ind only because o my inroducing he model vs. arge idea. This separaes hese heorems ino wo igures, making i much clearer, and allowing i o be reaed in wo pars. I have no seen his idea beore. 4. FURTHER GENERLIZTION; THE DISCRETE PRMETERS To generalize his class o heorems urher, I need o inroduce some consgrucion parameers, shown in he green rows o Table. We ve already discussed inernal variables are and k as well as n. In he model, p deermines which verex is used irs (ha is, when k=). Nex, q is he model verex separaion parameer or sage : i q =, consrucion riangles come rom adjacen verices o model. I q = (he green model in Fig. ), riangles come rom verices k and k+. Noe ha or Theorems and, q =, bu his is no so in oher resuls. Qy. Range Meaning n 3 n # sides o polygons in model and arge n Sage (o consrucion) couner k k n Verex couner, incremens during scan o model and arge p p n On he model, which verex is used irs (ha is, when k=). q q n On he model, he spacing o each riangle s base verices. r r n On he arge, which verex is used irs (ha is, when k=). s s n On he arge, he spacing o each riangle s base verices. Table : Discree parameers and variables In Theorems and we always use r = ; ha is, we sar (when k=) placing riangles on he arge a verex. Parameers r and s are o he arge as p and q are in he model. In Theorems and we also have s =, so riangles go on adjacen arge verices in all sages. During he raverse or a given sage, as k incremens, he model and arge riangles may be placed on verices as shown in he examples o Table. Example here corresponds o Fig. above. s k incremens, all subscrips always incremen by. 5. THE LGEBR Example (see Fig. ) Example p =, q =, r =, s =. k p =, q =, r =3, s =4. Z Z Y,,, Z Z 3 Y,,3,3 Z Z Y,,, Z Z 4 Y,3,4,4 Z Z 3 Y,3,, Z 3 Z Y,4,, Z 3 Z 4 Y,4,3,3 3 Z 4 Z Y,,, Z 4 Z Y,,4,4 4 Z Z Y,,, Table : Z- verex correspondences The marices are based on he similariy o arge and model riangles. In Fig. a, equaing corresponding side raios gives he equaion shown. This igure shows a general model and is associaed arge riangle (in blue), or sage, a a paricular place k in he raversal; i also shows he nex riangle, corresponding o k+, wih dashed lines. On he le, he railing 4

5 verex is Z p+k because he raversal sars a model verex p and so ar we have raversed k verices. The leading verex is jus q places beyond, because ha s he spacing being used a sage. Similarly, a he same ime in he arge we sared a verex r and have gone hrough k verices here. The leading verex is, by deiniion, s verices beyond. The general ranser o a riangle rom model o arge or given, k, p, q, r, and s is shown in Eq. c ( sub-subscrips are omied): Z p+k Z p+q+k Y,r+s+k,r+k,r+k Eq. c: Generalized verex correspondence In Fig. a, similariy o he model and arge blue riangles gives he equaion shown. Z k+p+ d Z p+k Z p+k- -,r+s+k Z p+q+k n Z p+q+k+ Y Model -,r+s+k+ Targe,k+r+ d,k+r n -,r+k -,r+k- Fig. a: Similar riangles and raios o sides Tha equaion is solved or he new arge verex,r+k in erms o he known model and he wo arge - verices o he previously derived sage. Wih he subsiuion i=r +k (mod n), he general marix enry becomes ha o Eq. 3. Y Z p r + i ( M ) ij = i j = i Z p r + q + i Z p r + i ( M ) ij = Z Y Z p r + i p r + q + i Z p r + q + i (M ) ij = oherwise. Equaion 3: Marix elemens i j = i + s (modn) 6. MORE GENERLITY; SERCHING OVER THE DISCRETE PRMETERS I waned o ry o ind he mos general heorem o his ype, limied only by he resricion ha he consrucion be described by p, q, r, and s. I searched or parameer ses which yielded inal polygons which degeneraed o single poins, or ones whose shape was consan wih changes in. I waned he heorems o be broad enough o include non-regular models (wih he resricion ha no wo verices o Z coincide, because o he denominaors in Eq. 3). I also waned o allow general and independen posiions o he common verices Y, as well as compleely general saring arge polygons. Each o he our parameer ses (p), (q), (r), and (s) has has n members, where a member o (p) or (r) can ake on n values, and a 5

6 member o (q) or (s) can ake on n values. For a complee, naïve search, we would have o go hrough (n n ) (n) combinaions, which even or he smalles ineresing case o n=5 is abou 5 billion. Thereore I looked or ways o reduce he search space. Eq. 3 shows ha p and r appear only as (p r )(mod n), so wihou loss o generaliy we can assume p =. Nex, noice ha adding a consan c o all r amouns o simply relabeling he arge. Then i we wan o invesigae a cerain se (r, r,, r n- ), we jus se c = r and consider only ses wih r =. There are n remaining r parameers, each admiing n values r n, so (r) as a whole has n n possible values. In each o he ses q and s, n, each member o which can ake on n values, so he possible combinaions are (n) n- in number. Togeher, q, r, and s can ake (n) n- n n values. Even or n=5 his amouns o , over 8 million values, and or n=7, i s over 36 rillion. Clearly, a general search is possible only or n=5. The main purpose o he search was o ind all combinaions ha gave a inal produc marix Q n having all rows idenical. Compuing his inal marix requires inding he produc o n marices, each n.n; i done numerically, his involves doing n 3 muliplicaions and addiions. Iniially I considered esablishing he equal-row propery purely symbolically, bu Mahemaica ook much oo long (using a GHz PC) even or n=5. 7. CONTINUOUS PRMETERS; SERCHING Besides he discree parameers o Table, he poenial heorem I m looking or has he ollowing coninuous posiion parameers, each one a complex number. See Table 3. Z k : Verices o model, n in all, arbirary excep none may coincide. Y : Common verices o model, n in all. Values are arbirary. k : Verices o iniial arge, n in all. Values are arbirary. Table 3: Coninuous parameers In he compuer search or good values o he discree parameers, I used high-precision numeric ess wih independen random values or hese 6n coninuous reals. I checked he equal-row numerically, which ran a leas a housand imes aser han a symbolic es would. When I go his se up correcly, i was able o run over all combinaions o (q), (r), and (s) or n=5. The resuls were surprising: he combinaions giving row equaliy, and hereore a successul proposiion, had he unexpeced rules shown in Table 4: # Rule or consrain. p = ( n) and r = can be assumed.. Se (s), arge verex spacing, is he independen variable, wih q = q(s) and r = r(s). 3. ll s ( n) are relaively prime o n. 4. The las member o he se (s), s n-, can have any value s n n 5. For x n, all values in (s s n- ) mus be eiher x or nx Examples: Valid ses or n=5 are (,4,4, y), (4,4,, y), (,3,, y), (3,3,3, y), ec. For n = 7, s hrough s n- can be any mixure o ;6, ;5, or 3;4. 6. (q) is given by q = s (mod n). Example or n = 5: I s = (,4,,3), q = (,,3,4)*(,4,,3) = (,3,3,). For n = 7, i s = (,,5,,5,), q = (,,3,4,5,6)*(,,5,,5,) = (,4,,,4,6). Table 4: Consrains on q and s or Theorem 3 Limiing he search wih hese consrains drasically cu he number o combinaions o be ried, and i made he n = 7 case searchable. The inal sep was o ind a rule or valid ses o r 6

7 as a uncion o s. Iniially he values o r giving row equaliy, and hereore a valid hypohesis, had no apparen paern. Bu knowing ha is values had o be relaed o s in some logical way, I kep looking or his relaion. Obviously he number o possible relaionships would be huge, especially since I had no idea wha one would look like. Wih he above consrains, he number o r, s combinaions o ry was now n n- n-, which or n=7 is only.8 million; however each es sill involved 343 muliplies and many oher operaions, so was sill slow. I waned o say wih prime n, because according o rule 3 above, ha would allow me o ry more ses (s). 8. COMPLETING THEOREM 3 Table 5 shows a ypical able o numbers or n=7 ha I worked wih. I seleced a ew rows rom he much larger oupu o he acual compuer search. In his secion o he oupu daa, s,,s 5 are resriced o values and 6, alhough combinaions o,5 and 3,4 are also valid. # q,, q 6 r,, r 6 s,, s 6 g =r /s (mod 7).,,3,4,5,6,,,,,,,,,,,,,,,.,5,4,4,,4,6,6,,5,,6,6,,6,3,,,,,5 3.,5,4,4,,6,6,6,,5,3,6,6,,6,,,,,,3 4.,5,3,4,,3,6,,,4,6,6,,,6,4,,,,3,5 5.,,4,4,,3,,5,,4,6,,6,,6,4,,,,3,5 6. 6,5,4,4,5,3,,,3,3, 6,6,6,,,4,,,3,3,5 7.,,3,3,5,5,,,4,,,,,6,,,,,3,,5 8.,5,3,3,5,5,6,,5,,,6,,6,,,,,,,5 9. 6,,4,4,,,,6,,5,6 6,,6,,6,5,,,,,5. 6,,4,3,5,,,6,6,3,6 6,,6,6,,5,,,,3,5 Table 5: Some successul combinaions or n = 7 In row, all s =, so adjacen verices in he arge are used a every sage. ll r =, so ha arge verex is always used irs. In each sage, one addiional model verex is skipped. Row in he able characerizes he PDN and Theorems and. Rows - show ha s 6 can have several values, and q is given by q = *s in all cases. Rows and 3 show ha changing one member o s, say s 3, changes only q 3, bu he digis o r a and o he righ o he changed s 3 are aeced. Examining he rows, I saw ha r 5 +r 6 = s 5 +s 6 (mod 7). This urned ou o be rue or all examined cases. (In rows hrough 5, r 6 = g 6 s 6, bu his relaion is no always rue.) his poin I sill had no uncion or he oher members o r. Bu knowing ha q is relaed o s by a modular produc q = *s (mod n), I made a guess and ook he modular quoien g = r / s (mod 7), namely ha number saisying r = g s (mod n). The essenial (and surprising observaion) was ha or a given <n, g is exacly he number o members o s, s - ) no equal o s. For example, in row 5 (yellow) o Table 5, g = because s is no equal o s ; g 3 = because he se (s,s ) has members no equal o s 3 ; g 5 =3 since he se (s,,s 4 ) has 3 elemens no equal o 6. lhough g 5 = 5, his is no used because o Rule 4 above. This rule, conirmed by numerous oher cases, allows us o add he inal consrain on (r) o he proposiion. nicipaing being able o prove his, I will call i Theorem 3. The ull se o consrains on he discree ses (p), (q), (r), and (s) are shown in Table 6. Furher generalizaions along hese lines seem o be obviaed by he requiremen ha all consrucion sages be irreversible. 7

8 # Rule or consrain. p = ( n) and r = can be assumed w.l.o.g.. Se (s) is he independen variable, wih q=q(s) and r=r(s). 3. ll s ( n) are relaively prime o n. 4. s n- can have any value s n n (subjec o Rule 3) 5. For x n, all values in he se (s s n- ) mus be eiher x or nx (subjec o Rule 3) Example: I n = 5, valid ses are (,4,4,y), (4,4,,y), (,3,,y), (3,3,3,y), ec. For n = 7, s hrough s n- can be any mixure o ;6, ;5, or 3;4. 6. (q) is given by q = s (mod n). Example or n = 5: I s = (,4,,3), q = (,,3,4)*(,4,,3) = (,3,3,). For n = 7, i s = (,,5,,5,), q = (,,3,4,5,6)*(,,5,,5,) = (,4,,,4,6). 7. new 8. new For n, r = g s (mod n), where g is he number o elemens in (s, s - ) which are no equal o s. Example: s = (,,5,,5,4) g = (,,,,3,5) r = (,,3,,,x). (Rule 8 gives r 6.) The las member o he se (r), r n, is ound rom r n- + r n- = (s n- + s n- )(mod n). Example: r n- = s n- + s n- r n- = (5+4)(mod 7) = = r 6. Table 6: Consrains on q, r, and s or Theorem 3. There are oher expressions or Rule 7, bu I have ound none simpler han he above. I ook me much longer o ind hese unexpeced consrains, especially Rules 7 and 8, han his descripion implies. Discovery involved saring a hese ables o numbers or a long ime; I kep hinking o he movie version o John Nash. I m unclear as o why his srange se o rules holds, and exacly wha i means geomerically. In paricular, i seems odd ha he rule or r n- is dieren rom he rule or r hrough r n-. I s possible ha rules 7 and 8 amoun only o relabeling he model and/or he arge in cerain ways, bu I have no seen his ye. In reurn or his complexiy, we have a good PDN generalizaion, namely Theorem 3, which holds or any n>. For prime n, he resriced se (s) can have n3 (n) values, wih (q) and (r) ixed uncions o (s). (The irs and las s s can have any value s, s n n, and he oher n3 s s can each have wo values dependen on s.) s poined ou previously, he heorem also involves 3n coninuous complex parameers. Even or n=7, he independen discree parameers number 576 and here are 64 independen reals. Table 6a gives he number o combinaions o be esed under various assumpions. (Having ound he consrains or Theorem 3, searching is no longer necessary; he las column gives he number o parameer combinaions making he heorem valid.) Search ype Formula n=5 n=7 Over all (p),(q),(r),(s) n n (n) n.56* 3.* 9 Resricions on (p),( r) n n (n) n- 8.9* * 3 Wih Theorem 3 rules (n ) (n3) Table 6a: Number o combinaions o be ried 8

9 9. EXMPLE OF THEOREM 3 FOR n=7 Table 7 gives he parameers or his example. s beore, is he consrucion sage variable. Rule 4 in Table 6 says ha s n- can have any value. Rule 5 is obeyed because he oher s are eiher or 7. Rule 6 is ollowed because q =s, q =s (mod n), ec. Including he dependen variables in (g), noe ha he parameers obey Rule 7, r =g s (mod n), excep or r 6 which is given by Rule 8. = p q r s g 5 Table 7: p,q,r,s or he n=7 Figure. Fig. 3 shows only he and 5 polygons, and he inal s claimed, 5 is similar o he model, and says so i we change. I we move Y or Z 6, all inermediae polygons change bu 5 is unchanged in shape. I don ye have a proo o his proposiion, bu i s almos cerainly correc, and I m working o prove i. My proo o Theorem proceeds by inducion on parial marix producs, compued and examined wih he help o Mahemaica. Proo o Theorem 3 will probably also be compuer-aided. Z' 3 Z' Y Z' Z' 4 Z' 6 3 Z' 5 Z' 6 Model ngle Model ngle Model 3 ngle 3 Model 4 ngle 4 Model 5 ngle 5 Model 6 ngle Key ps ps ps 3 ps 4 ps 5 ps 6 ps poly poly poly 3 poly 4 poly 5 poly 5 Z 4 Z 3 Z Z Maser model Z Z 5 Z6 =3 456 p= q= r=5 334 s=5 554 Fig. 3: Theorem 3 or p, q, r, s as shown. 9

10 . REMRKS Table 8 summarizes he heorems and shows heir curren saus. Name Models Z Common Verices Y Targe verices or riangles Curren saus PDN Regular Y = Z cenroid s = Published in 98 Theorem Irregular Y Z cenroid Y = Y u s = Proved, Monhly, Mar. 3 Theorem Irregular Y Z cenroid Y Y u s = Proved bu no published Theorem 3 Irregular Y Z cenroid Y Y u s, consrains on (q), (r), (s) No proved or published Table 8. Summary o heorems I is possible ha Theorem 3 does no really conain new geomery, bu can be derived rom Theorem by relabeling he model and/or arge, perhaps wih some algebraic subsiuions. noher poin is ha wihou he compuer search, I would no know wheher he p, q, r, s paradigm conained urher geomeric ruhs han I have repored here. s hings sand, Theorem 3 is he las generalizaion along his line; however I have ound oher similar proposiions, which I inend o prove and publish evenually. This ield seems o be very erile in new resuls.. SOME RELTED QUESTIONS IN COMBINTORIL GEOMETRY. Wha is he number d m,n o opologically disinc n-gons obainable by merging ses o wo or more adjacen verices o an m-gon, wih n<m? Concepeually, pu a black bead on every edge o be collapsed and a whie bead on he edges beween he verices kep separae. The n-gon coun can be shown o be equivalen o he number o wo-color braceles (necklaces allowing reversal) having mn black and n whie beads; d m,n is given by 537 in he EIS. Examples are d 5,3 =, d 6,4 =3, and d,5 =38, so here are wo disinc ways o reduce a penagon o a riangle, hree dieren reducions o a hexagon o a quadrilaeral, ec. (Thanks o Claude Chaunier or showing me his.) These reducions creae new, srange, perhaps ugly heorems or n-gons rom consrucions or m-gons, wih n<m.. For n=5, he number o line inersecions or crossings, excluding verices, can be,,,3, and 5. I denoe his se (c 5 ) = (,,,3,5), so he cardinaliy o he se is c 5 = 5. For he 7- gon in Fig. 3, c x = 8, so 8 (c 7 ). Wha is c 7? For a given n, wha values can appear in he se (c n ), and is he ineger sequence c n in he EIS? Where? 3. Coun he number e n o opologically disinc polygons o n sides. In my deiniion, a opological change is deined by passing a verex hrough a line. Mirroring is no a change. For n = 3,4,5, e n =,,6 respecively. The sel-crossing polygon o Fig. 3 is one o perhaps several dozen 7-gons. This is probably in he EIS, bu where? Is here a beer deiniion o disinc? 4. Is here an n-gon in R 3 whose projecions ino R assume all disinc n-gons? For wha values o n is his possible, in addiion o he obvious cases n = 3 and 4? For how large an n are experimens easible? I ve always waned o pose a diicul new problem. Is his one? REFERENCES. S. B. Gray, Generalizing he Per-Douglas-Neumann heorem on n-gons, merican Mahemaical Monhly (3), -7.