Geodesics on Regular Polyhedra with Endpoints at the Vertices

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1 Arnol Mth J (2016) 2: DOI /s z RESEARCH CONTRIBUTION Geoesis on Regulr Polyher with Enpoints t the Verties Dmitry Fuhs 1 To Sergei Thnikov on the osion of his 60th irthy Reeive: 3 Otoer 2015 / Revise: 23 Otoer 2015 / Aepte: 3 Mrh 2016 / Pulishe online: 23 Mrh 2016 Institute for Mthemtil Sienes (IMS), Stony Brook University, NY 2016 Astrt In reent work of Dvis et l (2016), the uthors onsier geoesis on regulr polyher whih egin n en t verties (n o not touh other verties) The ses of regulr tetrher n ues re onsiere The uthors prove tht (in these ses) geoesi s ove never egins t ens t the sme vertex n ompute the proilities with whih geoesi emnting from given vertex ens t every other vertex The min oservtion of the present rtile is tht there exists lose reltion etween the prolem onsiere in Dvis et l (2016) n the prolem of lssifition of lose geoesis on regulr polyher onsiere in rtiles (Fuhs n Fuhs, Mos Mth J 7: , 2007; Fuhs, Geom Dei 170: , 2014) This pproh yiels ifferent proofs of result of Dvis et l (2016) n permits to otin similr results for regulr other n iosher (in prtiulr, suh geoesi never ens where it egins) Keywors Regulr polyher Geoesi segments Enpoints 1 Introution A geoesi on surfe of polyheron is, y efinition, lolly shortest urve whih my trnsverslly interset eges, ut oes not ontin verties esies, possily, the enpoints A geoesi is stright within the fes n t every intersetion with n ege, the opposite ngles forme y the geoesi n the ege (in the two fes tthe to this ege) re equl If geoesi hs two enpoints t verties, we ll it simple geoesi segment (A simple geoesi segment is llowe to hve self-intersetions) B Dmitry Fuhs fuhs@mthuviseu 1 University of Cliforni, Dvis, USA

2 202 D Fuhs v u Fig 1 The plnr evelopment of regulr tetrheron All the results of Dvis et l (2016) mentione elow re given with proofs, whih, t lest in the se of the ue, re not the sme s in Dvis et l (2016) 2 Tetrheon Theorem 21 (Dvis et l 2016, Corollry 38) A simple geoesi segment strting t some vertex of regulr tetrheron never ens t the sme vertex n ens t the three other verties with equl proilities 1 Proof Consier the plnr evelopment of the regulr tetrheron with verties,,, (see Fig 1) A line segment in this plne strting t the leftmost point mrke n ening t one of the verties within the ngle shown in Fig 1 hs the form p u + q v where p n q re non-negtive integers Nmely, it ens t vertex mrke, ifp n q re oth even, t vertex mrke, ifp is o n q is even, t vertex mrke, ifp is even n q is o, n t vertex mrke, ifp n q re oth o It orrespons to simple geoesi segment if n only if p n q re reltively prime; in prtiulr, if it ens t, then it oes not orrespon to ny simple geoesi segment Theorem follows 3 Otheron Theorem 31 A simple geoesi segment strting t some vertex of regulr otheron never ens t the sme vertex It ens t the opposite vertex with the proility 1 4 n t the eh of the other verties with the proility 3 16 Proof Figure 2 shows regulr otrheron (left) n its plnr evelopment The ltter is multivlue in the sense tht every vertex,, is, tully, two verties ( 1 Here n elow, speking of the proility with whih geoesi segment with given strting point ens t some vertex, I men the symptoti proility for the set of geoesis of oune length

3 Geoesis on Regulr Polyher with Enpoints t the Verties 203 v u Fig 2 A regulr otheron n its (multivlue) plnr evelopment Fig 3 The segment p u + q v overe y the tringles of the tiling represents n, et), every ege represents four eges ( represents,,, n, et), n every tringulr fe represents eight tringulr fes Thus every line in the plne emnting, sy, from the leftmost vertex n not following the eges represents (polygonl) line on the surfe of the otheron, ut following this line we my nee to reple y, y, n/or y For positive reltively prime p, q, vetor p u + q v represents simple geoesi segment on the surfe ening t or,ifp q 0 mo 3, ening t or,if p q 1 mo 3 n ening t or,ifp q 1mo 3 The most importrnt thing we nee to prove is tht it never ens t Let p > q > 0, GCD(p, q) = 1, p q mo 3 Figure 3 shows prllelogrm spnne y p u n q v (on the piture, p = 13, q = 7) We onsier the igonl segment overe y tringles of the tiling The verties re lele oring to the following rules The left lower tringle is ; if two tringles shre sie, then the two verties not on this sie re lele y the sme letter, one with prime, one without prime, like n, n, et We re intereste in verties lele y or The form sequene,,,, where the neighors re verties of the tringles shring sie not ontining these verties These sies re mrke (enirle) on the piture We nee to show tht the numer of mrke sies is o It is 13 on the piture Let us onsier the generl se Let r = q The following segments re mrke (oorintes re given in the p q sis {u, v}):

4 204 D Fuhs (0, 1) (1, 0) (1, 2) (2, 1) ([r, [r+1) ([r + 1, [r) ([r+2,[r) ([r+2,[r+1) ([r+3,[r+1) ([2r+2,[2r) ([2r+2,[2r+1) ([2r+3,[2r+1) ([2r+4,[2r) ([4r+3,[4r+1) ([4r+4,[4r) ([4r+5, [4r) ([4r+5, [4r + 1)([5r+5, [5r) ([5r+5, [5r+1) ([(p q 4)r+p q 3, [(p q 4)r+1) ([(p q 4)r + p q 2, [(p q 4)r) ([(p q 2)r+p q 3, [(p q 2)r+1) ([(p q 2)r+p q 2, [(p q 2)r) ([(p q 2)r+p q + 3, [(p + q 2)) ([(p q 1)r+p q 1, [(p + q 1)r+1) ([(p q 1)r+p q, [(p q 1)r+1) ([(p q 1)r+p q, [(p q 1)r) ([(p q)r+p q 1, [(p q)r) ([(p q)r+p q, [(p q)r 1) [the lst segment is, tully, (p 1, q) (p, q 1) In this listing of segments, we see the lternting of groups of prllel segments n stir-like hins The groups of prllel segments ontin, respetively, [r+1, [4r+[2r+1,,[(p q 2)r (p q 4)r+1, q [(p q 1)r items; the stir-like hins ontin, respetively, 2([2r [r) + 1, 2([5r [4r) + 1,,2([(p + q 1)r [(p + q 2)r+1items The totl is [r+[2r [4r+[5r [(p q 2)r+[(p q 1)r+q where the numer of 1 s is p q A simple omputtion shows tht this sum is ([ 3 p [ q ) , whih is ertinly o 3 3 We rrive t the following result No simple geoesi segment emnting from ens t If this segment is etermine y reltively prime p, q, then it ens t

5 Geoesis on Regulr Polyher with Enpoints t the Verties 205 v u Fig 4 The ue n plnr evelopment of the segment p u + q v with p = 3, q = 2 if n only if p q mo 3; symptotilly, the proility of this event is 1 4 The remining 3 4 is eqully (euse of the symmetry) istriute etween,,, n 4 Cue We egin this setion with formulting some results from Fuhs n Fuhs (2007) n Fuhs (2014) We lel the verties of the ue s shown in Fig 4, left An ritrry urve on the surfe of the ue not pssing through the verties my e presente s urve in the plne furnishe with the stnr squre lttie, n there rises leling of the verties long this urve Importnt remrk: verties of the lttie whose oorintes hve the sme prity get the lels from the set {,,, }, n verties whose oorintes hve ifferent prities get the lels from the set {,,, } (mrke in Fig 4) Stright segments in the plnr evelopment orrespon to geoesis on the surfe of the ue A segment [(, ), ( + p, + q) with integer, n reltively prime p, q orrespons to simple geoesi segment on the ue joining two verties of the ue, n ll suh geoesi segments n e otine in this wy If we shift the segment to prllel segment strting t the point (ε, 0) with smll ε>0, then we get geoesi on the ue, whih my e (n, tully, lwys is) not lose: the segment in Fig 4, right, shifte little it to the right, orrespons to geoesi strting ner n ening ner To get lose geoesi, we nee to repet the segment on the plne ertin (miniml) numer n C (p, q) of times Theorem 41 (Fuhs n Fuhs 2007, Theorem 44) (1) For ll p, q, n C (p, q) = 2, 3, or 4; (2) if p n q re oth o, then n C (p, q) = 3; (3) if one of p, q is even, then n C (p, q) = 2 or 4 To formulte neessry result from Fuhs (2014), we nee some nottion Let S e the set of pirs (p, q) of integers suh tht p is even, q is o, n p, q o not

6 206 D Fuhs Fig 5 The funtion V ( p, q) hve ommon ivisors > 1 [thus, for exmple, (0, 1), (2, 1) S,ut(0, 3) / S The group Ɣ 2 of integer 2 2 mtries ongruent to the ientity moulo 2 ts trnsitively in S; it is known tht Ɣ 2 is free group with genertors A = [ 12 01, B = [ Let H Ɣ 2 e the sugroup generte y A 2, B 2, ABA n BAB It is prove in Fuhs (2014) tht H hs inex 3 in Ɣ 2 n not norml Theorem 42 [Fuhs 2014, Theorem 22, Prt (3) The group H hs two orits in S, n these orits re {n C (p, q) = 2} n {n C (p, q) = 4} The symptoti size of the seon of these orits is twie the symptoti size of the first one We will nee here some itionl fts For every pir (p, q) of reltively prime integers the segment p u+q v etermines simple geoesi segment on the surfe of the ue strting t Denote y V (p, q) the enpoint of this geoesi For exmple, Fig 4 shows tht V (3, 2) = A smll tle of vlues of the funtion V is shown in Fig 5 The funtion V hs lot of symmetries, oth visile n hien First of ll, if, for given (p, q), the pir (p, q ) is one of ( p, q), (p, q), (q, p) or ( q, p), then V (p, q ) is esily relte to V (p, q) For exmple, if q > 0, then V ( p, q) is otine from V (p, q) y the trnsformtion, ( inee, ll we nee for this trnsition, is the replement of the squre y the squre ) et It is importnt tht ll these trnsformtions preserve (n ) To esrie some other symmetries, onsier the following two mtries from H: C = B 2 = [ 10 41, D = AB 1 A = [ 34 23

7 Geoesis on Regulr Polyher with Enpoints t the Verties 207 Fig 6 To proof of Lemm 43 Lemm 43 (1) V (C( p, q)) = V ( p, q) (2) The trnsformtion V ( p, q) V (D(p, q)) ts in the following wy:,, Proof Essentilly, this theorem is prove in Fuhs (2014) The proof is ontine in Fig 6 [orrowe from Fuhs (2014) We n reple the stnr funmentl squre of the lttie y one of the prllelogrms (0, 0), (1, 4), (1, 5), (0, 1) or (0, 0), (3, 2), (7, 5), (4, 3) The leling of verties shown in Fig 6 provies lels for the verties of these two prllelogrm In the left igrm, it is gin, n this proves Prt (1) In the right igrm, it is whih mens tht the whole leling is trnsforme y the rottion of the ue whih mps the fe into the fe This is the rottion of the ue y 120 roun the igonl, s esrie in Prt (2) Now we formulte the min result of this setion Theorem 44 Consier simple geoesi segment on the ue emnting from the vertex ; suppose tht it orrespons to some reltively prime p, q (1) (Dvis et l 2016, Corollry 515, Theorem 517) The vertex V (p, q) nnot e the sme s ; it is one of,, with the proility 4 27 for eh; one of,, with the proility 1 9 for eh n it is with the proility 2 9 (2) The vertex V (p, q) is one of,,, if n C (p, q) = 4, is one of,,,if n C (p, q) = 3, n is,ifn C (p, q) = 2 Proof As in the proof of Theorem 31, the min thing we nee to prove is tht simple geoesi segment eginning t vertex of the ue never ens t the sme vertex It is ler from the remrk in the eginning of the setion tht if it the enpoint of the simple geoesi segment oinies with the eginning, then this segment orrespons to the pir (p, q) with p, q eing oth o

8 208 D Fuhs Fig 7 Verties of the iosheron f e e f Return to Theorem 44 Notie tht for ritrry reltively prime o p, q there exists sequene of trnsformtions esrie efore n in Lemm 43 whih reue the pir (p, q) to (1, 1) Inee, omining the trnsformtion C with sign hnges n swpping of oorintes, we n reue the generl se to the se of positive p, q with p < q < 2p [esies the se (p, q) = (1, 1); n this trnsformtion oes not inrese the minimum of solute vlues p, q Then we pply D; (p, q) eomes (4p 3q, 3p 2q), n 2p < 2q < 4p p < 3p 2q < q, or 3p 2q < p; thus the minimum of p, q ereses Repeting this proeure suffiiently mny times, we rrive t (p, q) = (1, 1) Thus n ritrry (p, q) n e otine from (1, 1) y hin of inverse trnsformtion, n, sine V (1, 1) = n no one of our trnsformtions onnets with nything else, we n onlue tht V (p, q) = Finlly, onsier n ritrry simple geoesi segment σ strtingt; let its plnr evelopment is p u + q v Tke the geoesi segment σ prllel to σ n strting t point of the ege lose to Ifσ ens t,, or then repet σ 4 times; if σ ens t,,or, then repet σ 3 times; n if σ ens t, then repet σ 2 times Oviously, we get geoesi on the ue ening t point of one of the eges,, or But in the lst two ses, to get lose geoesi, we nee to repet lrey repete 4, or 3, or 2 times σ 3 more times, whih woul men tht n C (p, q) = 12 or 9, or 6, in ontrition to Prt (3) of Theorem 41 Thus, the geoesi σ repete 4, or 3, or 2 times is lose n we onlue tht if V (p, q) is,,or, then n C (p, q) = 4, if V (p, q) is, or, then n C (p, q) = 3 n if V (p, q) is, then n C (p, q) = 2 Theorem 44 follows 5 Iosheron We lel the 12 verties of the regulr iosheron s shown in Fig 7 Notie tht, here prime mens opposite : the vertex is opposite to, the vertex is opposite to, et

9 Geoesis on Regulr Polyher with Enpoints t the Verties 209 f e f f e f e e f e f e f Fig 8 To proof of Lemm 53 As in the ses of the tetrheron n the otheron, geoesis on the surfe of the iosheron (not pssing through the verties) re presente y stright lines in the plne furnishe y the stnr tringulr tiling A non-self-repeting lose geoesi orrespons to the segment p u + q v with reltively prime integer p, q repete ertin numer n I (p, q) of times Theorem 51 (Fuhs n Fuhs 2007, Theorem 61) The numer n I (p, q) tkes vlues 2, 3, n 5 The following informtions of the funtion n I re ontine in Fuhs (2014) Let S = {(p, q) Z Z GCD(p, q) = 1}/(p, q) ( p, q) The group PSL(2, Z) = SL(2, Z)/{±I } trnsitively ts in S LetH e the sugroup of PSL(2, Z) generte y K = [ Theorem 52 (Fuhs 2014, Theorem 31), L = [ , n M = [ (1) The group H hs inex 10 in P SL(2, Z) (2) The group H hs three orits in S, n these orits re {n I (p, q) = 2}, {n I (p, q) = 3}, n {n I (p, q) = 5} (3) The symptoti sizes of these three orits re relte s 2:3:5 As in the se of the ue, we nee some enhnement of Theorem 52 For reltively prime p, q,letv (p, q) e the vertex of the iosheron, whih is the enpoint of the simple geoesi segment eginning t n orresponing to the plnr segment p u + q v Lemm 53 (1) V (M(p, q)) = V (p, q) (2) The trnsformtion V (p, q) V (K (p, q)) is the rottion of the iosheron e f, e f (3) The trnsformtion V (p, q) V (L(p, q)) is the rottion of the iosheron inverse to the rottion in (2): f e, f e

10 210 D Fuhs Proof As in the proof of Lemm 43, the trnsformtions onsist in replements of the funmentl tringle of the tiling s shown in Fig 8 [mostly orrowe from Fuhs (2014) We see tht the trnsformtions in Prts (1) (3) of lemm orrespon to the replement of the funmentl tringle (showe in Fig 8) y, respetively, tringles,, n f(rwn in olfe lines in Fig 8) Lemm follows The following sttement is the min result of this setion Theorem 54 Consier simple geoesi segment on the iosheron emnting from the vertex n ening t some vertex h; suppose tht it orrespons to some reltively prime p, q (1) The vertex h nnot e (2) The vertex h is one of the verties,,, e, f with the proilitiy 1 for eh; 10 it is one of the verties,,, e, f with the sme proility 3 for eh; n 50 it is with the proility 1 5 (3) The vertex h is one of,,, e, f,ifn I (p, q) = 5, it is one of,,, e, f,if n I (p, q) = 3, n it is,ifn! (p, q) = 2 Proof A to Lemm 53 tht V ( p, q) is lso otine from V (p, q) y rottion, this time e f, e f Consier the group H SL(2; Z) generte y the mtries I, K, L, M The tion of the group H in S hs three orits, the inverse imges the orits of H in S with resopet to the projetion S S Lemm 53, supplemente y the lst remrk, shows tht the sets T A ={(p, q) V (p, q) A with A eing one of the four sets {}, {,,, e, f }, {,,, e, f }, { } re invrint with respet to H Sine the three orits re represente y the pirs (1, 1), (1, 2), (1, 4) n V (1, 1) =, V (1, 2) =, V 1, 4) =, We onlue tht T {} is empty, while the three other sets T A re the three orits of H, n Theorem 52 yiels ientifitions T {,,,e, f } = n 1 I (5), T {,,,e, f } = n 1 I (3), T { } = n 1! (2) Theorem follows 6 Doeheron I nnot sy muh out this se Still it seems very likely tht on the surfe of the oeheron there exist simple geoesi segments whih egin n en t the sme vertex By mens of omputer experiments, I hve foun vriety of likely exmples of this The shortest one is shown in Fig 9 It is 22-gonl geoesi emnting from the vertex A uner the ngle α = tn φ 57φ with the sie (φ = 1618 is the golen rtio) Aoring to the omputer, the istne etween the 22n ege n A oes not exee (we ssume tht the length of the sie is 1) In the sme

11 Geoesis on Regulr Polyher with Enpoints t the Verties 211 α A Fig 9 A simple geoesi segment AA on the oeheron time, the istnes etween the other eges n other verties re never less thn 002 This mens tht in the unlikely event tht the 22n ege oes not hit A, we n mke it ening t A y smll perturtion of α Aknowlegments This work ws one uring my sty t the Mx Plnk Institute in Bonn I m grteful to the Institute for its hospitlity Referenes Dvis, D, Dos, V, Tru, C, Yng, J: Geoesi trjetories on regulr polyher Disrete Mth (2016) (To pper) Fuhs, D: Perioi illir trjetories in regulr polygons n lose geoesis on regulr polyher Geom Dei 170, (2014) Fuhs, D, Fuhs, E: Close geoesis on regulr polyher Mos Mth J 7, (2007)