Also vilble t http://mc-journl.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 1 (017) 415 44 Sphericl qudrngles with three equl sides nd rtionl ngles Abstrct Kris Coolset Deprtment of Applied Mthemtics Computer Science nd Sttistics, Ghent University, Krijgsln 81 S9, B 9000 Gent, Belgium Received 14 June 016, ccepted 5 June 016, published online 19 Februry 017 When the condition of hving three equl sides is imposed upon (convex) sphericl qudrngle, the four ngles of tht qudrngle cnnot longer be freely chosen but must stisfy n identity. We derive two simple identities of this kind, one involving rtios of sines, nd one involving rtios of tngents, nd improve upon n erlier identity by Ueno nd Agok. The simple form of these identities enble us to further investigte the cse in which ll of the ngles re rtionl multiples of nd produce full clssifiction, consisting of 7 infinite clsses nd 9 spordic exmples. Aprt from being interesting in its own right, these qudrngles ply n importnt role in the study of sphericl tilings by congruent qudrngles. Keywords: Sphericl qudrngle, rtionl ngle, sphericl tiling. Mth. Subj. Clss.: 51M09, 5C0, 11Y50 1 Introduction In generl there will be n infinite number of non-congruent sphericl qudrngles with given (ordered) qudruple of ngles α,,, δ, provided tht < α + + + δ < 6. By imposing restrictions on the sides of qudrngle this is reduced to finite number. In this pper we shll investigte the cse of convex qudrngle ABCD with (t lest) three equl sides, sy AB = BC = CD = nd derive two simple identities which must be stisfied by the ngles of tht qudrngle s consequence of this restriction (cf. Theorem.1). A similr, but more complicted identity for this cse ws lredy published by Ueno nd Agok in [3]. We shll show tht our identities re stronger (cf. Section ) in sense to be mde cler below. E-mil ddress: kris.coolset@ugent.be (Kris Coolset) cb This work is licensed under http://cretivecommons.org/licenses/by/3.0/
416 Ars Mth. Contemp. 1 (017) 415 44 The identity of Ueno nd Agok rose in the serch for tilings of the sphere by congruent qudrngles. In tht context it is prticulrly relevnt to consider qudrngles ll of whose ngles re rtionl multiples of (henceforth simply clled rtionl ngles). Indeed, consider vertex P of such tiling. P belongs to certin number (sy N A ) of qudrngles in the tiling for which P corresponds to vertex A of the qudrngle. Likewise there will be N B qudrngles for which P corresponds to B, nd similr numbers N C, N D for C nd D. Becuse the sum of ll ngles in P must be, we find N A α + N B + N C + N D δ =, (1.1) where α,,, δ re the corresponding ngles of the qudrngle (see Figure 1 for nming conventions). A different vertex P of the tiling will led to similr identity, but generlly with different vlues of N A, N B, N C, N D. We my tret the set of identities (1.1) tht rise from ll vertices of given tiling s system of equtions with unknowns α,,, δ. Note tht ll coefficients in these equtions re integers, while every right hnd side is equl to. In prticulr, if this system hs rnk 4, there will be exctly one solution nd it will consist entirely of rtionl ngles. In Theorem 3. we give full clssifiction of ll convex sphericl qudrngles with three equl sides whose ngles re rtionl. There turn out to be 7 infinite fmilies of such qudrngles nd 9 spordic exmples. The proof of Theorem 3. hinges on the fct tht our identity (3.) cn be rewritten s n equlity between two products of two sines nd tht instnces of such identities with rtionl ngles were lredy clssified by Myerson [] (cf. Theorem 3.1). A α B b C δ D A B x y C D Figure 1: Nming conventions for sphericl qudrngle ABCD with three equl sides. Reltions between the ngles In wht follows we shll consider sphericl qudrngle ABCD with corresponding ngles α,,, δ, sides,,, b nd digonls x, y s indicted in Figure 1. The sphericl qudrngle will be clled convex if it stisfies 0 <, b, x, y, α,,, δ <. In prticulr this mens tht ll constituent sphericl tringles ABC, ABD, ACD nd BCD re proper nd stisfy the clssicl lws of sphericl trigonometry. We tke the following inequlities for such qudrngles from [1, Lemm.1]: α + δ < +, α + δ < +, α + < + δ, + δ < + α. (.1)
K. Coolset: Sphericl qudrngles with three equl sides nd rtionl ngles 417 As mentioned in the introduction, we lso hve E = α + + + δ > 0, (.) where E denotes the sphericl excess of the qudrngle, which is equl to the re of the qudrngle on unit sphere. Finlly, we note tht α = δ if nd only if =, cf. [1, Lemm.3]. Theorem.1. In convex sphericl qudrngle ABCD with three equl sides, the following identities hold: or equivlently, sin(α ) sin tn ( δ ) tn δ = sin(δ ) sin, (.3) = tn ( α ) tn α. (.4) Proof. Consider the equilterl sphericl tringle ABC. The (polr) cosine rule for side BC yields cos = cos φ + cos φ cos sin φ sin = cot φ 1 + cos sin = cot φ cot, (.5) where φ = BAC = ACB s indicted in Figure. The sine rules for side AC in both A φ B x φ b C δ D ABC nd ACD, yield nd hence Figure : The sphericl tringles ABC nd ACD. sin sin x = sin φ sin, sin δ sin x sin(α φ) =, sin sin δ sin(α φ) = = sin α cot φ cos α. sin sin φ Multiplying by cot nd using sin = cos sin nd (.5), yields sin δ sin = sin α cos cos α cot, whence sin δ + cos α sin cos = sin α sin. (.6)
418 Ars Mth. Contemp. 1 (017) 415 44 By repeting the rgument bove for tringles ABD nd BCD, or equivlently, by interchnging α δ nd, we find cos = We shll now use (.6.7) to compute sin α + cos δ sin sin δ sin. (.7) def = cos sin α sin δ sin α + cos δ = cos sin α sin δ + cos α sin δ in two wys. From (.6) we obtin = csc sin δ + csc cos α sin sin δ + cos α sin δ = (csc ) 1 sin δ + cot cos α sin δ + cos α = cot sin δ + cot cos α sin δ + cos α = By symmetry, from (.7) we obtin = ( cot sin α + cos δ ). ( cot sin δ + cos α ). (Note tht the originl definition of remins unchnged when interchnging α nd δ.) Combining both vlues of we end up with two possibilities: cot sin δ + cos α = ± ( cot sin α + cos δ ). (.8) We consider both cses seprtely. The second cse will turn out to be impossible. Cse 1. Assume there is plus sign in the right hnd side of (.8). Then (.8) rewrites to cot sin δ cos δ = cot sin α cos α, (.9) which is equivlent to (.3). In generl, if x 1 /y 1 = x /y, then lso (x 1 y 1 )/(x 1 + y 1 ) = (x y )/(x + y ). Applying this to (.3) yields sin(δ ) sin sin(δ ) + sin = sin(α ) sin sin(α ) + sin, which trnsforms to nd hence ) cos δ sin( δ ) sin δ cos( δ ) = cos α sin( α sin α cos( α ), tn ( δ ) tn δ = tn ( α ) tn α.
K. Coolset: Sphericl qudrngles with three equl sides nd rtionl ngles 419 Cse. Assume there is minus sign in the right hnd side of (.8), i.e., cot sin δ + cos α = cot sin α cos δ. This formul cn be obtined from the formul for the first cse by replcing α by α nd δ by δ. As consequence, we now hve the following identities: sin( δ ) sin = sin( α ) tn ( sin, δ ) tn ( ) δ = tn ( α ) tn α, equivlent to sin(δ + ) sin = sin(α + ) sin, tn δ tn ( δ + ) = tn ( α + ) tn α. Becuse the tngent function is monotonous in the intervl [0, /[, the ltter is only possible if = = 0 or if one of 1 (δ + ), 1 (α + ) lies outside tht intervl. And becuse of the signs, this implies tht both vlues must belong to the intervl ] 1, [. Hence < α +, + δ. Now α + < + δ by (.1). Hence α + < + δ nd hence α < δ. By symmetry, lso δ < α, contrdiction. In [3] Ueno nd Agok derived the following identity for sphericl qudrngles with three equl sides: (1 cos ) cos α (1 cos )(1 cos ) cos α cos δ + (1 cos ) cos δ + cos cos + sin α sin sin sin δ 1 = 0. (.10) We hve Lemm.. Formul (.10) is equivlent to cot sin δ cot sin α = ±(cos α cos δ). (.11) Proof. We express cos nd sin in terms of cot s follows: sin = cot cot + 1, cos = cot 1 cot + 1, 1 cos = cot + 1 (.1) nd similr for cos nd sin. Also note tht cos cos 1 = (cot 1)(cot 1) (cot + 1)(cot + 1) 1 = cot cot (cot + 1)(cot (.13) + 1). Applying (.1.13) to the left hnd side of (.10) trnsforms it into cos α cot + 1 4 cos α cos δ (cot + 1)(cot + 1) + cos δ cot + 1 cot + cot 4 sin α cot (cot + 1)(cot + cot sin δ + 1) (cot + 1)(cot + 1)
40 Ars Mth. Contemp. 1 (017) 415 44 which fter multiplying by the common denomintor nd dividing by, reduces to cos α (cot ) + 1 cos α cos δ + cos δ (cot ) + 1 cot cot + sin α cot cot sin δ = cos α cos α cos δ + cos δ + cos α cot + cos δ cot cot cot + sin α cot cot sin δ ( = (cos α cos δ) cot sin δ cot ) sin α. Remrk tht choosing the minus sign in (.11) yields our formul (.9) from the proof of Theorem.1. This shows tht Theorem.1 is stronger thn the result of Ueno nd Agk, s they lso llow solutions with plus sign in the right hnd side of (.11). 3 Rtionl ngles In wht follows we shll investigte sphericl qudrngles with three equl sides with the dditionl property tht the four ngles α,,, δ re rtionl. Our min tool is the following theorem from []. Theorem 3.1 (Myerson). For ll we hve All other solutions of sin 6 sin = sin sin ( ). (3.1) sin x 1 sin x = sin x 3 sin x 4 (3.) with rtionl numbers x 1, x, x 3, x 4 such tht 0 < x 1 < x 3 x 4 < x 1/, re given in Tble 1. x 1 x x 3 x 4 1/1 8/1 1/14 3/14 1/14 5/14 /1 5/1 4/1 10/1 3/14 5/14 1/0 9/0 1/15 4/15 /15 7/15 3/0 7/0 1/30 3/10 1/15 /15 1/15 7/15 1/10 7/30 1/10 13/30 /15 4/15 Tble 1: Nongeneric solutions to (3.). x 1 x x 3 x 4 4/15 7/15 3/10 11/30 1/30 11/30 1/10 1/10 7/30 13/30 3/10 3/10 1/15 4/15 1/10 1/6 /15 8/15 1/6 3/10 1/1 5/1 1/10 3/10 1/10 3/10 1/6 1/6
K. Coolset: Sphericl qudrngles with three equl sides nd rtionl ngles 41 Althought our condition (.3) is lmost direct mtch with equtions (3.1) nd (3.) of the theorem, there re some dditionl complictions tht must be tken into ccount. Most importntly, Theorem 3.1 imposes extr conditions on the ngles x 1,..., x 4 which re too stringent for the ngles α,,, δ of (.3). First, ll ngles in Theorem 3.1 must lie in the intervl ]0, /[ while of the four ngles in (.3) only nd stisfy this restriction, while the other two (α, δ ) re only known to lie in the intervl ] /, [. This mens tht we hve to renormlize these ngles by using the identities sin( x i ) = sin x i, sin( x i ) = sin x i. As consequence, we shll lwys need to consider the following five cses: {α,,, δ }, { α +,,, δ }, {x 1, x, x 3, x 4 } = {α,,, δ + }, { α +,,, δ + }, { α,,, δ}. (3.3) (Note tht α < 0 utomticlly implies δ < 0 becuse the signs of both sides of (.3) must be the sme.) Furthermore Theorem 3.1 ssumes specific ordering of the vribles x 1, x, x 3, x 4, which gin we cnnot gurntee. In principle we must therefore consider eight different wys to ssign the ngles on the right hnd side of (3.3) to the ngles of the left hnd side, yielding 40 possibilities in totl. We cn reduce this mount by hlf by tking into ccount the symmetry α δ,. Finlly, Theorem 3.1 does not consider the trivil cses where one (nd then t lest two) of the ngles is zero, or where {sin x 1, sin x } = {sin x 3, sin x 4 }. Tking ll of this into considertion leds to Theorem 3.. Consider convex sphericl qudrngle with three equl sides, with ngles nd sides s indicted in Figure 1. If the ngles α,,, δ re rtionl multiples of, then they must stisfy one of the following properties, or property derived from these by interchnging α δ nd : 1. α = nd = δ (nd ll four sides re equl),. α = δ nd =, 3. α = nd δ =, with α + δ <, 4. α = 3, = 3 nd δ = 3, with < < 3. 5. α = 6 +, = nd δ = +, with 3 < <, 6. α = 6 +, = nd δ = + 3 (= 3α), with 4 15 < < 3, 7. α = 6 +, = nd δ = 3 3, with < < 5 6, 8. (spordic cses) α/, /, /, δ/ re s listed in Tble.
4 Ars Mth. Contemp. 1 (017) 415 44 α/ / / δ/ 9/4 8/1 3/7 3/4 31/4 8/1 3/7 3/4 5/6 8/1 5/7 17/4 37/4 8/1 5/7 17/4 5/6 3/7 0/1 17/4 11/4 5/7 0/1 1/6 9/4 5/7 0/1 3/4 49/60 4/15 7/10 17/60 53/60 4/15 7/10 17/60 7/10 4/15 13/15 7/30 49/60 3/10 14/15 17/60 3/30 1/3 14/15 3/10 11/15 7/15 3/5 8/15 13/15 7/15 3/5 8/15 17/30 7/15 14/15 3/10 Tble : Spordic cses of Theorem 3.. α/ / / δ/ 5/6 8/15 3/5 19/30 3/30 8/15 3/5 19/30 5/6 8/15 11/15 17/30 9/10 8/15 11/15 17/30 3/60 8/15 9/10 13/60 31/60 8/15 9/10 19/60 17/30 8/15 13/15 11/30 31/60 3/5 5/6 3/60 11/15 3/5 13/15 8/15 19/30 3/5 14/15 13/30 5/6 3/5 14/15 17/30 19/60 7/10 14/15 13/60 37/60 7/10 14/15 9/60 3/30 11/15 14/15 19/30 Proof. (The proofs of the inequlities in the sttement of the Theorem re left to the reder. They re immedite consequences of (.1.).) We split the proof into three prts. Prt 1. We first consider the trivil cses. The only ngles in (.3) which re llowed to be zero, re α nd δ. This corresponds to cse 3 in the sttement of this theorem. Next, {sin x 1, sin x } = {sin x 3, sin x 4 } corresponds to either ( sin α ) = sin ( nd sin δ ) = sin, (3.4) or ( sin α ) ( = sin δ ) nd sin = sin. (3.5) Eqution (3.4) further splits into 4 different cses: α = nd δ =, α = ± nd δ =, α = nd δ = ±, α = ± nd δ = ±. The first cse corresponds to cse 1 in the sttement of this theorem, the other three re not llowed becuse then α = ± or = ±. Similrly, eqution (3.5) splits into the following cses: α = δ α = ± δ + α = δ α = ± δ + nd nd nd nd =, =, =, =.
K. Coolset: Sphericl qudrngles with three equl sides nd rtionl ngles 43 The first of these reduces to α = δ nd =, i.e., cse in the sttement of this theorem. The second implies =, α + δ = + which is disllowed by (.1). The third leds to α + = + δ, gin forbidden by (.1). The lst yields either α = δ or α = δ, which gin is not llowed. Prt. Let us now consider formul (3.1). As mentioned bove, this formul must be pplied to our problem in 0 different wys, 4 permuttions of the ngles in (3.1) ech time mtched to the 5 cses listed in (3.3). In Tble 3 we list ech of these 0 possibilities (columns 1 4), nd the corresponding vlues of α,,, δ (columns 5 8). In ech of the four tbles, columns 1 4 contin the sme vlues but correspond to different ngles of (3.1), s indicted in the column heders. Tble 3: The generic cse of Theorem 3.. α α 6 α + α + α 6 α α α + α + α α δ δ 3 3 3 α + + δ < δ + 3 3 3 + α + δ < + < 3 but α + + δ > > 3 δ 3 3 α + + δ = δ + 3 3 + α + δ > + δ 3 3 + α < 0 α δ δ + 3 6 + α + + δ = δ + + 7 3 6 α + δ = + δ 3 3 3 6 + δ + 3 3 7 3 6 α + δ > + δ 3 3 6 α + + δ < 6 α α α + α + α 6 α α α + α + α α δ δ 6 + + δ + 6 + + 3 δ 5 6 + + α + δ > + δ + 5 6 + + 3 α + δ > + δ 6 + + 3 α + + δ < α δ δ 3 3 δ + 3 + + δ > + α δ 4 3 3 α + > + δ δ + 4 3 + α + δ > + δ 3 α + + δ < It turns out tht 16 of these options re disllowed by the inequlities (.1.). We list the corresponding detils in the right hnd column of ech tble. The four possibilities tht remin re listed s cses 4 7 in the sttement of the theorem.
44 Ars Mth. Contemp. 1 (017) 415 44 Prt 3. For the spordic exmples from Tble 1 we could proceed in the sme mnner s in prt of this proof. Although there re some shortcuts which could be tken to void to hve to consider ech of the 0 15 = 300 cses seprtely, we thought it less error prone to enlist the help of computer. Recll tht i sin m n belongs to the cyclotomic field Q(ζ n) where ζ n is primitive n-th root of unity. Modern computer lgebr systems cn do exct rithmetic over such cyclotomic fields, hence we my use such system to directly check ll instnces of eqution (.3) in which α, /, /, δ re integrl multiples of n nd re in the required rnge. From Tble 1 we my derive ll vlues of n which we re required to try: for ech row let n denote twice the lest common multiple of the four denomintors. In fct, mny rows will yield the sme vlue of n nd it turns out to be sufficient to do the computtions only for n = 84 nd n = 10. We used this method to obtin the vlues in Tble. (The source code for these computtions is vilble from http://cgt.ugent.be/rtqud/.) The sme method, with n = 1p, p prime > 7, ws used to verify the results of prt of this proof. Note tht in cse 3 of Theorem 3. the qudrngle is union of three disjoint congruent tringles with ngles α, δ nd /3 cf. Figure 3. α δ 3 3 δ α 3 α δ Figure 3: The specil cse α =, δ =. References [1] Y. Akm nd N. Vn Cleemput, Sphericl tilings by congruent qudrngles: Forbidden cses nd substructures, Ars Mth. Contemp. 8 (015), 97 318, http://mc-journl.eu/ index.php/mc/rticle/view/581/767. [] G. Myerson, Rtionl products of sines of rtionl ngles, Aequtiones Mth. 45 (1993), 70 8, doi:10.1007/bf0184446. [3] Y. Ueno nd Y. Agok, Exmples of sphericl tilings by congruent qudrngles, Mem. Fc. Integrted Arts nd Sci., Hiroshim Univ., Ser IV 7 (001), 135 144, doi:10.1507/781.