Spherical folding tessellations by kites and isosceles triangles IV

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1 Also ille t ISSN - (printed edn.), ISSN - (electronic edn.) ARS MATHEMATICA CONTEMPORANEA (0) Sphericl folding tesselltions y kites nd isosceles tringles IV Ctrin P. Aelino, Altino F. Sntos Pólo CMAT-UTAD, Centro de Mtemátic d Uniersidde do Minho Uniersidde de Trás-os-Montes e Alto Douro, UTAD, Quint de Prdos, Vil Rel, Portugl Receied July 0, ccepted Septemer 0, pulished online Septemer 0 Astrct The clssifiction of the dihedrl folding tesselltions of the sphere nd the plne whose prototiles re kite nd n equilterl tringle were otined in []. Recently, this clssifiction ws extended to isosceles tringles so tht the clssifiction of sphericl folding tesseltions y kites nd isosceles tringles in three cses of djcency ws presented in [,, ]. In this pper we finlize this clssifiction presenting ll the dihedrl folding tesselltions of the sphere y kites nd isosceles tringles in the remining three cses of djcency, tht consists of fie spordic isolted tilings. A list contining these tilings including its comintoril structure is presented in Tle. Keywords: Dihedrl f-tilings, comintoril properties, sphericl trigonometry, symmetry groups. Mth. Suj. Clss.: C0, 0B, B0 Introduction By folding tesselltion or folding tiling of the Eucliden sphere S we men n edge-toedge pttern of sphericl geodesic polygons tht fills the whole sphere with no gps nd no oerlps, nd such tht the underlying grph hs een lency t ny ertex nd the sums of lternte ngles round ech ertex re. Folding tilings (f-tiling, for short) re strongly relted to the theory of isometric foldings on Riemnnin mnifolds. In fct, the set of singulrities of ny isometric folding corresponds to folding tiling, see [] for the foundtions of this suject. The study of this specil clss of tesselltions ws initited in [] with complete clssifiction of ll sphericl monohedrl folding tilings. Ten yers ltter Ueno nd Agok E-mil ddresses: celino@utd.pt (Ctrin P. Aelino), folgdo@utd.pt (Altino F. Sntos) c This work is licensed under

2 0 Ars Mth. Contemp. (0) [] he estlished the complete clssifiction of ll tringulr sphericl monohedrl tilings (without ny restriction on ngles). Dwson hs lso een interested in specil clsses of sphericl tilings, see [0], [] nd [], for instnce. The complete clssifiction of ll sphericl folding tilings y rhomi nd tringles ws otined in 00 []. A detiled study of the tringulr sphericl folding tilings y equilterl nd isosceles tringles is presented in []. Sphericl f-filings y two noncongruent clsses of isosceles tringles in prticulr cse of djcency were recently otined []. Here we discuss dihedrl folding tesselltions y sphericl kites nd isosceles sphericl tringles. A sphericl kite K (Figure ) is sphericl qudrngle with two congruent pirs of djcent sides, ut distinct from ech other. Let us denote y (α, α, α, α ), α > α, the internl ngles of K in cyclic order. The length sides re denoted y nd, with <. From now on T denotes sphericl isosceles tringle with internl ngles β nd γ (γ β), nd length sides c nd d, see Figure. We shll denote y Ω(K, T ) the set, up to isomorphism, of ll dihedrl folding tilings of S whose prototiles re K nd T. K d T c d c Figure : A sphericl kite nd sphericl isosceles tringle Tking into ccount the re of the prototiles K nd T we he As α > α we lso he α + α + α > nd β + γ >. α + α >. We egin y pointing out tht ny element of Ω (K, T ) hs t lest two cells congruent to K nd T, respectiely, such tht they re in djcent positions nd in one nd only one of the situtions illustrted in Figure. After certin initil ssumptions re mde, it is usully possile to deduce sequentilly the nture nd orienttion of most of the other tiles. Eentully, either complete tiling or n impossile configurtion proing tht the hypotheticl tiling fils to exist is reched. In the digrms tht follow, the order in which these deductions cn e mde is indicted y the numering of the tiles. For j, the loction of tiling j cn e deduced directly from the configurtions of tiles (,,..., j ) nd from the hypothesis tht the configurtion is prt of complete tiling, except where otherwise indicted.

3 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV c d d K d T K d T K c T d c d I II III K K K d d c d T c IV c T d V d T d VI Figure : Distinct cses of djcency of K nd T The cses of djcency I, II nd III he lredy een nlyzed in [,, ]. In this pper we consider the remining cses of djcency IV, V nd V I. Cse of Adjcency IV Suppose tht ny element of Ω (K, T ) hs t lest two cells congruent, respectiely, to K nd T, such tht they re in djcent positions s illustrted in Figure IV. As = d, using trigonometric formuls, we otin cos γ( + cos β) sin γ sin β = α cos + cos α cos α. (.) sin α sin α Concerning the ngles of the tringle T we he necessrily one of the following situtions: γ > β or γ < β. In the following susections we consider seprtely ech one of these cses.. γ > β In this cse we he γ > nd, c < = d. Proposition.. Under the conditions ssumed in this section, there is single folding tiling, L, such tht α =, α + γ = nd α = β =. Plnr nd D representtions of L re gien in Figure. Proof. Suppose tht ny element of Ω (K, T ) hs t lest two cells congruent, respectiely, to K nd T, such tht they re in djcent positions s illustrted in Figure IV.

4 Ars Mth. Contemp. (0) Osering Figure, tile cnnot e kite this cse ws lredy nlyzed in [] (cse.) nd, under the current conditions, gie rise to no f-tilings. Consequently, side of length c of ech tringle must e djcent to side of length c of nother tringle. Moreoer, we he α α > α. In fct, if α > α nd Figure : Locl configurtions (i) α + γ = (Figure ), we rech contrdiction t ertex, s α + ρ >, for ll ρ {α, α }; (ii) α +γ < (Figure ), t ertex we he necessrily α +γ +kα =, k. But (α + α + γ) + (α + γ + α ) > (α + α + α ) >, which is impossile. Figure : Locl configurtions At ertex we he α + γ = or α + γ <.. Suppose firstly tht α + γ = (Figure ). At ertex we he kα =, with k. As (α + γ) + (α + α + α ) > (α + α ) + (γ + β + β) >, it follows tht k = ( α = ). With the leling of Figure, if (i) θ = α (Figure ), then t ertex we he necessrily α + kβ =, k, nd so α > = α > γ α > β (note tht α + β + α > ). But then tile must e tringle, which is impossile; (ii) θ = β (Figure ), then t ertex it follows tht α + kβ =, k (note tht we must he α > β). But t ertex we he γ + γ < nd γ + γ + ρ >, for ll ρ. (iii) θ = γ, we get the configurtion illustrted in Figure. Now, if

5 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV 0 cse (i) cse (ii) Figure : Locl configurtions cse (iii) 0 0 cse (iii) Figure : Locl configurtions θ = α (Figure ), we he necessrily α + kα =, with k, nd α > = α > γ > β > α (α + β + α > ). But then, the other sum of lternte ngles t ertex must e greter or equl thn α + β + β >, which is contrdiction ( (α + γ) + (α + α ) + (α + β + β) > (α + α + α ) + (β + γ + γ) > ). θ = β (Figure ), t ertex we he γ + γ + kα =, k, nd contrdiction rises t ertex s α + ρ >, for ll ρ {α, α }. (c) θ = γ, t ertex we he θ {β, α }. In the first cse, illustrted in Figure, we rech contrdiction t ertex. On the other hnd, if θ = α, due to the ngles inoled in the sums of lternte ngles t ertex, we must he α = β. Tking into ccount the tringle nd the kite s res, it follows tht γ + β + β = γ + α + α = (Figure ). At ertex we he α + β < nd α + β + ρ >, for ll ρ {α, α, α, β, γ}. (d) θ = α, tking into ccount the nlysis of the preious cses, t ertex we he kα = kβ =, k. Due to the kite s re, it follows tht γ < β nd consequently cos β < cos ( ) γ. Using eqution (.), we conclude tht β >, nd so k =. The lst configurtion is then extended to the one illustrted in Figure. We shll denote this f-tiling y L. Its D

6 Ars Mth. Contemp. (0) 0 cse (iii) 0 cse (iii)(c) Figure : Locl configurtions 0 Figure : Locl configurtion occurring in cse (iii)(c) representtion is gien in Figure.. Suppose now tht α + γ < (Figure ). Agin, due to the nlysis mde in [] (cse.), we use the fct tht side of length c of ech tringle must e djcent to side of length c of other tringle. At ertex we must he α + γ + kα =, with k. Neertheless, we rech contrdiction t ertex (Figure 0) since there is no wy to stisfy the ngle-folding reltion round this ertex.. γ < β In this cse we he β > nd < = d < c. Proposition.. Under the conditions ssumed in this section, there is single folding tiling, J, such tht α =, α + γ =, γ = nd β + β + α =. Plnr nd D representtions of J re gien in Figure.

7 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV plnr representtion D representtion Figure : f-tiling L Figure 0: Locl configurtion occurring in cse Proof. Suppose tht ny element of Ω (K, T ) hs t lest two cells congruent, respectiely, to K nd T, such tht they re in djcent positions s illustrted in Figure IV. As c, we get the configurtion illustrted in Figure, nd, t ertex, we he cse Figure : Locl configurtions

8 Ars Mth. Contemp. (0) α + γ = or α + γ <.. Suppose firstly tht α + γ = (Figure ). Note tht the conditions α > α α nd α > α > α led to contrdiction t ertex, s α + ρ >, for ll ρ {α, α }. Therefore α α > α. Now, if (i) α + α =, then β + β + kα =, k, nd so α > α = > β > γ > α. Consequently, γ = (s β <, we he γ > ). Then, the lst configurtion is extended to the one illustrted in Figure. We shll denote this f-tiling y J. Its D representtion is gien in Figure plnr representtion D representtion Figure : f-tiling J (ii) α + α <, then kα =, k, β + β + kα =, k, nd so α > > β > α > γ > α. As γ >, we he necessrily k = (Figure ). Now, if t ertex we he k > (Figure ), one of the ngles θ, θ or θ must e α. But then we rech contrdiction t ertex, or, respectiely, s α + ρ >, for ll ρ {α, α }. On the other hnd, if k =, we get the configurtion illustrted in Figure (note tht t ertex we cnnot he γ+γ+γ =, s = α > γ). At ertex we rech similr contrdiction s in the cse k >.. Suppose now tht α + γ < (Figure ). If α > α α or α > α > α, t ertex we must he α + kγ =, with k, nd consequently t ertex it follows tht α +α, nd so α nd α +α >. But then n incomptiility on the sides rises t ertex. If α α > α, nd (i) θ = α (Figure ), then θ must e β, otherwise we get, t ertex, θ = α nd α + ρ >, for ll ρ {α, α }. Neertheless, n impossiility cnnot e oided t ertex since we otin β + γ + ρ >, for ll ρ {α, α }.

9 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV Figure : Locl configurtions occurring in cse (ii) 0 0 cse (ii) cse (i) Figure : Locl configurtions (ii) θ = γ nd θ = β (Figure ), then γ < nd β >. At ertex we must he β+α, howeer (α +γ+γ)+(β+α ) = (β+γ+γ)+(α +α ) > is impossile. θ = γ, it follows tht α + kγ =, k, s illustrted in Figure. But ny choices for θ nd θ led to contrdiction. Cse of Adjcency V Proposition.. Ω(K, T ) is composed y single folding tiling, M, such tht α =, α + β = nd γ = α =. For plnr representtion see Figure 0. Its D representtion is gien in Figure.

10 Ars Mth. Contemp. (0) Figure : Locl configurtions occurring in cse (ii) Proof. Suppose tht ny element of Ω (K, T ) hs t lest two cells congruent, respectiely, to K nd T, such tht they re in djcent positions s illustrted in Figure V. The cse nlyzed in [] (cse.) gie rise to no f-tilings including two cells in these djcent positions, nd so side of length c of ech tringle must e djcent to side of length c of nother tringle.. If α > α, then α > nd we get the configurtion of Figure. If α + β = (Figure ), we he α = (ertex ), nd so α + α >, justifying the choice for θ. But t ertex we otin contrdiction s α + γ + γ > ((α + α ) + (α + β) + (α + γ + γ) = (α + α + α ) + (β + γ + γ) > ). Figure : Locl configurtions occurring in cse If α + β <, then α + kβ =, with k (note tht α + α > ). Consequently, γ > β nd α > β. Osering Figure, we conclude tht there is no wy to stisfy the ngle-folding reltion round ertex (α + α > α + α >, α + α >, α + γ + ρ >, for ll ρ, nd θ = β implies θ = γ nd γ + γ + ρ >, for ll ρ).. Suppose now tht α α (Figure ). It follows tht α > α > β nd γ >.. If β > γ, then t ertex we must he α +β +kα =, with k, or α +β =. In the first cse we he α > α > β > γ > α (Figure ). As θ or θ must e α, we get n impossiility t ertex or, respectiely.

11 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV cse cse Figure : Locl configurtions 0 Figure : Locl configurtions occurring in cse. Therefore α +β =. At ertex we cnnot he α +β = = α +α, s illustrted in Figure, otherwise t ertex we get α +γ +kα =, k, nd contrdiction rises t ertex. Consequently, we get the configurtion illustrted in Figure. Note 0 0 Figure : Locl configurtions occurring in cse. tht t ertex we cnnot he γ + γ + kα =, k, nor γ + γ + γ + kα =, k,

12 0 Ars Mth. Contemp. (0) otherwise we otin similr contrdiction s efore (in fct we cnnot he two ngles α djcent). Osere lso tht we he necessrily α + α =, s α + α + α >. Now, θ = α, θ = γ or θ = β... If θ = α (Figure ), t ertex we must he α + α =, which implies α = β. Neertheless, contrdiction rises t ertex since we get α + γ + kα >, for ll k... If θ = γ (Figure 0), t ertex we otin β +γ +kα =. But t ertex we get α +γ+kα =, which is not possile s (α +γ+α )+(α +β)+(α +α ) > (α + α + α ) + (β + γ + γ) > f-tiling M Figure 0: Locl configurtions occurring in cse... If θ = β, the lst configurtion is extended to the one illustrted in Figure 0. We shll denote this f-tiling y M. Its D representtion is gien in Figure. Figure : f-tiling M

13 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV. Suppose now tht β < γ (Figure ). In this cse we he γ > nd θ = β or θ = α. Figure : Locl configurtions occurring in cse. If θ = β (α + β, see Figure ), then t ertex we must he γ + γ + kα =, with k 0. As we seen efore, s two ngles α in djcent positions led to contrdiction, we must he γ + γ =. Moreoer, θ cnnot e α, otherwise we would otin θ = α nd, t ertex, α +γ >. The cse θ = β lso leds to contrdiction s γ + γ = nd ertex cnnot he lency four. Finlly, if θ = α, we otin the configurtion illustrted in Figure. At ertex we rech contrdiction s (α + β + α ) + (α + γ) + (α + α ) > (α + α + α ) + (β + γ + γ) >. Figure : Locl configurtion occurring in cse. Cse of Adjcency V I Suppose tht ny element of Ω (K, T ) hs t lest two cells congruent, respectiely, to K nd T, such tht they re in djcent positions s illustrted in Figure V I. As = c, using trigonometric formuls, we otin cos β + cos γ sin γ = α cos + cos α cos α. (.) sin α sin α

14 Ars Mth. Contemp. (0) Remrk.. The cses nlyzed in [] nd [] gie rise to no f-tilings including two cells in these djcent positions, nd so side of length c of ech tringle must e djcent to side of length c of nother tringle. Proposition.. Ω(K, T ) iff (i) α + γ =, α =, γ + γ + α = nd β =, or (ii) α + γ =, α = β = nd γ + γ + α =. In the first cse, there is single f-tiling denoted y N. A plnr representtion is gien in Figure nd D representtion is gien in Figure. In the second cse, there is single f-tiling, P. The corresponding plnr nd D representtions re gien in Figure nd Figure 0, respectiely. Proof. Concerning the ngles of the tringle T we he necessrily one of the following situtions: γ > β or γ < β. We consider seprtely ech one of these cses.. Suppose firstly tht γ > β. If α > α, then α > nd we get the configurtion of Figure. Due to the edge Figure : Locl configurtions occurring in cse lengths nd lso Remrk., cnnot he lency four nd so α +γ +kα =, k. Therefore, nlyzing ertices nd we conclude tht α + α < nd α, which is impossile tking into ccount the kite s re. Thus, α α > α (Figure ) nd θ = β or θ = γ. In the first cse, cnnot he lency four nd there is no wy to stisfy the ngle-folding reltion round this ertex. Consequently, θ = γ nd (i) if α + γ <, then α + γ + kα =, k (Figure ). At ertex we rech contrdiction s α + ρ >, for ll ρ {α, α }. (ii) if α + γ =, then the lst configurtion extends to the one illustrted in Figure. Now, if θ = β (Figure ), we otin contrdiction t ertex. On the other hnd, if θ = γ glol plnr representtion is chieed s illustrted in Figure. We denote such f-tiling y N. The corresponding D representtion is gien in Figure.

15 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV cse (i) 0 0 cse (ii) Figure : Locl configurtions occurring in cse f-tiling N Figure : Locl configurtions occurring in cse (ii). Suppose now tht γ < β. If α > α, then α > nd we get the configurtion of Figure. Due to the edge lengths nd lso Remrk., cnnot he lency four nd so α + α + kγ =, k. But then the other sum of lternte ngles must contin α + α >, which is not possile.

16 Ars Mth. Contemp. (0) Figure : f-tiling N Figure : Locl configurtions occurring in cse Therefore, α α > α nd kα =, k, nd we he α +γ = or α +γ <.. If α + γ =, with the leling of Figure, we he θ = γ or θ = β... If θ = γ, the lst configurtion is extended to the one illustrted in Figure. 0 0 f-tiling P Figure : Locl configurtions occurring in cse.

17 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV... If θ = γ, t ertex we he α + γ + γ = or α + γ + γ + γ =. Note tht we cnnot he more ngles α round, s two ngles of this type in djcent positions led to n impossiility, s seen efore. The condition α + γ + γ = implies α + α =, nd we get the configurtion illustrted in Figure. We denote this f-tiling y P, whose D representtion is gien in Figure 0. Figure 0: f-tiling P On the other hnd, if α + γ + γ + γ = (Figure ), the ngles θ nd θ cnnot e α otherwise we rech contrdiction t ertices nd, respectiely. But this implies tht t ertex we he two ngles α in djcent positions, which is not lso possile. 0 0 Figure : Locl configurtions occurring in cse.... If θ = β, then t ertex we he β + γ + kα =, k, which leds to contrdiction s illustrted in Figure (see ertex )... If θ = β, we otin similr impossiility s in the preious cse.. If α + γ < (Figure ), then θ {β, γ}.

18 Ars Mth. Contemp. (0) Figure : Locl configurtions occurring in cse. If θ = β (Figure ), then α + β + kα =, k. It follows tht the other sum of lternte ngles t ertex must e greter or equl to α + γ + γ >, which is n impossiility. If θ = γ nd (i) θ = γ (Figure ), then β > α >, which implies tile. At ertex we otin β + γ + kα =, k, giing rise to two ngles α in djcent positions, which leds to contrdiction, s seen preiously. Figure : Locl configurtions occurring in cse. (ii) θ = α (Figure ), ertex hs lency six or greter thn six. In the first cse, we otin two ngles α in djcent positions, which is not possile. In the lst cse, we he necessrily θ = γ, nd so β > α >. Consequently, contrdiction rises t ertex or.

19 C. P. Aelino nd A. F. Sntos: Sphericl folding tesselltions y kites nd isosceles tringles IV Concerning to the comintoril structure of ech tiling otined, we he tht (i) the symmetries of the f-tilings L, J, M nd N tht fix ertex of lency four nd surrounded y (α, α, α, α ) re generted y reflection nd y the rottion through n ngle round the xis y ±. On the other hnd, for ny ertices nd of this type, there is symmetry sending into. It follows tht the symmetry group hs exctly = elements nd it forms the group of ll symmetries of the cue - the octhedrl group, sometimes referred s C S. (ii) the f-tiling P hs only two ertices surrounded y (α, α, α, α ), sy the north nd south poles. The symmetries of P tht fix the north pole re generted y reflection nd y the rottion through n ngle round the zz xis, giing rise to sugroup isomorphic to D (the dihedrl group of order ). Now, the reflection on the equtor is lso symmetry of P, nd so it follows tht the symmetry group of P is isomorphic to C D. Summry In Tle is shown list of the sphericl dihedrl f-tilings whose prototiles re sphericl kite nd n isosceles sphericl tringle, K nd T, of internl ngles (α, α, α, α ), nd (β, γ, γ), respectiely, in cses of djcency IV, V nd V I. Our nottion is s follows: γ is the solution of eqution (.), with α =, α = γ nd α = β = ; β is the solution of eqution (.), with α =, α = γ nd α = β ; β is the solution of eqution (.), with α =, α = β nd α = γ = ; γ is the solution of eqution (.), with α =, β =, α = γ nd α = γ ; γ is the solution of eqution (.), with α = β =, α = γ nd α = γ. V is the numer of distinct clsses of congruent ertices; N is the numer of tringles congruent T nd N is the numer of kites congruent to K (used in the dihedrl f-tilings); G(τ) is the symmetry group of ech tiling τ Ω (K, T ). f-tiling α α α β γ V N N G(τ) L γ J M β β β γ C S C S β C S N γ γ γ C S P γ γ γ C D Tle : Comintoril structure of dihedrl f tilings of S y sphericl kites nd isosceles tringles in cses of djcency IV, V nd V I References [] C. Aelino nd A. Sntos, Sphericl nd plnr folding tesselltions y kites nd equilterl tringles, Austrlsin Journl of Comintorics, (0), 0.

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