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Hyperbolic Tessellations by Saccheri and Lambert Quadrilaterals

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Hyperbolic Tessellations by Saccheri and Lambert Quadrilaterals Hyperbolic tessellations by saccheri and lambert quadrilaterals Domingo Gámez1 Miguel Pasadas1 Rafael Pérez1 Ceferino Ruiz2 [email protected] [email protected] [email protected] [email protected] 1Department of Applied Mathematics, University of Granada, 18071 Granada 2Department of Geometry and Topology, University of Granada, 18071 Granada Mathematical Subject Classification (2000): 51M10 – 20F55 INTRODUCTION Theorem 2 Let F be a convex polygon of k sides ORTHOGONALITY CONSTRUCTIONS and interior angles ¼=ni, with ni N; ni 2, 2 = The construction of Saccheri and Lambert qua- Let X be the hyperbolic plane with the topology for each i = 1; 2; :::; k, with vertices Pi and sides drilaterals involve the resolution of the orthog- onality problems. These constructions can be induced by the hyperbolic metric. The model we determined by the segments Pi 1Pi where P0 = 2 classi…ed in three types [3]: shall use is the open disc unit of Poincar¶e D . P , satisfying k ¼=n < (k ¡ 2)¼. Let ¾ be k i=1 i i 1.- Determination of the orthogonal line to Let ¡ be a subgroup of the group of hyperbolic the re°ection on the line contai¡ning the segment isometries of X, Iso(X). The action of ¡ on X is P another one given through a point of it Pi 1Pi for each i = 1; 2; :::; k. Then the group ¡ the natural application © : ¡ X X, given ¡ £ ¡! generated by the re°ections ¾1; ¾2; :::; ¾k (polyg- ®~ = 0 ®~ = 0 by ©(°; z) = °(z). For each z X we denote the onal group) is an NEC group. 6 set £ = °(z) : ° ¡ as the2orbit of z by the f 2 g In addition, the family °(F ) : ° ¡ consti- action of ¡ upon X. The action of a subgroup tutes a ¡-tessellation of Xf. 2 g ¡ on X gives rise to the following relationship: z; w X; z w ° ¡ : w = °(z). 8 2 v , 9 2 KALEIDOSCOPIC SACCHERI AND LAMBERT QUADRILATERALS This relationship v is an equivalence relation on X, whose equivalence classes are the orbits of the The Saccheri quadrilaterals have two consecu- elements of X by the action of ¡. The quotient tive right angles on a side named base, and two set X= v will be designated as ¡ X and is called equal and acute angles opposite to the base. the orbit space. n The Lambert quadrilateral is a quadrilateral with three right angles. The union of two Lambert De¯nition 1 Let ¡ be a subgroup of Iso(X). ¡ quadrilaterals is a Saccheri quadrilateral: that is, being ~ the angle determined by the euclidean ~ ~ is said to be an NEC group, that is, a Non- it has two adjacent right angles and two equal tangent line to at the point with the euclid- ean line = 0. Euclidean Crystallographic Group, if it is dis- acute angles. And, conversely, if in a Saccheri crete (with compact open topology) and the orbit quadrilateral we trace the unique perpendicular 2.- Determination of the unique orthogonal li- space is compact. line that is common to the base and the opposite ne to a pair of ultraparallel lines, and . side, we obtain two Lambert quadrilaterals, con- If ¡ is an NEC group, then its rotations have gruent by means of re°ection with respect to the multiple integer angles of 2¼=n, with n N, and common perpendicular. they do not contain limit rotations that2is, rota- Theorem 3 ([7]) For every R > 0 and n N, tions whose center lies on the in¯nity line. 2 The fundamental regions comprise the smallest n > 2, there exists a unique Saccheri quadrilat- tile that allow us to tessellate by means of the eral with the base R and acute angles ' = ¼=n, action of an NEC group. For this purpose we unique up to congruence, that tessellates the hy- establish the following de¯nition. perbolic plane. De¯nition 2 ([3]) Let F X be closed and ¡ an Example 1 As an example NEC group. F is said½to be a fundamental re- illustrating this situation, gion for ¡ if: we show below tessella- i) for each z X there exists ° tion of a Poincar¶e disc 2 2 by means of a Saccheri ¡ so that °(z) F ¼ 2 quadrilateral for ' = and 3.- Determination of the orthogonal line to ii) if z F is such that °(z) F , 3 another one given through an outer point . with 2° = 1, then z; °(z) ar2e in R = 1, created with the boundar6 y of F Hyperbol package for Math- ematica software that has iii) F = Closure(int(F )): been developed by the au- It is important to point out that this de¯nition thors. is an improvement upon those used in [10] and Theorem 4 ([6]) For each R > 0 and n N, n > [12], in the sense that we eliminate the cases of 2, there exists a unique Lambert quad2rilateral pathological regions having isolated points that (P; Q; Q0; P 0) with three right angles, side P; P 0 correspond with other points of the fundamental Lor length R, and acute angle Á = ¼=n, unique region under consideration. up to congruence, that tessellates the hyperbolic De¯nition 3 Let ¡ be a NEC group. A ¡{tesse- plane. llation of the hyperbolic plane X, with the fun- damental region F is a set of regions Fi i I Example 2 As an example REMARK f g 2 such that: illustrating this situation, i) Fi = X we show below two tessella- We refer the reader to [3], which presents an 1 i I tions of the Poincar¶e disc, electronic tool named Hyperbol ( ) whose compu- S2 ii) int(Fi) int(Fj) = ? for i = j created also with the Hyper- tational support is Mathematica software. This \ 6 tool consists of modules that allow us to draw iii) i I; ° ¡ : Fi = °(F ). bol package, using a Lam- 8 2 9 2 bert quadrilateral for Á = di®erent hyperbolic constructions in Poincare's models for the hyperbolic plane, usually denoted In the study of subgroups of NEC groups, as ¼=4, and R 0:881373. ' by H2 and D2. Such constructions include re- well as int he tessellations of X, the following °ections, rotations, translations, glide re°ections, proposition is important: and the orbits of a point. These isometries and Proposition 1 ([11]) Let ¡ by an NEC group and geometric loci act on the hyperbolic plane; and if a euclidean element appears in some representa- ¡0 a subgroup of ¡ with the index p ( ¡ : ¡0 = p). Let F ¡ by a fundamental region for ¡. Then ¡ tion of this plane, we shall note it in an explicit ¯ ¯ 0 has a fundamental region F ¡0 result¯ing fro¯m the way. ¡ union of p congruent replicas of F and besides 1Software available at http://www.ugr.es/local/ruiz/software.htm ¡ p = ¡0 . j j ¢ j j REFERENCES Polygonal groups are those that are generated [1] A.F. Beardon, Hyperbolic polygons and Fuchsian groups, J. London Math. Soc., Vol. 20, 247-255. 1979. [2] M. C. Escher, Regelmatige vlakverdeling, Stichting 'De Roos', Utrech, 1958. by re°ections in the sides of a polygon. Certain [3] D. G¶amez, Construcciones en Geometr¶³a Hiperb¶olica y Teselaciones mediante Grupos NEC Poligonales. hyperbolic polygons give rise to NEC groups, and Algoritmos de Automatizaci¶on, PhD Thesis, Granada, 2001. [4] D. G¶amez, M. Pasadas, R. P¶erez, C. Ruiz, Hyperbolic Plane Tesselations. Proccedings of the VI Journµees the hyperbolic plane is tessellated with them. Zaragoza-Pau de Mathematiques Appliquµees et Statistique, 257-264, Jaca, Spain, 1999. Example 3 We use the Saccheri quadrilat- [5] D. G¶amez, M. Pasadas, R. P¶erez, C. Ruiz, Regla y compas hiperb¶olicos electr¶onicos para teselar. Proc- cedings of the I Encuentro de Matem¶aticos Andaluces, Vol. 2, 467-474, Sevilla, 2000. eral that is obtained from the union of the Lemma 1 ([1]) Let ® ; ® ; :::; ® be real numbers [6] D. G¶amez, M. Pasadas, R. P¶erez, C. Ruiz, The Lambert quadrilateral and tesselations in the hyperbolic 1 2 k Lambert quadrilateral, of the above exam- plane. Int. Math. J., Vol. 2, 777-795, 2002. with 0 ®i < ¼ for each i = 1; 2; :::; k. Then, [7] D. G¶amez, M. Pasadas, R. P¶erez, C. Ruiz, The Saccheri Quadrilateral, Translations and Tessellations · ple, and its re°ection with respect to the in the Hyperbolic Plane, Int. J. Pure Appl. Math., Vol. 23, No. 4, 565-576, 2005. there exists a convex hyperbolic polygon F with [8] M. J. Greenberg, Euclidean and non{Euclidean geometries: development and history, 3rd Edition, W. line that contains the segment of length R. H. Freeman & Co., New York, 1993. angles ® ; ® ; :::; ® if, and only if, k ® < 1 2 k i=1 i This Saccheri quadrilateral is subjacent in [9] A. M. Macbeath, The classi¯cation of non{Euclidean plane crystallographic groups, Canadian J. Math., (k 2)¼: Vol. 19, 1192-1205, 1967. ¡ P the work of Escher in the Cirkellimiet IV [10] E. Mart¶³nez, Convex Fundamental Regions for NEC Groups, Arch. Math., Vol. 47, 457-464, 1986. (Circle limit IV), known as Angels and Dev- [11] R. P¶erez G¶omez, Espacios Crom¶aticos. PhD Thesis, Granada, 1992. [12] H. C. Wilkie, On Non-Euclidean Crystallographic Groups, Math. Z., Vol. 91, 87-102, 1966. T heorem 1 ([10]) Let ¡ be an N EC group. T hen, ils, as it is the union of three consecutive [13] Haags Gemeentemuseum, M. C. Escher (1898{1972). Regelmatige Vlakverdelingen in het Haags Gem- there exists a convex polygon F that constitutes mentemuseum (Regular divisions of the plane at the Haags Gemeentemuseum), Snoeck-Ducaju & Zoon, triangles, which contain the ¯gures appear- Gent, 1986. a fundam ental region for ¡.
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