Hyperbolic Tessellations by Saccheri and Lambert Quadrilaterals

Home , Lambert quadrilateral, Saccheri quadrilateral

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 orthogonality 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 euclidean 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 fundamental 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

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