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On the Dedekind Tessellation On the Dedekind tessellation Jerzy Kocik Department of Mathematics Southern Illinois University [email protected] Abstract The Dedekind tessellation – the regular tessellation of the upper half-plane by the Möbius action of the modular group – is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their complete algebraic characterization with the help of a representation of the modular group acting by Lorentz transformations on Minkowski space. This interesting example of the interplay of geometry, group theory and number theory leads also to convenient algorithms for computer drawing of the Dedekind tessellation. Key words: Dedekind tessellation, Minkowski space, Lorentz transformations, modular group. AMS Classification: 52C20, 51F25, 11A05 1 Introduction Figure (1.1) shows the well-known regular tessellation of the Poincaré half-plane H, the upper half of the plane of complex numbers. It was first discussed in the 1877 paper by Richard Dedekind [1], which justifies the name Dedekind tessellation, but the first illustration appeared in Felix Klein’s paper [2] (see [6, 7] for these credits). Usually, the tessellation is understood in the context of the action of the modular group PSL±¹2; Zº on H as a model of non-Euclidean, hyperbolic geometry. It is viewed as a set of "ideal triangles” (triangles with arc sides) that results as an orbit of any of them via the action of the modular group. arXiv:1912.05768v1 [math.HO] 12 Dec 2019 Figure 1.1: The Dedekind tessellation 1 Recall that the group PSL¹2; Zº acts on the extended complex plane CÛ = C [ f1g (Riemann sphere) via Möbius transformations defined by a b az + b z 7! · z ≡ ; ad − bc = 1: (1.1) c d cz + d with a; b; c; d 2 Z. The group preserves the Poincaré half-plane and in particular the extended real line RÛ = R [ f1g. The Dedekind tessellation coincides with the orbit PSL¹2; Zº · ∆, where ∆ is any of the ideal triangles, but it is customary to choose the 2 1 triangle (jzj ≥ 1, jRe¹zºj ≤ 2 ), called the fundamental domain, shown shaded in Figure 2.1). One may ask what would be an efficient, practical, way to draw this pattern. (Pro- ducing the orbit of the modular group is hardly convenient, since the algorithm walks over circles in an inconsistent way.) Problem: Suppose we want to view the tessellation as a system of circles rather than ideal triangles. Is there any simple characterization of this system of “Dedekind circles” — denoted D — in terms of the list of the centers (the values of x 2 R) and the associated radii? Here it is: The main result: Algebraic characterization of the system of Dedekind circles: 1. The centers of the circles are rational and the curvatures of all circles are integral. If the center is k/n then the radius is 1/n. 2. For a fraction k/n to be the center of a Dedekind circle, the following necessary and sufficient conditions must be satisfied: ◦ either n is odd and n j ¹k2 − 1º ◦ or n is a multiple of 8 and ¹k2 − 1)/n is an odd integer ◦ or n = 0, in which case the circle is a vertical line at x = k/2, k odd. 3. The above rules exhaust the system of Dedekind circles. A simple algorithm to build the set of circles of arbitrary size emerges: For every cur- vature n (denominator) there is a finite set K¹nº of integers 0 ≤ k < n such that k/n is a center of a Dedekind circle (or a line if n = 2). These data correspond to circles in the interval »0; 1º. The rest of the pattern is reconstructed by the translational symmetry, since z ! z + 1 is an element of PSL¹2; Zº. Table 1.2 shows the data for curvatures up to n = 107. For instance the entry “n = 72, K¹nº = 19; 35; 37; 53" means that the interval »0; 1¼ contains four circles of radius r = 1/72 with centers at x = 19/72, x = 35/72, x = 37/72, and x = 53/72. In Figure 1.3, these fractions k/n are drawn as points ¹k; nº in N × N. 2 n K(n) n K(n) n K(n) nK(n)nK(n)nK(n) (vertical line at ½) 0 0 (vertical line at ½) 3737 1,1, 36 36 73 73 1, 721, 72 1 1 0 0 3939 1,1, 14, 14, 25, 25, 38 38 75 75 1, 26,1, 49, 26, 74 49, 74 3 3 1,1, 2 2 4040 11,11, 19, 19, 21, 21, 29 29 77 77 1, 34,1, 54, 34, 76 54, 76 5 5 1,1, 4 4 4141 1,1, 40 40 79 79 1, 781, 78 7 7 1,1, 6 6 4343 1,1, 42 42 80 80 9, 39,9, 41, 39, 71 41, 71 8 8 3,3, 5 5 4545 1,1, 19, 19, 26, 26, 44 44 81 81 1, 801, 80 9 9 1,1, 8 8 4747 1,1, 46 46 83 83 1, 821, 82 1111 1,1, 10 10 4848 7,7, 23, 23, 25, 25, 41 41 85 85 1, 16,1, 69, 16, 84 69, 84 1313 1,1, 12 12 4949 1,1, 48 48 87 87 1, 28,1, 59, 28, 86 59, 86 15 1, 4, 11, 14 51 1, 16, 35, 50 88 21, 43, 45, 67 1516 1,7, 4, 9 11, 14 5351 1,1, 52 16, 35, 50 89 88 1, 8821, 43, 45, 67 1617 7,1, 9 16 5553 1,1, 21, 52 34, 54 91 89 1, 27,1, 64, 88 90 1719 1,1, 16 18 5655 13,1, 27,21, 29, 34, 43 54 93 91 1, 32,1, 61, 27, 92 64, 90 1921 1,1, 18 8, 13, 20 5756 1,13, 20, 27, 37, 29,56 43 95 93 1, 39,1, 56, 32, 94 61, 92 2123 1,1, 8, 22 13, 20 5957 1,1, 58, 20, 37, 56 96 95 17, 47,1, 39,49, 56,79 94 2324 1,5, 22 11, 13, 19 6159 1,1, 60 58, 99 96 1, 10,17, 89, 47, 98 49, 79 2425 5,1, 11, 24 13, 19 6361 1,1, 8, 60 55, 62 101 1, 100 27 1, 26 64 31, 33 99 1, 10, 89, 98 25 1, 24 63 1, 8, 55, 62 103101 1, 1021, 100 2729 1,1, 26 28 6564 1,31, 14, 33 51, 64 31 1, 30 67 1, 66 104 27, 51, 53, 77 29 1, 28 65 1, 14, 51, 64 103 1, 102 32 15, 17 69 1, 22, 47, 68 105104 1, 29,27, 34, 51, 41, 53, 77 31 1, 30 67 1, 66 64, 71, 76, 104 33 1, 10 71 1, 70 105 1, 29, 34, 41, 3235 15,1, 6,17 29, 34 7269 19,1, 35,22, 37, 47, 53 68 107 1,106 33 1, 10 71 1, 70 64, 71, 76, 104 35 1, 6, 29, 34 72 19, 35, 37, 53 107 1, 106 Figure 1.2: Table of admissible circles with curvatures n < 108 Figure 2.2: Table of admissible circles with curvature n less than 108. k n Figure 2.3: “Canals on Mars” – is there any visible pattern to the fractions? Fractions k/n are the Figurepositions 1.3: of “Canalsthe centers on of Mars" the Dedekind’s — which circles. of the visibleOnly k patterns< n are presented. are accidental and which have number-theoretic significance? Fractions k/n representing the positions of the centers of the Dedekind circles are presented as points ¹n; kº. Only k < n are shown. 3 A geometric object that comes to mind is the Apollonian Window [4], an Apollonian gasket made of circles whose curvatures are also integral [5, 3]. The intriguing algebraic- geometric interactions in the Apollonian disk packing occur also in the case of the Dedekind system, and we shall use similar tools to investigate them and to prove the main result. 3 2 Dedekind circles in Minkowski space Circles (and lines as their special case) may be mapped to the Minkowski space M R3;1, where their image lies on the unit space-like hyperboloid [3]. In particular, a circle centered at ¹x; yº with radius r is mapped to the vector 2 xÛ 3 6 7 6 yÛ 7 C = 6 7 (2.1) 6 β 7 6 7 6 γ 7 4 5 where β = 1/r is the curvature, and xÛ = x/r and yÛ = y/r are “reduced coordinates". The last entry, “co-curvature" γ, has the geometric meaning of curvature of the image of the circle under inversion in the unit circle at the origin. In the case of a line, the curvature vanishes, β = 0, and the correspondence is 2 xÛ 3 6 7 6 yÛ 7 C = 6 7 ) Line 2xxÛ + 2yyÛ = γ (2.2) 6 0 7 6 7 6 γ 7 4 5 In general, the co-curvature may be easily determined from the fact that the vector is normalized, jCj2 = −1, where the quadratic form in M is defined by hC; Ci = jCj2 = −xÛ2 − yÛ2 + βγ (2.3) (non-diagonalized basis), i.e. γ = ¹xÛ2 + yÛ2 − 1)/β. We shall also use a more suggestive representation of the circle vectors that we call circle symbols [3], namely 2 xÛ 3 6 7 6 yÛ 7 xÛ; yÛ xÛ; yÛ 6 7 = = (2.4) 6 β 7 β; γ β 6 7 6 γ 7 4 5 where the co-curvature may optionally be dropped in case of a proper circle (not line).
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