The Densest Geodesic Ball Packing by a Type of Nil Lattices

W. Heisenberg’s famous real matrix group provides a noncommutative translation group of an affine 3-space. The Nil geometry which is one of the eight homogeneous Thurston 3-geometries, can be derived from this matrix group. E. Molnár proved in [2], that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere PS(V4,V 4,R). In our work we will use this projective model of the Nil geometry. In this paper we investigate the geodesic balls of the Nil space and compute their volume (see (2.4)), introduce the notion of theNil lattice, Nil parallelepiped (see Section 3) and the density of the lattice-like ball packing. Moreover, we determine the densest lattice-like geodesic ball packing (Theorem 4.2). The density of this densest packing is ≈ 0.78085, may be surprising enough in comparison with the Euclidean result π √ 18 ≈ 0.74048. The kissing number of the balls in this packing is 14. 1. On the Nil geometry The Nil geometry can be derived from the famous real matrix group L(R) discovered by Werner Heisenberg. The left (row-column) multiplication of Heisenberg ∗The research of the author is supported by the project of INTAS-NIS Ref. No. 06-10000138776. 0138-4821/93 $ 2.50 c © 2007 Heldermann Verlag 384 J. Szirmai: The Densest Geodesic Ball Packing by a Type of Nil Lattices matrices 1 x z 0 1 y 0 0 1 1 a c 0 1 b 0 0 1  = 1 a+ x c+ xb+ z 0 1 b+ y 0 0 1  (1.1) defines translations L(R) = {(x, y, z) : x, y, z ∈ R} on the points of the space Nil = {(a, b, c) : a, b, c ∈ R}. These translations are not commutative in general. The matrices K(z) C L of the form K(z) 3 1 0 z 0 1 0 0 0 1  7→ (0, 0, z) (1.2) constitute the one parametric centre, i.e. each of its elements commutes with all elements of L. The elements of K are called fibre translations. Nil geometry of the Heisenberg group can be projectively (affinely) interpreted by the right translations on points as the matrix formula (1; a, b, c)→ (1; a, b, c)  1 x y z 0 1 0 0 0 0 1 x 0 0 0 1  = (1;x+ a, y + b, z + bx+ c) (1.3) shows, according to (1.1). Here we consider L as projective collineation group with right actions in homogeneous coordinates. We will use the Cartesian homogeneous coordinate simplex E0(e0), E ∞ 1 (e1), E ∞ 2 (e2), E ∞ 3 (e3), ({ei} ⊂ V with the unit point E(e = e0 + e1 + e2 + e3)) which is distinguished by an origin E0 and by the ideal points of coordinate axes, respectively. Moreover, y = cx with 0 < c ∈ R (or c ∈ R \ {0}) defines a point (x) = (y) of the projective 3-sphere PS (or that of the projective space P where opposite rays (x) and (−x) are identified). The dual system {(e)}, ({e} ⊂ V 4) describes the simplex planes, especially the plane at infinity (e) = E∞ 1 E ∞ 2 E ∞ 3 , and generally, v = u 1 c defines a plane (u) = (v) of PS (or that of P). Thus 0 = xu = yv defines the incidence of point (x) = (y) and plane (u) = (v), as (x)I(u) also denotes it. Thus NIL can be visualized in the affine 3-space A (so in E) as well [4]. In this context E. Molnár [2] has derived the well-known infinitesimal arclength square at any point of Nil as follows (dx) + (dy) + (−xdy + dz) = (dx) + (1 + x)(dy) − 2x(dy)(dz) + (dz) =: (ds) (1.4) Hence we get the symmetric metric tensor field g on Nil by components, furthermore its inverse: gij := 1 0 0 0 1 + x −x 0 −x 1  , g := 1 0 0 0 1 x 0 x 1 + x  . (1.5) J. Szirmai: The Densest Geodesic Ball Packing by a Type of Nil Lattices 385 The translation group L defined by formula (1.3) can be extended to a larger group G of collineations, preserving the fibering, that will be equivalent to the (orientation preserving) isometry group of Nil. In [3] E. Molnár has shown that a rotation trough angle ω about the z-axis at the origin, as isometry of Nil, keeping invariant the Riemann metric everywhere, will be a quadratic mapping in x, y to z-image z as follows: r(O,ω) : (1;x, y, z)→ (1; x, y, z); x = x cosω − y sinω, y = x sinω + y cosω, z = z − 1 2 xy + 1 4 (x − y) sin 2ω + 1 2 xy cos 2ω. (1.6) x→ x′ = x, y → y′ = y, z → z′ = z − 1 2 xy to (1; x′, y′, z′)→ (1; x′, y′, z′)  1 0 0 0 0 cosω sinω 0 0 − sinω cosω 0 0 0 0 1  = (1;x”, y”, z”), with x”→ x = x”, y”→ y = y”, z”→ z = z” + 1 2 x”y”, (1.7) i.e. to the linear rotation formula. This quadratic conjugacy modifies the Nil translations in (1.3), as well. We shall use the following important classification theorem. Theorem 1.1. (E. Molnár [3]) (1) Any group of Nil isometries, containing a 3dimensional translation lattice, is conjugate by the quadratic mapping in (1.7) to an affine group of the affine (or Euclidean) space A = E whose projection onto the (x, y) plane is an isometry group of E. Such an affine group preserves a plane → point polarity of signature (0, 0,±0,+). (2) Of course, the involutive line reflection about the y axis (1; x, y, z)→ (1;−x, y,−z), preserving the Riemann metric in (1.5), and its conjugates by the above isometries in (1) (those of the identity component) are also Nil isometries. There does not exist orientation reversing Nil isometry. The geodesic curves of the Nil geometry are generally defined as having locally minimal arc length between their any two (near enough) points. The equation systems of the parametrized geodesic curves g(x(t), y(t), z(t)) in our model can be determined by the general theory of Riemann geometry (see [4]): We can assume, that the starting point of a geodesic curve is the origin because we can transform a curve into an arbitrary starting point by translation (1.1); x(0) = y(0) = z(0) = 0; ẋ(0) = c cosα, ẏ(0) = c sinα, ż(0) = w; −π ≤ α ≤ π. 386 J. Szirmai: The Densest Geodesic Ball Packing by a Type of Nil Lattices The arc length parameter s is introduced by s = √ c2 + w2 · t, where w = sin θ, c = cos θ, − 2 ≤ θ ≤ π 2 , i.e. unit velocity can be assumed. Remark 1.1. Thus we have harmonized the scales along the coordinate axes (see also formula (3.2) and Remark 3.3). The equation systems of a helix-like geodesic curve g(x(t), y(t), z(t)) if 0 < |w| < 1 [4]: x(t) = 2c w sin wt 2 cos (wt 2 + α ) , y(t) = 2c w sin wt 2 sin (wt 2 + α ) , z(t) = wt · { 1 + c 2w2 [( 1− sin(2wt+ 2α)− sin 2α 2wt ) + ( 1− sin(2wt) wt ) − ( 1− sin(wt+ 2α)− sin 2α 2wt )]} = = wt · { 1 + c 2w2 [( 1− sin(wt) wt ) + (1− cos(2wt) wt ) sin(wt+ 2α) ]} . (1.8) In the cases w = 0 the geodesic curve is the following: x(t) = c · t cosα, y(t) = c · t sinα, z(t) = 1 2 c · t cosα sinα. (1.9) The cases |w| = 1 are trivial: (x, y) = (0, 0), z = w · t. In Figure 1 it can be seen −1.5 −1.0 −0.5 0.0 0 0.0 0.5 5 10