PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 129–146 DOI: https://doi.org/10.2298/PIM1817129M

TOP DENSE HYPERBOLIC BALL PACKINGS AND COVERINGS FOR COMPLETE COXETER ORTHOSCHEME GROUPS

Emil Molnár and Jenő Szirmai

Abstract. In n-dimensional hyperbolic space H^{n} (n ⩾ 2), there are three types of spheres (balls): the sphere, horosphere and hypersphere. If n = 2, 3 we know a universal upper bound of the ball packing densities, where each ball’s volume is related to the volume of the corresponding Dirichlet–Voronoi (D-V) cell. E.g., in H^{3} a densest (not unique) horoball packing is derived from the {3, 3, 6} Coxeter tiling consisting of ideal regular simplices T_{reg}^{∞} with dihedral angles \frac{π}{3} . The density of this packing is δ_{3}^{∞} ≈ 0.85328 and this provides a very rough upper bound for the ball packing densities as well. However, there are no “essential" results regarding the “classical" ball packings with congruent balls, and for ball coverings either.

The goal of this paper is to find the extremal ball arrangements in H^{3} with “classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. In Theorems 1.1 and 1.2 we formulate also conjectures for the densest ball packing with density 0.77147 … and the loosest ball covering with density 1.36893 …, respectively. Both are related with the extended Coxeter group (5, 3, 5) and the so-called hyperbolic football manifold. These facts can have important relations with fullerenes in crystallography.

1. Introduction

Ball packing problems concern the arrangements of non-overlapping equal balls packed in space. Usually, space is the classical three-dimensional Euclidean space E^{3}. However, ball packing problems can be generalized to the other 3-dimensional Thurston geometries (see e.g., [12]).

In an n-dimensional space of constant curvature E^{n}, H^{n}, S^{n} (n ⩾ 2) let d_{n}(r) be the density of n + 1 congruent balls of radius r mutually touching one another with respect to the simplex spanned by the centres of the balls. Fejes Tóth and Coxeter [5] conjectured that in an n-dimensional space of constant curvature the density of packing balls of radius r cannot exceed d_{n}(r). This conjecture has been proved by Rogers [15] in Euclidean n-space. The 2-dimensional case has been solved

2010 Mathematics Subject Classification: 52C17. Key words and phrases: hyperbolic geometry, ball packings and coverings.

by Fejes Tóth [5]. In a three-dimensional space of constant curvature the problem has been investigated (and solved it in some sense) by Böröczky and Florian [2] and it has been studied by Böröczky [1] for n-dimensional spaces of constant curvature (n ⩾ 4). The upper bound d_{n}(∞) for H^{n} (n = 2, 3) is attained for a regular horoball packing, that is, a packing by horoballs inscribed in the cells of a regular honeycomb of \overline{H}^{n} (i.e., H^{n} is closed by its ideal points, or ends).

In \overline{H}^{3} there is exactly one horoball packing with horoballs in the same type whose Dirichlet–Voronoi cells give rise to a regular honeycomb described by the Schläfli symbol {6, 3, 3}. Its dual {3, 3, 6} consists of ideal regular simplices T_{reg}^{∞} with dihedral angles \frac{π}{3} building up a 6-cycle around each edge of the tessellation. The density of this packing is δ_{3}^{∞} ≈ 0.8532885328 that attends at other horoball packings as well [9]. We have considered some new aspects related to the horoball and hyperball packings in [16, 17, 18, 11, 10].

However, there are no “essential" results regarding the “classical ball packings and coverings" with congruent balls. What are the extremal ball arrangements in H^{n} and what are their densities?

The goal of this paper is to study the above problems in H^{3} with “classical balls". We consider periodic congruent ball packings and coverings (for simplicity and for good constructions, only) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. We formulate two theorems and conjectures for the densest ball packing with density 0.77147 …, and for the loosest ball covering with density 1.36893 …, respectively.

Definition 1.1. For a given packing of H^{n} (n ⩾ 2) each ball’s volume is related to the volume of its D-V cell, (i.e., the closed domain whose any point lies not farther from the given ball’s centre than from any other centre of the ball system), then take the infimum of these ratios for all balls; this is called the density of the given packing. Then the supremum of these infima is taken for all ball packings of H^{n} to get the maximum packing density and the densest packing of H^{n} (if such a packing exists at all).

Theorem 1.1. Among the extended complete Coxeter orthoscheme groups the group (5, 3, 5), extended by a half-turn, provides the ball centre orbit of A_{3} for the densest ball packing by its football D-V-cell {5, 6, 6} and inscribed ball, (see Fig. 1 and 3). The density of this ball packing is 0.77147 … with radius r = 0.95142 ⋯= A_{3}F_{03}.

This is the conjectured maximal density for all ball packings of H^{3}.

Definition 1.2. For a given covering of H^{n} each ball’s volume is related to the volume of its D-V cell, than take the supremum of these ratios for all balls (this is called the density of the given covering); then the infimum of these suprema is taken for all ball coverings of H^{n} to get the minimal covering density and the loosest covering of H^{n} (if such a covering exists at all).

Theorem 1.2. Among the above groups the same (with half-turn extended (5, 3, 5) group) provides the same (congruent) ball centre orbit of A_{3} for the loosest ball covering, again by its football D-V-cell {5, 6, 6} but with its circumscribed ball, see Figs. 1 and 3. The density of this ball covering is 1.36893 … with radius R = 1.12484 … = A_{3}F_{12}.

This is the conjectured minimal density for all ball coverings of H^{3}.

Our numerical results are collected in Tables, where we give only the relevant data (because of the page restriction of this paper).

Our systematic computations will give the Proof of Theorems 1.1-1.2.

We will use the well-known Beltrami-Cayley-Klein model of H^{3} with the classical projective metric calculus (see e.g., [12]).

2. The projective model and complete orthoschemes

We use for H^{3} (and analogously for H^{n}, n > 3) the projective model in the Lorentz space E^{1,3} whose real vector space V^{4} equipped with the bilinear form of signature (1, 3), ⟨x, y⟩ = −x^{0}y^{0} + x^{1}y^{1} + x^{2}y^{2} + x^{3}y^{3}, where the non-zero vectors x = (x^{0}, x^{1}, x^{2}, x^{3}) ∈ V^{4} and y = (y^{0}, y^{1}, y^{2}, y^{3}) ∈ V^{4}, are determined up to real factors, for representing points of P^{n}(R). Then H^{3} can be interpreted as the interior of the quadric Q = {(x) ∈ P^{3}|⟨x, x⟩ = 0} =: ∂H^{3} in the real projective space P^{3}(V^{4}, V_{4}) (here V_{4} is the dual space of V^{4}). Namely, for an interior point Y (y) holds ⟨y, y⟩ < 0.

The points of the boundary ∂H^{3} in P^{3} are called points at infinity, or at the absolute of H^{3}. The points lying outside ∂H^{3} are said to be outer points of H^{3} relative to Q. Let (x) ∈ P^{3}, a point (y) ∈ P^{3} is said to be conjugate to (x) if ⟨x, y⟩ = 0 holds. The set of all points which are conjugate to (x) form a (polar) hyperplane pol(x) := {(y) ∈ P^{3}|⟨x, y⟩ = 0}. Thus the quadric Q induces a bijection (linear polarity V^{4} → V_{4}) from the points of P^{3} onto their polar hyperplanes.

The point X(x) and the hyperplane α(a) are incident if xa = 0 (x ∈ V^{4} ∖{0}, a ∈ V_{4}∖\{0\}). The constant k = \sqrt{−1/K} is the natural length unit in H^{3}, where K denotes the constant negative sectional curvature. In the following we may assume that k = 1.

2.1. Characteristic orthoschemes and their volumes. An orthoscheme O in H^{n} n ⩾ 2 in classical sense is a simplex bounded by n + 1 hyperplanes H_{0}, …, H_{n} such that H_{i}⊥H_{j} , for j ≠ i−1, i, i+1. Or, equivalently, the n+1 vertices of O can be labelled by A_{0}, A_{1}, …, A_{n} in such a way that span(A_{0}, …, A_{i}) ⊥ span(A_{i}, …, A_{n}) for 0 < i < n − 1.

Geometrically, complete orthoschemes of degree m = 0, 1, 2 can be described as follows:

(1) For m = 0, they coincide with the class of classical orthoschemes introduced by Schläfli. The initial and final vertices, A_{0} and A_{n} of the orthogonal edge-path A_{i}A_{i+1} , i = 0, …, n − 1, are called principal vertices of the orthoscheme (see Remark 4.1).

(2) A complete orthoscheme of degree m = 1 can be constructed from an orthoscheme with one outer principal vertex, one of A_{0} or A_{n}, which is simply truncated by its polar plane (see Fig. 1-2).

(3) A complete orthoscheme of degree m = 2 can be constructed from an orthoscheme with two outer principal vertices, A_{0} and A_{n}, which is doubly truncated by their polar planes pol(A_{0}) and pol(A_{n}) (see Figs. 1 and 2).

For the complete Coxeter orthoschemes O ⊂ H^{n}, we adopt the usual conventions and sometimes even use them in the Coxeter case: If two nodes are related by the weight cos \frac{π}{p}, then they are joined by a (p − 2)-fold line for p = 3, 4 and by a single line marked by p for p ⩾ 5. In the hyperbolic case if two bounding hyperplanes of S are parallel, then the corresponding nodes are joined by a line marked ∞. If they are divergent then their nodes are joined by a dotted line.

In the following, we concentrate only on dimensions 3 and on hyperbolic Coxeter– Schläfli symbol of the complete orthoscheme tiling P generated by reflections in the planes of a complete orthoscheme O. To every scheme there is a corresponding symmetric 4 × 4 matrix (b^{ij} ) where b^{ii} = 1 and, for i ≠ j ∈ {0, 1, 2, 3}, b^{ij} equals − cosα_{ij} with all angles α_{ij} between the faces i, j of O.

For example, (b^{ij} ) below is the so called Coxeter–Schläfli matrix with parameters (u; v; w), i.e., α_{01} = \frac{π}{u}, α_{12} = \frac{π}{v}, α_{23} = \frac{π}{w} to be discussed yet for hyperbolicity. Now only 3 ⩽ u, v, w come into account (see [6, 7]). Then

(2.1)  (b^{ij} ) = ⟨b^{i}, b^{j} ⟩ := ( \begin{matrix}1 & − cos \frac{π}{u} & 0 & 0 \\ − cos \frac{π}{u} & 1 & − cos \frac{π}{v} & 0 \\ 0 & − cos \frac{π}{v} & 1 & − cos \frac{π}{w} \\ 0 & 0 & − cos \frac{π}{w} & 1 \end{matrix} ) .

This 3-dimensional complete (truncated or frustum) orthoscheme O = W_{uvw} and its reflection group G_{uvw} is depicted in Fig. 2, and by the symmetric Coxeter– Schläfli matrix (b^{ij} ) in formula (2.1), furthermore by its inverse matrix (a_{ij} ) in formula (2.2).

(2.2) (a_{ij} ) = (b^{ij} )^{−1} = ⟨a_{i}, a_{j} ⟩ :=

= \frac{1}{B} ( \begin{matrix}sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v} & cos \frac{π}{u} sin^{2} \frac{π}{w} & cos \frac{π}{u} cos \frac{π}{v} & cos \frac{π}{u} cos \frac{π}{v} cos \frac{π}{w} \\ cos \frac{π}{u} sin^{2} \frac{π}{w} & sin^{2} \frac{π}{w} & cos \frac{π}{v} & cos \frac{π}{w} cos \frac{π}{v} \\ cos \frac{π}{u} cos \frac{π}{v} & cos \frac{π}{v} & sin^{2} \frac{π}{u} & cos \frac{π}{w} sin^{2} \frac{π}{u} \\ cos \frac{π}{u} cos \frac{π}{v} cos \frac{π}{w} & cos \frac{π}{w} cos \frac{π}{v} & cos \frac{π}{w} sin^{2} \frac{π}{u} & sin^{2} \frac{π}{u} − cos^{2} \frac{π}{v} \end{matrix} ) ,

where

B = det(b^{ij} ) = sin^{2} \frac{π}{u} sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v} < 0, i.e., sin \frac{π}{u} sin \frac{π}{w} − cos \frac{π}{v} < 0.

In what follows, we use the above orthoscheme whose volume is derived by the next Theorem of Kellerhals ([8], by the ideas of Lobachevsky):

Theorem 2.1 (Kellerhals). The volume of a three-dimensional hyperbolic complete orthoscheme O = W_{uvw} ⊂ H^{3} is expressed with the essential angles α_{01} = \frac{π}{u}, α_{12} = \frac{π}{v}, α_{23} = \frac{π}{w}, (0 ⩽ α_{ij} ⩽ \frac{π}{2} ) (Fig. 1.a, b) in the following form:

Vol(O) = \frac{1}{4}\{L(α_{01} + θ) − L(α_{01} − θ) + L(\frac{π}{2} + α_{12} − θ)

+ L(\frac{π}{2} − α_{12} − θ) + L(α_{23} + θ) − L(α_{23} − θ) + 2L(\frac{π}{2} − θ)\},

where θ ∈ [0, \frac{π}{2} ) is defined by

tan(θ) = \sqrt{cos^{2} α_{12} − sin^{2} α_{01} sin^{2} α_{23}} ; cosα_{01} cosα_{23} ,

and where L(x) := − ∫_{0}^{x} log|2 sint| dt denotes the Lobachevsky function (in J. Milnor’s interpretation).

The volume Vol(B(R)) of a ball B(R) of radius R can be computed by the classical formula of J. Bolyai:

(2.3) Vol(B(R)) = 2π(cosh(R) sinh(R) − R) = π(sinh(2R) − 2R)

= \frac{4}{3}πR^{3}( 1 + \frac{1}{5}R^{2} + \frac{2}{105}R^{4} + ⋯) .

3. Essential points in a complete (truncated) orthoscheme

Let A_{0}(a_{0}), A_{1}(a_{1}), A_{2}(a_{2}), A_{3}(a_{3}) be the vertices of the above complete orthoscheme W_{uvw} (see Fig. 1,2). The principal vertices A_{0} and A_{3} can be proper (a_{ii} < 0, i ∈ {0, 3}), boundary (a_{ii} = 0) or outer points (a_{ii} > 0) We exploit the logical symmetry 0, 1 ↔ 3, 2.

We distinguish the following main configurations of the principal vertices A_{0} and A_{3}:

1. A_{3} is proper or boundary point \frac{π}{u} + \frac{π}{v} ⩾ \frac{π}{2} .

1.i. A_{0} is proper or boundary point \frac{π}{v} + \frac{π}{w} ⩾ \frac{π}{2} .

1.s.i u = w, F_{03}F_{12} is half turn axis, h is the half turn changing 0 ↔ 3, 1 ↔ 2. Here a "half orthoscheme" JQEB_{13}F_{12}B_{02}F_{03}A_{2} will be the fundamental domain of G_{u=w,v} .

1.ii A_{0} is outer \frac{π}{v} + \frac{π}{w} < \frac{π}{2} , then a_{0}(a_{0}) = CLH is its polar plane.

2. A_{3} is outer point \frac{π}{u} + \frac{π}{v} < \frac{π}{2} , then a_{3}(a_{3}) = JEQ is its polar plane. 2.i. A_{0} is proper or boundary point \frac{π}{v} + \frac{π}{w} ⩾ \frac{π}{2} . 2.ii A_{0} is also outer \frac{π}{v} + \frac{π}{w} < \frac{π}{2} , then a_{0}(a_{0}) = CLH is its polar plane. 2.s.ii u = w, F_{03}F_{12} is half turn axis, h is the half turn changing 0 ↔ 3,

1 ↔ 2. Here a "half orthoscheme" JQEB_{13}F_{12}B_{02}F_{03}A_{2} will be the fundamental domain of G_{u=w,v} .

We obtain with easy calculations the following inportant lemmas (see Fig. 1,2):

Lemma 3.1. Let A_{0} be an outer principal vertex of the orthoscheme W_{uvw} and let a_{0}(a_{0}) = CLH be its polar plane where C = a_{0} ∩ A_{0}A_{1}, L = a_{0} ∩ A_{0}A_{2}, H = a_{0} ∩ A_{0}A_{3} whose vectors are

C(c) = a_{0} ∩ A_{0}A_{1}; c = a_{1} − \frac{a_{01}}{a_{00}} a_{0}, with

⟨c, c⟩ = \frac{(a_{11}a_{00} − a_{01}^{2})}{a_{00}} = ⟨c, a_{1}⟩ = \frac{sin^{2} \frac{π}{w}}{sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v}} = \frac{a_{11}}{a_{00}}

L(l) = a_{0} ∩ A_{0}A_{2}; l = a_{2} − \frac{a_{02}}{a_{00}} a_{0}, with

134 MOLNÁR AND SZIRMAI

(3.1) ⟨l, l⟩ = \frac{(a_{22}a_{00} − a_{02}^{2})}{a_{00}} = ⟨l, a_{2}⟩ = \frac{1}{sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v}} = \frac{1}{Ba_{00}}

H(h) = a_{0} ∩ A_{0}A_{3}; h = a_{3} − \frac{a_{03}}{a_{00}} a_{0}, with

⟨h, h⟩ = \frac{(a_{33}a_{00} − a_{03}^{2})}{a_{00}} = ⟨h, a_{3}⟩ = \frac{sin^{2} \frac{π}{v}}{sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v}} = \frac{sin^{2} \frac{π}{v}}{Ba_{00}}

Figure 1. Simple and double truncated complete orthoschemes

Lemma 3.2. Let A_{3} be an outer principal vertex of the orthoscheme W_{uvw} and let a_{3}(a_{3}) = JEQ be its polar plane where J = a_{3} ∩ A_{3}A_{0}, E = a_{3} ∩ A_{3}A_{1}, Q = a_{3} ∩ A_{3}A_{2} whose vectors are

J(j) = a_{3} ∩ A_{3}A_{0}; j = a_{0} − \frac{a_{03}}{a_{33}} a_{3}, with

⟨j, j⟩ = \frac{(a_{00}a_{33} − a_{03}^{2})}{a_{33}} = ⟨j, a_{0}⟩ = \frac{sin^{2} \frac{π}{v}}{sin^{2} \frac{π}{u} − cos^{2} \frac{π}{v}} = \frac{sin^{2} \frac{π}{v}}{Ba_{33}}

E(e) = a_{3} ∩ A_{3}A_{1}; e = a_{1} − \frac{a_{13}}{a_{33}} a_{3}, with

(3.2) ⟨e, e⟩ = \frac{(a_{11}a_{33} − a_{13}^{2})}{a_{33}} = ⟨e, a_{1}⟩ = \frac{1}{sin^{2} \frac{π}{u} − cos^{2} \frac{π}{v}} = \frac{1}{Ba_{33}}

Q(h) = a_{3} ∩ A_{3}A_{2}; q = a_{2} − \frac{a_{23}}{a_{33}} a_{3}, with

⟨q, q⟩ = \frac{(a_{22}a_{33} − a_{23}^{2})}{a_{33}} = ⟨q, a_{2}⟩ = \frac{sin^{2} \frac{π}{u}}{sin^{2} \frac{π}{u} − cos^{2} \frac{π}{v}} = \frac{a_{22}}{a_{33}} .

Lemma 3.3. The midpoint K(k) (see Fig. 1) of A_{2}Q can be determined by the following vector:

(3.3)  k = \frac{a_{2}}{\sqrt{−a_{22}}} + \frac{q}{\sqrt{−⟨q, q⟩}} with ⟨k, k⟩ = −2( 1 + \sqrt{\frac{1}{a_{33}}} ) .

Figure 2.

Especially if u = w, the midpoints F_{03} of A_{0}A_{3} and F_{12} of A_{1}A_{2} can play important roles, since F_{03}F_{12} will be the axis of half turn

h : 0 ↔ 3, 1 ↔ 2, i.e. A_{0} ↔ A_{3}, b^{0} ↔ b^{3}, A_{1} ↔ A_{2}, b^{1} ↔ b^{2}.

(Here a_{00} = a_{33} and a_{11} = a_{22} hold, of course.)

Lemma 3.4. The midpoints F_{03}(f_{03}) of A_{3}A − 0 and F_{12}(f_{12}) of A_{1}A_{2} (see Fig. 1) can be determined by the following vectors:

f_{03} = a_{0} + a_{3}, ⟨f_{03}, f_{03}⟩ = 2(a_{00} + a_{03}) < 0,
f_{12} = a_{1} + a_{2}, ⟨f_{12}, f_{12}⟩ = 2(a_{11} + a_{12}) < 0,

independently of A_{0} and A_{3} both are either proper, boundary or outer points.

The perpendicular foot point Y (y), dropped onto a plane (u) from point X(x) is given by

(3.4)  y = x − \frac{⟨x, u⟩}{⟨u, u⟩}u,

where (u) is the pole of the plane (u). In the considered cases the planes are the faces of the orthoscheme W_{uvw} , therefore the poles b_{∗}^{i} = b^{i} = b^{ij} a_{j} play important roles in the computations that are derived by matrix (2.1):

(3.5)

b^{0} = a_{0} − cos \frac{π}{u}a_{1}, b^{1} = − cos \frac{π}{u}a_{0} + a_{1} − cos \frac{π}{v}a_{2},
b^{2} = − cos \frac{π}{v}a_{1} + a_{2} − cos \frac{π}{w}a_{3}, b^{3} = − cos \frac{π}{w}a_{2} + a_{3}.

Figure 3.

E.g. the foot point F_{12}^{2}(f_{12}^{2}) of f_{12} = a_{1} + a_{2} onto the plane b^{2} (see Fig. 1) is

f_{12}^{2} = f_{12} + cb^{2} = (1 + cos \frac{π}{v})a_{1} + cos \frac{π}{w}a_{3}, by f_{12}^{2}b^{2} = 0,

thus for coefficient c = −1. We similarly get the other foot points on the simplex side planes b^{i} i ∈ {0, 1, 2, 3}.

Dividing points of edges of W_{uvw} (or of its half domain) e.g., D_{ij} ∈ A_{i}A_{j} (i < j ∈ {0, 1, 2, 3}) can also have special roles with more cumbersome calculations: d_{ij} = a_{i} + d_{ij} a_{j} will be interesting for appropriate coefficient d_{ij} . At least two locally minimal face distances from a D_{ij} occur for packing. Or at least two locally maximal vertex distances from a D_{ij} occur for covering. Maybe, we obtain different D_{ij} ’s for locally maximal packing and locally minimal covering, respectively.

We leave out these lengthy discussions here. Only F_{03}, F_{12} and K will be mentioned in Sect. 4 at cases c, d and h, respectively.

Face points F^{i} on b^{i} or F_{i} on a_{i} can play similar roles (for at least three locally minimal face distances and at least three locally maximal vertex distances, respectively) for searching locally extremal densities, but we do not discuss details here, because they will not be relevant for the optimal packing and covering, respectively.

Extremal “incentre" I in the body of W_{uvw} for packing and “circumcentre" G in W_{uvw} for covering (for at least four locally minimal face distances and at least four locally maximal vertex distances, respectively) will not be discussed as well, because they provide only local extrema (far from our (5, 3, 5) absolute optima (Table 1.s.i.a)), after lengthy but straightforward discussions.

4. Ball packings and coverings

For any fixed u, v, w, satisfying the hyperbolicity inequlities, we consider the competitive ball centres, as follow, for maximal packing density and minimal covering density. We indicate the centres where they occur, respectively. For the ball centre O we have roughly 8 competitive cases (Fig. 1):

a. A_{3} is the ball centre, | Stab_{A_{3}} G| = \frac{8uv}{4−(u−2)(v−2)} , b. A_{2} is the ball centre, | Stab_{A_{2}} G| = 4u, c. F_{03} is the ball centre, | Stab_{F_{03}} G| = 4v, if u = w, d. F_{12} is the ball centre | Stab_{F_{12}} G| = 8, if u = w, e. Q is the ball centre, | Stab_{Q} G| = 4u, f. J is the ball centre, | Stab_{J} G| = 4v, g. E is the ball centre, | Stab_{E} G| = 8, h. the midpoint K of A_{2}Q is the ball centre, | Stab_{K} G| = 2u.

K gives (only) estimates for the points of A_{2}Q, in general. Competitive packing ball radii r for r^{opt} with maximal density

(4.1) δ^{opt} = \frac{Vol(B(r^{opt}))}{| Stab_{O} G| Vol(W_{uvw} )} or δ^{opt} = \frac{Vol(B(r^{opt}))}{\frac{1}{2}| Stab_{O} G| Vol(W_{uvw} )} if u = w.

Competive covering ball radii R for R^{opt} with minimal density:

(4.2) Δ^{opt} = \frac{Vol(B(R^{opt}))}{| Stab_{O} G| Vol(W_{uvw} )} or Δ^{opt} = \frac{Vol(B(R^{opt}))}{\frac{1}{2}| Stab_{O} G| Vol(W_{uvw} )} if u = w.

4.1. Case 1.i.a. In this case A_{3} is proper O = A_{3} is the ball centre. A_{O} is either proper or boundary point. In the latter case covering is not defined. The optimal radii can be derived by the Beltrami–Cayley–Klein model (see Section 2, 3):

(4.3)

r = A_{3}A_{2}; cosh(A_{3}A_{2}) = \frac{−a_{23}}{\sqrt{a_{22}a_{33}}} = sin \frac{π}{u} cos \frac{π}{w} ; \sqrt{sin^{2} \frac{π}{u} − cos^{2} \frac{π}{v}} = \sqrt{1 − \frac{1}{a_{33}}} ,

R = A_{3}A_{0}; cosh(A_{3}A_{0}) = \frac{−a_{03}}{\sqrt{a_{00}a_{33}}} = \sqrt{1 − \frac{sin^{2} \frac{π}{v}}{Ba_{00}a_{33}}} ,

The densities of the ball packings and coverings for given parameters u, v, w can be computed by formulas (4.1) and (4.2), also later on.

Table 1.i.a, Packing, O = A_{3} , \frac{1}{u} + \frac{1}{v} > \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, 3, 6)  0.34657  0.04229  0.17861  0.17598
(4, 3, 5)  0.53064  0.03589  0.66207  0.38437
(4, 3, 6)  0.65847  0.10572  1.30405  0.25697
(5, 3, 4)  0.80846  0.03589  2.52145  0.58554
(5, 3, 5)  0.99639  0.09333  5.04848  0.45080
(5, 3, 6)  1.08394  0.17150  6.73795  0.32740
(3, 4, 4)  0.65848  0.07633  1.30405  0.35592
(3, 5, 3)  0.86830  0.03905  3.18663  0.68003
Table 1.i.a, Covering, O = A_{3} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} > \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  Δ^{opt}
(4, 3, 5)  1.22646  0.03589  10.41484  6.04641
(5, 3, 4)  1.22646  0.03589  10.41484  2.41856
(5, 3, 5)  1.90285  0.09333  58.62658  5.23495
(3, 5, 3)  1.38257  0.03905  16.16044  3.44864

4.2. Case 1.s.i.a. O = A_{3} is the ball centre and u = w. The density is related to the "half orthoscheme".

(4.4) r = min{A_{3}A_{2}, A_{3}F_{03}}; where cosh(A_{3}A_{2}) is in (4.3),

cosh(A_{3}F_{03}) = \sqrt{\frac{a_{03}}{2a_{33}} + \frac{1}{2}} = cosh( \frac{1}{2}A_{3}A_{0}) ,

R = A_{3}F_{12}; cosh(A_{3}F_{12}) = \frac{−(a_{13} + a_{23})}{\sqrt{2a_{33}(a_{12} + a_{22})}} = \sqrt{\frac{1}{2} + \frac{a_{03} − 1}{2a_{33}}} ,

Table 1.s.i.a, Packing, O = A_{3} , \frac{1}{u} + \frac{1}{v} > \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(5, 3, 5)  0.95142  0.09333  4.31988  0.77147
(3, 5, 3)  0.69129  0.03905  1.52220  0.64967

Table 1.s.i.a, Covering, O = A_{3} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} > \frac{1}{2}

(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(5, 3, 5)  1.12484  0.09333  7.66539  1.36893
(3, 5, 3)  0.89558  0.03905  3.53002  1.50661

4.3. Case 1.i.b. O = A_{2} is the ball centre.

(4.5)

r = A_{2}b^{2} =: A_{2}A_{2}^{2}; cosh(A_{2}b^{2})= \sqrt{1 − \frac{1}{a_{22}}}

R = max\{A_{2}A_{3}, A_{2}A_{0}\}; cosh(A_{2}A_{3}) = \sqrt{1 − \frac{1}{a_{33}}} ,

cosh(A_{2}A_{0}) = \frac{−a_{02}}{\sqrt{a_{00}a_{22}}} = cot \frac{π}{u} cos \frac{π}{v} ; \sqrt{sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v}} = \sqrt{1 − \frac{1}{Ba_{00}a_{22}}} .

Table 1.i.b, Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(4, 3, 5)  0.38360  0.03589  0.24350  0.42409
(5, 3, 4)  0.45682  0.03589  0.41631  0.58007
(5, 3, 5)  0.58157  0.09333  0.88150  0.47227
(4, 4, 3)  0.48121  0.07633  0.48886  0.40029
(3, 5, 3)  0.34346  0.03905  0.17377  0.37082
Table 1.i.b, Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} > \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  Δ^{opt}
(5, 3, 4)  0.84248  0.03589  2.88536  4.02028

4.4. Case 1.s.i.b. O = A_{2} is the ball centre. u = w. The density is related to the "half orthoscheme".

(4.6)

r = min{A_{2}b^{2}, A_{2}F_{12}}; where cosh(A_{2}b^{2}) in (4.5),

cosh(A_{2}F_{12}) = \sqrt{\frac{a_{12}}{2a_{22}} + \frac{1}{2}} = cosh( \frac{1}{2}A_{2}A_{1}) = \sqrt{\frac{1}{2} + \frac{cos \frac{π}{v}}{2 sin^{2} \frac{π}{u}}} ,

R = max\{A_{2}A_{3}, A_{2}F_{03}\}; cosh(A_{2}A_{3}) = \frac{−a_{23}}{\sqrt{a_{22}a_{33}}} =

cosh(A_{2}F_{03}) = \frac{−(a_{02} + a_{23})}{\sqrt{2a_{22}(a_{33} + a_{03})}} = cot \frac{π}{u} \sqrt{\frac{cos \frac{π}{v} + sin^{2} \frac{π}{u}}{2(1 − cos \frac{π}{v})}} .

Table 1.s.i.b, Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(5, 3, 5)  0.45682  0.09333  0.41631  0.44609

Table 1.s.i.b, Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}

(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(5, 3, 5)  0.99639  0.09333  5.04848  5.40954

4.5. Case 1.s.i.c. O = F_{03} is the ball centre, u = w.

(4.7)

r = F_{03}b^{0} = F_{03}b^{3}; where cosh(F_{03}b^{0})= \sqrt{1 + \frac{cos \frac{π}{v} − sin^{2} \frac{π}{u}}{2(1 − cos \frac{π}{v})}} ,

R = max\{F_{03}A_{3}, F_{03}A_{2}\} = F_{03}A_{3}; cosh(F_{03}A_{3}) = \sqrt{\frac{a_{03}}{2a_{33}} + \frac{1}{2}} .

140 MOLNÁR AND SZIRMAI Table 1.s.i.c, Packing, O = F_{03} , u = w, \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(4, 4, 4)  0.56419  0.22899  0.80163  0.43759
(3, 5, 3)  0.38360  0.03905  0.24350  0.62355
(3, 6, 3)  0.61795  0.16916  1.06673  0.52551

Table 1.s.i.c, Covering, O = F_{03} , u = w, \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} > \frac{1}{2}

(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 5, 3)  0.69129  0.03905  1.52220  3.89804

4.6. Case 1.s.i.d. O = F_{12} is the ball centre and u = w. (4.8)

r = F_{12}b^{1} = F_{12}b^{2}; where cosh(F_{12}b^{2})= \sqrt{1 + \frac{cos \frac{π}{v} − sin^{2} \frac{π}{u}}{2}} ,

R = A_{3}F_{12}; cosh(A_{3}F_{12}) = \frac{−(a_{13} + a_{23})}{\sqrt{2a_{33}(a_{12} + a_{22})}} = \sqrt{\frac{1}{2} + \frac{a_{03} − 1}{2a_{33}}} .

We do not obtain relevant arrangements for optima.

4.7. Case 1.ii.a. In this case A_{3} is proper O = A_{3} is the ball centre. A_{O} is outer, a_{0} = CLH (see Fig. 1) is its polar plane.

(4.9) r = min{A_{3}A_{2}, A_{3}H}; cosh(A_{3}A_{2}) is in (4.5),

cosh(A_{3}H) = \sqrt{1 − \frac{a_{03}^{2}}{a_{00}a_{33}}} = \begin{matrix}sin & \frac{π}{v} \\ √ & \overline{Ba_{00}a_{33}} \end{matrix} ,

R = A_{3}C; cosh(A_{3}C) = \frac{a_{01}a_{03} − a_{00}a_{13}}{\sqrt{a_{00}a_{33}(a_{11}a_{00} − a_{01}^{2})}} = \begin{matrix}cot & \frac{π}{w} & cos & \frac{π}{v} \\ √ & \overline{a_{00}a_{33}B} \end{matrix} =

= \sqrt{1 − \frac{1}{Ba_{11}a_{33}} − \frac{a_{03}^{2}}{a_{00}a_{33}}} .

Table 1.ii.a, Packing, O = A_{3} , \frac{1}{u} + \frac{1}{v} > \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, 3, w → ∞)  \begin{matrix}arsh (\sqrt{2}/2) ≈ \\ ≈ 0.65848 \end{matrix}  0.15266  1.30405  0.35592
(3, 4, 6)  0.96242  0.19616  4.49014  0.47687
(3, 4, w → ∞)  \begin{matrix}log(1 + \sqrt{2}) ≈ \\ ≈ 0.88137 \end{matrix}  0.25096  3.34793  0.27793
(3, 5, 4)  1.30631  0.21299  13.09457  0.51233
(3, 5, 5)  1.40036  0.26320  16.95557  0.53684
(3, 5, w → ∞)  \begin{matrix}−arsh ( \frac{−1}{\sqrt{2 − 2 cos(π/5)}} ) ≈ \\ ≈ 1.25850 \end{matrix}  0.33233  11.43025  0.28662
(4, 3, w → ∞)  \begin{matrix}log(1 + \sqrt{2}) ≈ \\ ≈ 0.88137 \end{matrix}  0.25096  3.34793  0.27793
(5, 3, w → ∞)  \begin{matrix}−arsh ( \frac{−1}{\sqrt{2 − 2 cos(π/5)}} ) ≈ \\ ≈ 1.25850 \end{matrix}  0.33233  11.43025  0.28662

TOP DENSE HYPERBOLIC BALL PACKINGS AND COVERINGS … 141

Table 1.ii.a, Covering, O = A_{3} , \frac{1}{u} + \frac{1}{v} > \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}

(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 3, 8)  1.30889  0.10721  13.18949  5.12591
(3, 4, 5)  1.53591  0.16596  24.17649  3.03486
(3, 5, 4)  1.93116  0.21299  62.56492  2.44790
(3, 5, 5)  2.08287  0.26320  88.11084  2.78974
(4, 3, 8)  1.81579  0.18790  47.88239  5.30903
(5, 3, 8)  2.37474  0.26094  166.52911  5.31816

4.8. Case 1.ii.b. O = A_{2} is the ball centre. (4.10)

r = min\{A_{2}b^{2}, A_{2}L\}; where cosh(A_{2}b^{2})= \sqrt{1 − \frac{1}{a_{22}}} ,

A_{2}a_{0} = A_{2}L; cosh(A_{2}L) = \sqrt{1 − \frac{a_{02}^{2}}{a_{00}a_{22}}} = \frac{1}{sinu} \sqrt{\frac{B}{sin^{2} \frac{π}{w} − cos^{2} \frac{π}{v}}} ,

R = max\{A_{2}C, A_{2}A_{3}, A_{2}H\}; cosh(A_{2}C) = \frac{a_{01}a_{02} − a_{12}a_{00}}{\sqrt{a_{00}a_{22}(a_{11}a_{00} − a_{01}^{2})}} =

= cos \frac{π}{v} ; sin \frac{π}{u} sin \frac{π}{w} \sqrt{a_{00}} = \sqrt{1 − \frac{1}{Ba_{11}a_{22}} − \frac{a_{02}^{2}a_{12}}{a_{00}a_{22}}} ,

cosh(A_{2}H) = \frac{a_{02}a_{03} − a_{23}a_{00}}{\sqrt{a_{00}a_{22}(a_{33}a_{00} − a_{03}^{2})}} = \begin{matrix}cot & \frac{π}{w} \\ sin & \frac{π}{v} & \sqrt{a_{00}} \end{matrix} ,

cosh(A_{2}A_{3}) is in (4.5).

Table 1.ii.b, Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} ⩾ \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, 4, 6)  0.60745  0.19616  1.01065  0.42934
(3, 4, 7)  0.64583  0.21218  1.22631  0.48164
(3, 3, w → ∞)  \begin{matrix}arch (2/\sqrt{3}) ≈ \\ ≈ 0.54931 \end{matrix}  0.15266  0.73740  0.40253
(3, 5, 5)  0.67390  0.26320  1.40355  0.44439
(3, 5, w → ∞)  \begin{matrix}arch (2/\sqrt{3}) ≈ \\ ≈ 0.54931 \end{matrix}  0.33233  0.73740  0.18491
(4, 3, w → ∞)  \begin{matrix}arch (\sqrt{6}/2) ≈ \\ ≈ 0.65848 \end{matrix}  0.25096  1.30405  0.32477

(5, 3, w → ∞) arch( \frac{\sqrt{1 + 4 sin^{2} (π/5)}}{sin(π/5)} ) ≈

≈ 0.77173

0.33233 2.16804 0.32619

(3, 6, 5)  0.72182  0.35992  1.74787  0.40469
(3, 6, w → ∞)  \begin{matrix}arch (2/\sqrt{3}) ≈ \\ ≈ 0.54931 \end{matrix}  0.42289  0.73740  0.14531
(6, 3, w → ∞)  \begin{matrix}arch (\sqrt{2}) ≈ \\ ≈ 0.88137 \end{matrix}  0.42289  3.34793  0.32987
(4, 4, w → ∞)  \begin{matrix}arch (\sqrt{2}) ≈ \\ ≈ 0.88137 \end{matrix}  0.45798  3.34793  0.45689

Table 1.ii.b, Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} > \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 3, 8)  1.16245  0.10721  8.60501  6.68843
(3, 4, 5)  1.14791  0.16596  8.23135  4.13311
(3, 5, 4)  1.30632  0.21299  13.09457  5.12335
(4, 3, 9)  1.55832  0.20295  25.59284  7.88150
(5, 3, 9)  1.80489  0.27783  46.67018  8.39914

4.9. Case 2.i.b. In this case A_{3} is outer, a_{3} = JEQ is its polar plane, A_{0} is proper or boundary point. O = A_{2} is the ball centre (see Fig. 1).

(4.11)

r = min\{A_{2}b^{2}, A_{2}a_{3}\}; cosh(A_{2}b^{2})= \sqrt{1 − \frac{1}{a_{22}}} ,

A_{2}a_{3} = A_{2}Q; cosh(A_{2}Q) = \frac{1}{\sqrt{a_{33}}} = \sqrt{1 + \frac{sin^{2} \frac{π}{u} cos^{2} \frac{π}{w}}{cos^{2} \frac{π}{v} − sin^{2} \frac{π}{u}}} ,

R = max\{A_{2}A_{0}, A_{2}J\}; cosh(A_{2}A_{0}) = \frac{−a_{02}}{\sqrt{a_{00}a_{22}}} ,

cosh(A_{2}J) = \frac{a_{03}a_{23} − a_{02}a_{33}}{\sqrt{a_{33}a_{22}(a_{33}a_{00} − a_{03}^{2})}} = \begin{matrix}cot & \frac{π}{u} & cot & \frac{π}{v} \\ √ & \overline{a_{00}} \end{matrix} .

Table 2.i.b. Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} ⩾ \frac{1}{2}

(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(7, 3, 3)  0.70133  0.08856  1.59395  0.64279
(7, 3, 4)  0.81624  0.16297  2.60141  0.57008
(7, 3, 5)  0.87514  0.23326  3.27033  0.50072
(5, 4, 3)  0.69129  0.16596  1.52220  0.45859
(4, 5, 3)  0.69129  0.21299  1.52220  0.44668
(7, 3, 6)  0.90817  0.31781  3.69768  0.41553
(5, 4, 4)  0.86233  0.34084  3.11499  0.45696
(4, 6, 3)  0.65848  0.31717  1.30405  0.25697

Table 2.i.b. Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} > \frac{1}{2}

(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(8, 3, 3)  1.12838  0.10721  7.75022  2.25901
(7, 3, 4)  1.36005  0.16297  15.19874  3.33068
(7, 3, 5)  1.88213  0.23326  55.88945  8.55728
(5, 4, 3)  1.28550  0.16596  12.34723  3.71986
(4, 5, 3)  1.61692  0.21299  29.64079  8.69789

The cases 2. i. e, f, g, h with O = Q, J, E, K, respectively, will be not relevant for optimal densities.

4.10. Case 2.ii.b. In this case A_{3} is outer, a_{3} = JEQ is its polar plane, A_{0} is outer with polar plane a_{0} = CLH. O = A_{2} is the ball centre (see Fig. 1).

(4.12)

r = min\{A_{2}b^{2}, A_{2}Q, A_{2}L\}; cosh(A_{2}b^{2})= \sqrt{1 − \frac{1}{a_{22}}} , cosh(A_{2}Q) = \frac{1}{\sqrt{a_{33}}} ,

R = max\{A_{2}C, A_{2}H, A_{2}J\}; cosh(A_{2}C) = \frac{a_{01}a_{02} − a_{12}a_{00}}{\sqrt{a_{00}a_{22}(a_{11}a_{00} − a_{01}^{2})}} ,

cosh(A_{2}H) = \begin{matrix}cot & \frac{π}{w} \\ sin & \frac{π}{v} & \sqrt{a_{00}} \end{matrix} , cosh(A_{2}J) = \begin{matrix}cot & \frac{π}{u} & cot & \frac{π}{v} \\ √ & \overline{a_{00}} \end{matrix} .

Table 2.ii.b. Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(7, 3, 7)  0.92836  0.38325  3.97899  0.37080
(7, 3, 8)  0.94156  0.41326  4.17152  0.36051
(7, 3, w → ∞)  arch( \begin{matrix}√ & \overline{5−4 cos 2 (π/7)} \\ √ & \overline{1−cos 2 (π/7)} \end{matrix} ) ≈ 0.98513  0.49195  4.85782  0.35267
(5, 4, 5)  0.91604  0.46190  3.80539  0.41193
5, 4, w → ∞)  arch( \frac{(3−2 cos 2 (π/5))}{\sqrt{2(1−cos 2 (π/5} )}) ≈ 1.01789  0.59404  5.42887  0.45694
Table 2.ii.b. Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(7, 3, 7)  2.28239  0.38325  136.50395  12.72062
(7, 3, 8)  2.16470  0.41326  105.59967  9.12600
(7, 3, 9)  2.16413  0.43171  105.46662  8.72492
(5, 4, 5)  1.83500  0.46190  50.08711  5.42186
(5, 4, 6)  1.79568  0.50747  45.66776  4.49957

The cases 2. ii. e, f, g, h with O = Q, J, E, K, respectively are not relevant for optimal densities.

4.11. Case 2.s.ii.b. O = A_{2} is the ball centre (see Fig. 1) and u = w. The density is related to the "half orthoscheme". (4.13)

r = min\{A_{2}b^{2}, A_{2}Q, A_{2}F_{12}\}; cosh(A_{2}b^{2})= \sqrt{1 − \frac{1}{a_{22}}} , cosh(A_{2}Q) = \frac{1}{\sqrt{a_{33}}} ,

cosh(A_{2}F_{12}) = \sqrt{\frac{1}{2} + \frac{cos \frac{π}{v}}{2 sin^{2} \frac{π}{u}}} , R = max\{A_{2}J, A_{2}F_{03}\};

cosh(A_{2}J) = \begin{matrix}cot & \frac{π}{u} & cot & \frac{π}{v} \\ √ & \overline{a_{00}} \end{matrix} , cosh(A_{2}F_{03}) = cot \frac{π}{u} \sqrt{\frac{cos \frac{π}{v} + sin^{2} \frac{π}{u}}{2(1 − cos \frac{π}{v})}} .

Table 2.s.ii.b. Packing, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, v → ∞, 3)  arch ( \frac{\sqrt{42}}{6} ) ≈ 0.39768  0.44446  0.27191  0.10196
(4, v → ∞, 4)  arch ( \frac{\sqrt{6}}{2} ) ≈ 0.65848  0.63434  1.30405  0.25697
(5, 5, 5)  0.74746  0.57271  1.95548  0.34144
(5, v → ∞, 3, 5)  \begin{matrix}arch( \sqrt{2 − cos ( \frac{π}{5} )} ) ≈ \\ ≈ 0.55832 \end{matrix}  0.73015  0.77585  0.10626
(6, 4, 6)  0.78340  0.55557  2.27605  0.34140
(6, v → ∞, 6)  arch ( \frac{\sqrt{5}}{2} ) ≈ 0.48121  0.78465  0.48890  0.05192
Table 2.s.ii.b. Covering, O = A_{2} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 8, 3)  1.49263  0.32610  21.63099  11.05541
(4, 5, 4)  1.43911  0.43062  18.80212  5.45785
(5, 4, 5)  1.40493  0.46190  17.16523  3.71622
(6, 4, 6)  1.47083  0.55557  20.43783  3.06559

4.12. Case 2.s.ii.c. O = F_{03} is the ball centre (see Fig. 1) and u = w. (4.14)

r = min\{F_{03}b^{0}, F_{03}J\}; cosh(F_{03}b^{0})= \sqrt{1 + \frac{cos \frac{π}{v} − sin^{2} \frac{π}{u}}{2(1 − cos \frac{π}{v})}} ,

cosh(F_{03}J) = \begin{matrix}sin & \frac{π}{v} \\ √ & \overline{2Ba_{33}(a_{33} + a_{03})} \end{matrix} , R = max\{F_{03}A_{2}, F_{03}Q\};

cosh(F_{03}A_{2}) = cot \frac{π}{u} \sqrt{\frac{cos \frac{π}{v} + sin^{2} \frac{π}{u}}{2(1 − cos \frac{π}{v})}} , cosh(F_{03}Q) = \begin{matrix}cot & \frac{π}{u} & cos & \frac{π}{v} \\ √ & \overline{2Ba_{33}(a_{33} + a_{03})} \end{matrix} .

Table 2.s.ii.c. Packing, O = F_{03} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, 7, 3)  0.78871  0.27899  2.32647  0.59564
(3, 8, 3)  0.72041  0.32610  1.73696  0.33290
(4, 5, 4)  0.80846  0.43062  2.52145  0.58554
(5, 4, 5)  0.72146  0.46190  1.74508  0.47226
(6, 4, 6)  0.69217  0.55557  1.52838  0.34388
Table 2.s.ii.c. Covering, O = F_{03} , \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 8, 3)  1.26607  0.32610  11.68147  2.23886
(4, 5, 4)  1.22646  0.43062  10.41484  2.41856
(5, 4, 5)  1.28483  0.46190  12.32350  3.33500
(5, 5, 5)  1.51882  0.57271  23.14218  4.04082
(6, 4, 6)  1.43252  0.55557  18.47661  4.15712

4.13. Case 2.s.ii.d. O = F_{12} is the ball centre (see Fig. 1) and u = w. (4.15)

r = min\{F_{12}b^{1}, F_{12}a_{3}\}; cosh(F_{12}b^{1})= \sqrt{1 + \frac{cos \frac{π}{v} − sin^{2} \frac{π}{u}}{2}} ,

cosh(F_{12}a_{3}) = \sqrt{1 + \frac{cos^{2} \frac{π}{u} (cos \frac{π}{v} + sin^{2} \frac{π}{u})}{2(cos^{2} \frac{π}{v} − sin^{2} \frac{π}{u})}} , R = max\{F_{12}J, F_{12}Q\};

cosh(F_{12}J) = \frac{cos \frac{π}{u} (1 + cos \frac{π}{v})}{sin \frac{π}{v} \sqrt{2a_{33}(cos \frac{π}{v} + sin^{2} \frac{π}{u})}} , cosh(F_{12}Q) = \sqrt{\frac{cos \frac{π}{v} + sin^{2} \frac{π}{u}}{2a_{33} sin^{2} \frac{π}{u}}} .

This case is not relevant.

4.14. Case 2.s.ii.e. O = Q is the ball centre (see Fig. 1) and u = w.

(4.16)

r = min\{QA_{2}, QE, \frac{1}{2}QC\}; cosh(QA_{2}) = \frac{1}{\sqrt{a_{33}}} , cosh(QE) = \begin{matrix}cos & \frac{π}{v} \\ sin & \frac{π}{u} \end{matrix} ,

cosh( \frac{1}{2}QC) = \sqrt{\frac{1}{2} + \frac{a_{12}}{2a_{22}a_{33}}} , R = max\{QF_{03}, QF_{12}\};

cosh(QF_{03}) = \begin{matrix}cot & \frac{π}{u} & cos & \frac{π}{v} \\ √ & \overline{2Ba_{33}(a_{33} + a_{03})} \end{matrix} , cosh(QF_{12}) = \sqrt{\frac{cos \frac{π}{v} + sin^{2} \frac{π}{u}}{2a_{33} sin^{2} \frac{π}{u}}} .

Table 2.s.ii.e. Packing, O = Q, \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  r^{opt}  V ol(W_{uvw} )  V ol(B(r^{opt} ))  δ^{opt}
(3, v → ∞, 3)  arch ( \frac{2\sqrt{3}}{3} ) ≈ 0.54931  0.44446  0.73740  0.27652
(4, 8, 4)  0.76429  0.56369  2.10109  0.46592
(4, 9, 4)  0.73883  0.57923  1.88365  0.40650
(4, v → ∞, 4)  arch ( \frac{\sqrt{6}}{2} ) ≈ 0.65848  0.63434  1.30405  0.25697
(5, 5, 5)  0.77537  0.57271  2.20130  0.38436
(5, v → ∞, 5)  \begin{matrix}arch( \frac{1}{2} \sqrt{2 − cos ( \frac{π}{5} )^{2}} ) ≈ \\ ≈ 0.55832 \end{matrix}  0.73015  0.77585  0.10626
(6, 4, 6)  0.78340  0.55557  2.27605  0.34140
Table 2.s.ii.e. Covering, O = Q, \frac{1}{u} + \frac{1}{v} < \frac{1}{2} , \frac{1}{v} + \frac{1}{w} < \frac{1}{2}
(u, v, w)  R^{opt}  V ol(W_{uvw} )  V ol(B(R^{opt} ))  Δ^{opt}
(3, 8, 3)  1.17362  0.32610  8.90096  4.54920
(3, 9, 3)  1.23555  0.35444  10.69512  5.02911
(4, 5, 4)  1.22646  0.43062  10.41484  3.02321
(4, 8, 4)  1.33092  0.56369  14.02565  3.11023
(5, 4, 5)  1.28483  0.46190  12.32350  2.66800
(5, 5, 5)  1.42880  0.57271  18.29491  3.19444
(6, 4, 6)  1.40674  0.55557  17.24885  2.58725
(6, 5, 6)  1.63628  0.64650  31.09468  4.00806
(6, 6, 6)  1.83634  0.69130  50.24429  6.05670

The cases 2. s. ii. f, g, h with O = J, E, K, respectively are not relevant for optimal densities.

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Budapest University of Technology and Economics Institute of Mathematics Department of Geometry Budapest Hungary

emolnar@math.bme.hu
szirmai@math.bme.hu