Coverings with horo- and hyperballs generated by simply truncated orthoschemes
CCoverings with horo- and hyperballs generatedby simply truncated orthoschemes
Mikl´os Eper and Jen˝o SzirmaiBudapest University of Technology andEconomics Institute of Mathematics,Department of GeometryFebruary 26, 2020
Abstract
After having investigated the packings derived by horo- and hyper-balls related to simple frustum Coxeter orthoscheme tilings we con-sider the corresponding covering problems (briefly hyp-hor coverings)in n -dimensional hyperbolic spaces H n ( n = 2 , − and 3 − dimensional hyperbolic spaces hyp-hor coverings that are generated by simply truncated Coxeter or-thocheme tilings and we determine their thinnest covering configu-rations and their densities.We prove that in the hyperbolic plane ( n = 2) the density of theabove thinnest hyp-hor covering arbitrarily approximate the univer-sal lower bound of the hypercycle or horocycle covering density √ π and in H the optimal configuration belongs to the { , , } Coxetertiling with density ≈ . .
280 due to L. Fejes T´oth andK. B¨or¨oczky.Moreover, we study the hyp-hor coverings in truncated orthosche-mes { p, , } (6 < p < , p ∈ R ) whose density function attains itsminimum at parameter p ≈ . ≈ . a r X i v : . [ m a t h . M G ] F e b Mikl´os Eper and Jen˝o Szirmai covering density than the former determined ≈ . H . The packing and covering problems with solely horo- or hyperballs (horo- orhypespheres) are intensively investigated in earlier works in n -dimensional( n ≥
2) hyperbolic space H n .In n -dimensional hyperbolic space H n ( n ≥
2) there are 3 kinds of ”balls(spheres)”: the classical balls (spheres), horoballs (horospheres) and hyper-balls (hyperspheres).In this paper we consider the coverings with horo- and hyperballs andtheir densities in 2- and 3-dimensional hyperbolic space where the coveringsare derived from simply truncated Coxeter orthoscheme tilings.A Coxeter simplex is an n -dimensional simplex in X ∈ { S n , H n , E n } withdihedral angles either submultiples of π or zero. The group generated byreflections on the sides of a Coxeter simplex is called a Coxeter simplexreflection group. Such reflections determine a discrete group of isometries of X with the Coxeter simplex as its fundamental domain; hence such groupsgenerate a tessellation of X .First we shortly survey the previous results related to this topic.1. On horoball packings and coverings
In the case of periodic ball or horoball packings and coverings, the localdensity defined e.g. in [3] can be extended to the entire hyperbolicspace. This local density is related to the simplicial density functionthat we generalized in [21] and [22]. In this paper we will use suchdefinition of covering density.In the n -dimensional space X ∈ { E n , S n , H n } of constant curvature( n ≥ d n ( r ) to be the densityof n + 1 spheres of radius r mutually touching one another with respectto the regular simplex spanned by the centers of the spheres. L. FejesT´oth and H. S. M. Coxeter conjectured that the packing density ofballs of radius r in X cannot exceed d n ( r ). Rogers [14] proved thisconjecture in Euclidean space E n . The 2-dimensional spherical case overings with horo- and hyperballs . . . Theorem 1.1 (K. B¨or¨oczky)
In an n -dimensional space of constantcurvature, consider a packing of spheres of radius r . In the case ofspherical space, assume that r < π . Then the density of each spherein its Dirichlet–Voronoi cell cannot exceed the density of n + 1 spheresof radius r mutually touching one another with respect to the simplexspanned by their centers. In hyperbolic space H , the monotonicity of d ( r ) was proved by B¨or¨oczkyand Florian in [4].This upper bound for packing density in hyperbolic space H is ≈ . H where the ideal centers of horoballslie on the absolute figure of H ; for example, they may lie at the verticesof the ideal regular simplex tiling with Coxeter-Schl¨afli symbol { , , } .From this regular ideal tetrahedron tiling can be derived the knownleast dense ball or horoball covering configuration (see [6])with density ≈ . H men-tioned above is not unique. We gave several new examples of horoballpacking arrangements based on totally asymptotic Coxeter tilings thatyield the B¨or¨oczky–Florian upper bound [4].Furthermore, in [21], [22] we found that by allowing horoballs of differ-ent types at each vertex of a totally asymptotic simplex and generalizingthe simplicial density function to H n for ( n ≥ n ≥
3. For example, in H the locally optimal packing den-sity is ≈ . ≈ . H n . Further open problems and conjectures on 4-dimensionalhyperbolic packings are discussed in [5]. Using horoball packings in H , allowing horoballs of different types, we found seven counterexam-ples (realized by allowing up to three horoball types) to one of L. FejesT´oth’s conjectures stated in his foundational book Regular Figures. Mikl´os Eper and Jen˝o Szirmai
In [11] and [12] we continued our investigations of ball packings, inhyperbolic spaces of dimensions n = 5 . . .
9. Using horoball packings,allowing horoballs of different types when applicable, we found severalinteresting and dense packing configuratons with respect to the Coxetersimplex cells.The second-named author has several additional results on globally andlocally optimal ball packings in the eight Thurston geomerties arisingfrom Thurston’s geometrization conjecture see e.g. [16], [27].2.
On hyperball packings and coverings
In hyperbolic plane H the universal upper bound of the congruenthypercycle packing density is π , proved by I. Vermes in [32]. He ini-tiated this topic and determined also the universal lower bound of thecongruent hypercycle covering density, in [33], equal to √ π .In [23] and [24] we have analysed the regular prism tilings (simpletruncated Coxeter orthoscheme tilings) and the corresponding optimalhyperball packings in H n ( n = 3 , , n -dimensional hyperbolic regular prism honey-combs and the corresponding coverings by congruent hyperballs andwe determined their least dense covering. Furthermore, we formulatedconjectures for the candidates of the least dense covering by congruenthyperballs in the 3- and 5-dimensional hyperbolic space.In [18] we discussed congruent and non-congruent hyperball packings tothe truncated regular tetrahedron tilings. These are derived from thetruncated Coxeter simplex tilings { , , p } (7 ≤ p ∈ N ) and { , , , , } in 3- and 5-dimensional hyperbolic space, respectively. We determinedthe densest packing arrangement and its density with congruent hy-perballs in H and determined the smallest density upper bounds ofnon-congruent hyperball packings generated by the above tilings.In [17] we deal with such packings by horo- and hyperballs (brieflyhyp-hor packings) in H n ( n = 2 , H that canbe derived from truncated tetrahedron tilings. We proved that if thetruncated tetrahedron is regular { , , p } , but we allow also 6 < p ∈ R , overings with horo- and hyperballs . . . ≈ . H . However, we described a hyperball packing construction,by the regular truncated tetrahedron tiling under the extended Coxetergroup { , , } with maximal density ≈ . H into truncatedtetrahedra. Therefore, in order to get a density upper bound for hy-perball packings, it is sufficient to determine the density upper boundof hyperball packings in truncated simplices.In [20] we proved, that the density upper bound of the saturated con-gruent hyperball packings, related to corresponding truncated tetra-hedron cells is locally realized in a regular truncated tetrahedon withdensity ≈ . { , , p } (6 < p ∈ N ) and { , , p } (4 < p ∈ N )in hyperbolic space H . If we allow p ∈ R as well, then the locally dens-est (non-congruent half) hyperball configuration belongs to the trun-cated cube with density ≈ . H . We determined the extendable densest non-congruent hyperball packing arrangement related to the truncated cubetiling { , , } with density ≈ . { , , } with density ≈ . Mikl´os Eper and Jen˝o Szirmai
In this paper we deal with the coverings with horo- and hyperballs (brieflyhyp-hor coverings) in the n -dimensional hyperbolic spaces H n ( n = 2 , − and 3 − dimensional hyperbolic spaces hyp-horcoverings that are generated by complete Coxeter tilings of degree 1 i.e.the fundamental domains of these tilings are simple frustum orthoschemeswith a principal vertex lying on the absolute quadric and the other principalvertex is outer point. We determine their thinnest covering configurationsand their densities. These considered Coxeter tilings exist in the 2 − , 3 − and5 − dimensional hyperbolic spaces (see [7]) and have given by their Coxeter-Schl¨afli graph in Fig. 1. We prove that in the hyperbolic plane n = 2 the p qq q5 r n=2n=3n=5 p Î N =[p,3,6], p 7[p,4,4], p 5[p,6,3], p 4 Figure 1: Coxeter-Schl¨afli graph of Coxeter tilings of degree 1.density of the above hyp-hor coverings arbitrarily approximate the universalupper bound of the hypercycle or horocycle packing density √ π and in H the thinnest hyp-hor configuration belongs to the { , , } Coxeter tiling withdensity ≈ . { p, , } (6 < p < , p ∈ R ). Its density function is attained its minimum forparameter p ≈ . ≈ . ≈ . H . overings with horo- and hyperballs . . . For H n we use the projective model in the Lorentz space E ,n of signature(1 , n ), i.e. E ,n denotes the real vector space V n +1 equipped with the bilinearform of signature (1 , n ): (cid:104) x , y (cid:105) = − x y + x y + · · · + x n y n where the non-zero vectors x = ( x , x , . . . , x n ) ∈ V n +1 and y = ( y , y , . . . , y n ) ∈ V n +1 , are determined up to real factors, for representing points of P n ( R ). Then, H n can be interpreted as the interior of the quadric Q = { [ x ] ∈ P n |(cid:104) x , x (cid:105) =0 } =: ∂ H n in the real projective space P n ( V n +1 , V n +1 ).The points of the boundary ∂ H n in P n are called points at infinity of H n ,the points lying outside ∂ H n are said to be outer points of H n relative to Q .Let P ([ x ]) ∈ P n , a point [ y ] ∈ P n is said to be conjugate to [ x ] relative to Q if (cid:104) x , y (cid:105) = 0 holds. The set of all points which are conjugate to P ([ x ])form a projective (polar) hyperplane x = pol ( x ) := { [ y ] ∈ P n |(cid:104) x , y (cid:105) = 0 } . Thus the quadric Q induces a bijection (linear polarity V n +1 → V n +1 ) fromthe points of P n onto its hyperplanes.The distance s of two proper points [ x ] and [ y ] is calculated by the for-mula: cosh sk = −(cid:104) x , y (cid:105) (cid:112) (cid:104) x , x (cid:105)(cid:104) y , y (cid:105) . (2.1) A n -dimensional tiling P (or solid tessellation, honeycomb) is an infinite set ofcongruent polyhedra (polytopes) that fit together to fill all space ( H n ( n (cid:61) d can be described asfollows:1. For d = 0, they coincide with the class of classical orthoschemes in-troduced by Schl¨afli. The initial and final vertices, A and A n of theorthogonal edge-path A i A i +1 , i = 0 , . . . , n −
1, are called principalvertices of the orthoscheme.2. A complete orthoscheme of degree d = 1 can be interpreted as anorthoscheme with one outer principal vertex, say A n , which is trun- Mikl´os Eper and Jen˝o Szirmai cated by its polar plane pol ( A n ) (see Fig. 2 and 3). In this case theorthoscheme is called simply truncated with outer vertex A n .3. A complete orthoscheme of degree d = 2 can be interpreted as anorthoscheme with two outer principal vertices, A , A n , which is trun-cated by its polar hyperplanes pol ( A ) and pol ( A n ). In this case theorthoscheme is called doubly truncated. We distinguish two differenttypes of orthoschemes but I will not enter into the details (see [8]).In general the complete Coxeter orthoschemes were classified by Im Hofin [7] by generalizing the method of Coxeter and B¨ohm, who showed thatthey exist only for dimensions ≤
9. From this classification it follows, thatthe complete orthoschemes of degree d = 1 exist up to 5 dimensions.In this paper we consider the orthoschemes of degree 1 where the initialvertex A lies on the absolute quadric Q . These orthoschemes and the cor-responding Coxeter tilings exist in the 2-, 3 − and 5 − dimensional hyperbolicspaces and are characterized by their Coxeter-Schl¨afli symbols and graphs(see Fig. 1).In n -dimensional hyperbolic space H n ( n ≥
2) it can be seen that if S is a complete orthoscheme of degree d = 1 (with vertices A A A . . . A n − P P P . . . P n − ) a simply frustum orthoscheme (here A n is a outer vertex of H n then the points P , P , P , . . . , P n − lie on the polar hyperplane π of A n ).We consider the images of S under reflections on its side facets. Theunion of these n -dimensional orthoschames (having the common π hyper-plane) forms an infinite polyhedron denoted by G . G and its images underreflections on its ,,cover facets” fill hyperbolic space H n without overlap andgenerate n -dimensional tilings T . The constant k = (cid:113) − K is the natural length unit in H n . K will be theconstant negative sectional curvature. In the following we assume that k = 1 . n -dimensionalCoxeter orthoschemes
1. 2-dimensional hyperbolic space H In the hyperbolic plane a simple frustum orthoscheme is a Lambertquadrilateral with exactly three right angles and its fourth angle isacute πq ( q ≥
3) (see Fig. 1 and 3). In our case the Lambert quadrilateral overings with horo- and hyperballs . . .
V ol ( S ) = π . (2.2)2. 3-dimensional hyperbolic space H :Our polyhedron A A A P P P is a simple frustum orthoscheme withouter vertex A (see Fig. 5.a) whose volume can be calculated by thefollowing theorem of R. Kellerhals [8]: Theorem 2.1
The volume of a three-dimensional hyperbolic completeorthoscheme (except Lambert cube cases) S is expressed with the essen-tial angles α , α , α , (0 ≤ α ij ≤ π ) (Fig. 1 and 2) in the followingform: V ol ( S ) = 14 {L ( α + θ ) − L ( α − θ ) + L ( π α − θ )++ L ( π − α − θ ) + L ( α + θ ) − L ( α − θ ) + 2 L ( π − θ ) } , (2.3) where θ ∈ [0 , π ) is defined by the following formula: tan( θ ) = (cid:112) cos α − sin α sin α cos α cos α and where L ( x ) := − x (cid:82) log | t | dt denotes the Lobachevsky function. For our prism tilings T pqr we have: α = πp , α = πq , α = πr . The equidistant surface (or hypersphere) is a quadratic surface that lies ata constant distance from a plane in both halfspaces. The infinite body ofthe hypersphere is called a hyperball. The n -dimensional half-hypersphere ( n = 2 ,
3) with distance h to a hyperplane π is denoted by H hn . The volumeof a bounded hyperball piece H hn ( A n − ) bounded by an ( n − Mikl´os Eper and Jen˝o Szirmai A n − ⊂ π , H hn and by hyperplanes orthogonal to π derived from the facetsof A n − can be determined by the formulas (2.4) and (2.5) that follow fromthe suitable extension of the classical method of J. Bolyai ([2]): V ol ( H h ( A )) = V ol ( A ) sinh ( h ) , (2.4) V ol ( H h ( A )) = 14 V ol ( A ) [sinh (2 h ) + 2 h ] , (2.5)where the volume of the hyperbolic ( n − A n − lying in the plane π is V ol n − ( A n − ). A horosphere in H n ( n ≥
2) is a hyperbolic n -sphere with infinite radiuscentered at an ideal point on ∂ H n . Equivalently, a horosphere is an ( n − H n centeredat O (1 , , , . . . , T (1 , , . . . , S (1 , , . . . , s ) is derived from the equation of thethe absolute sphere − x x + x x + x x + · · · + x n x n = 0, and the plane x − x n = 0 tangent to the absolute sphere at T . The general equation ofthe horosphere in cartesian coordinates is the following:2 (cid:0)(cid:80) n − i =1 h i (cid:1) − s + 4 (cid:0) h n − s +12 (cid:1) (1 − s ) = 1 . (2.6) In n -dimensional hyperbolic space any two horoballs are congruent in theclassical sense. However, it is often useful to distinguish between certainhoroballs of a packing. We use the notion of horoball type with respect to thepacking as introduced in [22]. The intrinsic geometry of a horosphere is Euclidean, so the ( n − A of a polyhedron A on the surface of the horosphere canbe calculated as in E n − . The volume of the horoball piece H ( A ) determinedby A and the aggregate of axes drawn from A to the center of the horoballis ([2]) V ol ( H ( A )) = 1 n − A . (2.7) overings with horo- and hyperballs . . . We consider the usual Beltrami-Cayley-Klein ball modell of H centered at O (1 , ,
0) with a given vector basis e i ( i = 0 , ,
2) and set the 2-dimensionalCoxeter orthoscheme A A A in this coordinate system with coordinates A (1 , , A (1 , , A (1 , a , A is lying on the absolute quadric Q and the other principalvertex A is an outer point of the model, so 0 < a < , a ∈ R .The polar line of the outer vertex A is π = u (1 , − a , T . By the trun-cation of the orthoscheme A A A by the polar line π we get the Lam-bert quadrilateral A A P P (see Fig. 2), where the further vertices are: π ∩ A A = P (1 , a, − a ); π ∩ A A = P (1 , a, H without overlap, hence we getthe previously described 2-dimensional Coxeter tilings, given by the Coxetersymbol [ ∞ ] (see Fig. 1). The tilings contain the free parameter a , so wedenote the tilings by T a , and the Lambert quadrilaterals A A P P by F a ,which serve as the fundamental domain of the above tilings.a) b)Figure 2: a) C a -type hyp-hor covering at present a = 0 . , t = 0 . C a -typehyp-hor covering at present a = 0 . , t = 0 . F a , by the follows:1. The center of the horocycle can only be the vertex A . Let the inter-section of the horocycle with A A line S (1 , , s ) ( − < s <
1) and2
Mikl´os Eper and Jen˝o Szirmai with A P line T (1 , ta, − ta ) (0 < t < a ). We denote by H a ( t ) thehorocycle-piece determined by points A , S , T (see Fig. 2).2. Let A P be the base straight line of a hypercycle and M the intersec-tion point of the horo- and hypercycle lies on the A P or P P side of F a (see Fig. 2).3. Let the intersection of the hypercycle with the positive segment of A A line S (1 , , s ) (0 < s <
1) and with P P line R (1 , a, r ) (0 < r < √ − a ). We denote by H a ( t ) the hypercycle-piece settled by points P , R, S , A (see Fig. 2).We can see, that if the horo- and hypercycles satisfy the above requirements,than they cover F a . Thus the images of H a ( t ) and H a ( t ) under reflectionon the sides of F a provide a hyp-hor covering of hyperbolic plane H . Thefundamental domain F a (i.e. parameter a ) and point M (i.e. parameter t )determine the covering. We distinguish two main types of hyp-hor coverings,denoted by C a ( t ) if M ∈ A P and by C a ( t ) if M ∈ P P (see Fig. 2). Definition 3.1
The density of the above hyp-hor coverings C ia ( t ) ( i = 1 , are: δ ( C ia ( t )) = V ol ( H a ( t )) + V ol ( H a ( t )) V ol ( F a )It is obvious, that if the point M lies on the perimeter of F a , the densityof the covering is smaller, than it lies out of F a . Thus we get the coveringswith minimal densities in the above two main cases. C a ( t ) . In this case M ∈ A P is the intersection point of the cycles, so M = (1 , ta, − ta ) (0 < t ≤ S can be expressed using (2.6) andthe distance of M and S can be calculated by (2.1), thus we can determinethe volume of H a ( t ) by formula (2.6). The length of A P and the distanceof M and the x − axis can be calculated also by (2.1), thus we can determinethe volume of H a ( t ) by formula (2.4). We obtain by Definition 3.1, that thedensity of C a ( t ) can be expressed by the following formula: δ ( C a ( t )) = arccosh (cid:16) √ − a (cid:17) − ta a √ t − t − a t + 2 sinh (cid:16) arccosh (cid:16) ta + t − t − ta (cid:17)(cid:17) π overings with horo- and hyperballs . . . < a <
1, 0 < t ≤ Theorem 3.2
Analysing the above density formula we obtain that lim a → (cid:18) C a (cid:18) (cid:19)(cid:19) = √ π and (cid:0) C a (cid:0) (cid:1)(cid:1) < √ π for parameter < a < (see Fig. 3a). That means,that in hyperbolic plane H the universal lower bound density of ball andhoroball coverings can be arbitrary accurate approximate with the densities δ (cid:0)(cid:0) C a (cid:0) (cid:1)(cid:1)(cid:1) of hyp-hor packings of type 1. a) b)Figure 3: a) The density function of hyp-hor covering C a in case t =0 . C a in case t ≈ . C a ( t ) . In this case M ∈ P P the intersection point of the cycles, so the intersectionpoint of the horocycle and line A P is (1 , ta, − ta ) (0 < t < a − a ),by the condition, that M lies on the positive segment of P P . We get thevolume of H a ( t ) just like in the previous section. The coordinates of M andthe h length of M P can be calculated by (2.6) and (2.1). We can determinethe volume of H a ( t ) by formula (2.4). We obtain by Definition 3.1, that thedensity of C a ( t ) can be expressed by the following formula: δ ( C a ( t )) = arccosh (cid:16) √ − a (cid:17) sinh h + 2 sinh (cid:16) arccosh (cid:16) ta + t − t − ta (cid:17)(cid:17) π Mikl´os Eper and Jen˝o Szirmai where 0 < a <
1, 0 < t < a − a . Theorem 3.3
Analysing the above density formula (using also numericalapproximation methods) we obtain that it provides its minimum in case t ≈ . , a → (see Fig. 3b), and the minimum value is √ π . That means, thatin hyperbolic plane H the universal lower bound density of ball coveringscan be arbitrary accurate approximate with the densities δ ( C a ) of hyp-horpackings of type 2. H H there are 3 infinite series of thesimple frustum Coxeter orthoschemes with vertex at the infinity, that arelisted in Fig. 1, and characterized in Section 2.1. The Coxeter-Schl¨afli symbolof these orthoschemes are { p, q, r } , where ( q, r ) = (3 , , (4 , , (6 , p is an appropriate integer parameter: p ≥ q, r ) = (3 , p ≥ q, r ) = (4 , p ≥ q, r ) = (6 , F ( q,r ) p , and its vertices aredenoted by A , A , A , P , P , P (see Fig. 5.a).We consider the usual Beltrami-Cayley-Klein ball modell of H centredat O (1 , , ,
0) with a given vector basis e i ( i = 0 , , ,
3) (see Section 2.1)and with the 3-dimensional complete Coxeter orthoscheme A A A A whichinitial principal vertex A is lying on the absolute quadric Q and the otherprincipal vertex A is an outer point of the model. By the truncation ofthe orthoscheme with π (the polar plane of A ) we get the proper vertices P k [ p k ] = π ∩ A k A , ( i = 0 , , p k ∼ c · a + a k for some c ∈ R . P k [ p k ] lies on a = pol ( a ) if and only if p k a = 0, thus: c · a a + a k a = 0 ⇔ c = − a k a a a (4.1) ⇔ p k ∼ − a k a a a · a + a k ∼ a k ( a a ) + a ( a k a ) (4.2)We consider the Coxeter-Schl¨afli matrix ( c ij ) of the orthoscheme, and itsinverse ( h ij ), where the elements of the matrices: c ij = a i a j , h ij = a i a j .The polar hyperplane of A is a , thus h k = a k a , hence by (4.2) p k = a k h − a h k . overings with horo- and hyperballs . . . F ( q,r ) p in the usual coordi-nate system with vertices: P (1 , , , P (1 , , y, P (1 , x, y, A (1 , , , A (1 , , y, z ), A (1 , x, y, z ) (see Fig. 5.a). We get the following equations,using the formulas (2.1), (4.2) and h ij = a i a j :cosh( d ( P P )) = h h − h h (cid:112) ( h h − h )( h h − h ) = 1 (cid:112) ( − − y ) , (4.3)cosh( d ( P P )) = h h − h h (cid:112) ( h h − h )( h h − h ) = 1 (cid:112) ( − − y + x ) , (4.4)cosh( d ( A P )) = (cid:115) h h − h h h = 1 − y (cid:112) ( − y + z )( − y ) , (4.5)cosh( d ( A P )) = (cid:115) h h − h h h = 1 − y − x (cid:112) ( − y + x + z )( − y + x ) . (4.6)We can determine the coordinates x, y, z k , ( k = 1 ,
2) by solving these equa-tions, and the volume of the orthoschemes F ( q,r ) p by Theorem 2.1.The images of the above orthoscheme F ( q,r ) p under reflections on its facetsfill the hyperbolic space H without overlap, so we get the Coxeter tiling T ( q,r ) p of H with fundamental domain F ( q,r ) p .We construct hyp-hor coverings to F ( q,r ) p using the following requirements:1. The center of the horoball can only be the ideal vertex A . Let S , T , Q the intersection points of the horoball with A P , A A , A A lines. Wedenote by H ( q,r ) p the horoball-piece determined by points A , S , T , Q (see Fig. 5.b).2. P P P plane can be the base hyperplane of a hyperball. Let S , V , R be the intersection points of the hyperball with the line segments of A P , A P , A P . We denote by H ( q,r ) p the hyperball-piece bounded bythe base hyperplane, the surface of the hyperball and the hyperplanesperpendicular to the base hyperplane derived from edges P P , P P , P P (see Fig. 5.b).6 Mikl´os Eper and Jen˝o Szirmai
3. The intersection curve (which is a circle parallel with [ xy ] plane in Eu-clidean sense) of the horo- and hyperball passes through one of the edgesof the orthoscheme A A , A A , A A , A P , A P , A P (see Fig. 5.a).We can see, that if the horo- and hyperballs satisfy the above require-ments, than they cover F ( q,r ) p if and only if they cover all the edges of F ( q,r ) p .Hence, if a covering arrangement covers the edges of the orthoscheme, thanthe images of H ( q,r ) p and H ( q,r ) p under reflection on the facets of F ( q,r ) p providea hyp-hor covering of hyperbolic space H , denoted by C ( q,r ) p . Definition 4.1
The density of the above hyp-hor coverings C ( q,r ) p is: δ ( C ( q,r ) p ) = V ol ( H ( q,r ) p ) + V ol ( H ( q,r ) p ) V ol ( F ( q,r ) p ) (4.7)It is obvious, that if the intersection curve passes through one of the edgeof F ( q,r ) p , the density of the covering is smaller, than it goes out of F ( q,r ) p .Thus we get the coverings with minimal densities if the above requiementshold. Based on the above, we have to distinguish and study six cases. • If the intersection curve of the balls passes through A P (see Fig. 5.a),then the balls touch each other, thus the hyp-hor covering is obviouslynot realized. • If the intersection curve of the balls intersects the edge A A (seeFig. 5.a), then we can parametrize their common point: T ( t ) = (1 , tx, ty, tz +(1 − t )) , t ∈ [0 , S ∈ P A , S ∈ P A points. If the horoballcovers A , we can determine the intersection points U , U by solvingthe corresponding equations. By inspecting the z -coordinates of U i ( i = 1 ,
2) in the model, we can see, that U is always higher than U ,which means (using the convexity of the ellipsoids) that they togetherdo not cover the edge A A . If the hyperball covers A , we can deter-mine the intersection points Q , Q by solving the corresponding equa-tions. By inspecting the z -coordinates of Q i ( i = 1 ,
2) in the model,we can see, that Q is always higher than Q , which means (using theconvexity of the ellipsoids) that they together do not cover the edge A A . Thus in this case the hyp-hor covering is not realized. overings with horo- and hyperballs . . . • If the intersection curve of the balls contains a point of A P edge (seeFig. 5. a) then we can parametrize the intersection point V : V ( v ) =(1 , , y, vz ) , v ∈ [0 , A , than the balls do not cover edge A P ,and if the hyperball covers A , than the balls do not cover edge A A .Thus, in this case the hyp-hor covering is not realized. A A edge In this case, A A edge has a common point with the intersection curveof the balls (see Fig. 5.a), so we can parametrize the intersection point Q : Q ( q ) = (1 , , qy, qz + (1 − q )) , q ∈ [0 , S , S ∈ P A . After that, we candetermine the intersection points T , T ∈ A A by solving the correspondingequations. We prove, that the balls cover the edges of the orthoscheme, sothe hyp-hor covering is realized in this case. P A A P is a 2-dimensionalCoxeter orthoscheme, thus A A is covered as we have seen in Section 3. Thehyperball covers A , and we can see, that the hyperbolic length of A P edgeis always bigger than the length of A P edge, so the hyperball covers A ,and because of its convexity A P , A P , A A edges as well. By inspectingthe z -coordinates of S i and T i ( i = 1 ,
2) in the model, we can see, that S is always “higher” than S and T is always “higher” than T , which means(using the convexity of the ellipsoids) that they together cover the edges A P and A A .We know the coordinates of points Q, T i , S i ( i = 1 , V ol ( H ( q,r ) p ), V ol ( H ( q,r ) p ) using (2.5), (2.7) and the density of the cover-ing using (4.7), which depends on free parameter q . Analysing this densityfunction we can compute the optimal densities (see Fig. 4.a). The results fortiling T (6 , p (which provides the lowest density in this case) are summarizedin the table below. Type of tiling δ min q T (6 , . . T (6 , . . T (6 , . . Mikl´os Eper and Jen˝o Szirmai a) b)Figure 4: a) The density function δ ( C (6 , ( q )) b) The density function δ ( C (3 , ( u )) A A edge Now, the intersection curve of the balls passes through A A (see Fig. 5.a), sowe can parametrize the intersection point U ∈ A A : U ( u ) = (1 , ux, y, uz +(1 − u ) z ) , u ∈ [0 , S , S points. After that we can determine the intersectionpoints V , V , Q , Q and T , T by solving the corresponding equations. Wecan prove similarly to the above case, that the balls cover the edges of theorthoscheme, so the hyp-hor covering is realized in this case. The horoballcovers A A , the hyperball covers A P , and together they cover A A (seeFig. 5.b). By inspecting the z -coordinates of S i , V i and T i ( i = 1 ,
2) in themodel, we can see in this case too, that the balls cover A P , A P , A A edges (see Fig. 5.b).We know points Q i , T i , S i ( i = 1 , V ol ( H ( q,r ) p ), V ol ( H ( q,r ) p ) using (2.5), (2.7) and the density of the covering using (4.7),which depends on free parameter u . Analysing this density function we cancompute the optimal densities (see Fig. 4 b). The results for tiling T (3 , p (which provides the lowest density in this case) are summarized in the nexttable. overings with horo- and hyperballs . . . F (3 , with smallest density ≈ . δ min u T (3 , . . T (3 , . . T (3 , . . Remark 4.2
To any parameter p (6 < p < , p ∈ R ) belongs a simplefrustum orthoscheme F (3 , p as well, therefore we can determine the densitiesof the corresponding hyp-hor coverings using the above computation method.The density function depends on free parameters u and p , and analysing thisfunction we get the minimal density in case p ≈ . with δ ≈ . .This hyp-hor covering is just locally optimal, because the corresponding tilingcan not be extended to H . A P edge In this case, A P passes through the intersection curve of the balls (seeFig. 5.a), so we can parametrize the intersection point of the curve and theedge: R ( r ) = (1 , x, y, rz ) , r ∈ [0 , Mikl´os Eper and Jen˝o Szirmai is very similar to the above two cases. We can determine the coordinates of Q i , T i , S i ( i = 1 ,
2) points, see that the horo- and hyperball cover the edges,so the hyp-hor covering is realized, and compute the density of the coveringby (4.7). The results for tiling T (4 , p (which provides the smallest density inthis case) are summarized in the next table.Type of tiling δ min r T (4 , . . T (4 , . . T (4 , . . Theorem 4.3 In H , among the hyp-hor coverings generated by simple trun-cated orthoschemes, the C (3 , covering configuration (see Subsection 4.3) pro-vides the lowest covering density ≈ . . The above density is smallerthan the so far known lowest covering density ≈ . in the -dimensionalhyperbolic space, which was described by L. Fejes T´oth and K. B¨or¨oczky. Theorem 4.4
In hyperbolic space H the function δ ( C (3 , p ) (6 < p < , p ∈ R ) attains its mimimum in case p ≈ . , with density δ ≈ . ,but the corresponding hyp-hor covering can not be extended to the entirety ofhyperbolic space H . We note here, that the discussion of the densest horoball packings in the n -dimensional hyperbolic space n ≥ References [1] Bezdek, K. Sphere Packings Revisited,
Eur. J. Combin. , (2006), 864–883.[2] Bolyai, J. The science of absolute space , Austin, Texas (1891). overings with horo- and hyperballs . . .
Acta Math. Acad. Sci. Hungar. , (1978), 243–261.[4] B¨or¨oczky, K. - Florian, A. ¨Uber die dichteste Kugelpackungim hyperbolischen Raum, Acta Math. Acad. Sci. Hungar. , (1964), 237–245.[5] Fejes T´oth, G. - Kuperberg, G. - Kuperberg, W. Highly Satu-rated Packings and Reduced Coverings, Monatsh. Math. , (1998), 127–145.[6] Fejes T´oth, L. Regular Figures, Macmillian (New York) , 1964.[7] Im Hof, H.-C. Napier cycles and hyperbolic Coxeter groups,
Bull. Soc. Math. Belgique , (1990) , 523–545.[8] Kellerhals, R. On the volume of hyperbolic polyhedra, Math.Ann. , (1989) , 541–569.[9] Kozma, R.T., Szirmai, J. Optimally dense packings for fullyasymptotic Coxeter tilings by horoballs of different types,
Monatsh. Math. , (2012), 27–47.[10] Kozma, R.T., Szirmai, J. New Lower Bound for the OptimalBall Packing Density of Hyperbolic 4-space, Discrete Comput.Geom. , (2015), 182-198, DOI: 10.1007/s00454-014-9634-1.[11] Kozma, R.T., Szirmai, J. New horoball packing density lowerbound in hyperbolic 5-space, Geometriae Dedicata , (2019), DOI:10.1007/s10711-019-00473-x.[12] Kozma, R.T., Szirmai, J. Horoball Packing Density LowerBounds in Higher Dimensional Hyperbolic n -space for 6 ≤ n ≤ Submitted manuscript , (2019), arXiv:1907.00595.[13] Moln´ar, E. The Projective Interpretation of the eight 3-dimensional homogeneous geometries,
Beitr. Algebra Geom., , (1997), 261–288.[14] Rogers, C.A. Packing and Covering, Cambridge Tracts in Math-ematics and Mathematical Physics 54, Cambridge UniversityPress , (1964).2
Mikl´os Eper and Jen˝o Szirmai [15] Szirmai, J. The optimal ball and horoball packings to theCoxeter honeycombs in the hyperbolic d -space, Beitr. AlgebraGeom., (2007), 35–47.[16] Szirmai, J. The densest geodesic ball packing by a type of Nillattices,
Beitr. Algebra Geom., (2007), 383–397.[17] Szirmai, J. Packings with horo- and hyperballs generated by sim-ple frustum orthoschemes,
Acta Math. Hungar. , (2017),365–382, DOI:10.1007/s10474-017-0728-0.[18] Szirmai, J. Density upper bound of congruent and non-congruent hyperball packings generated by truncated regularsimplex tilings, Rendiconti del Circolo Matematico di PalermoSeries 2 , (2018), 307–322, DOI: 10.1007/s12215-017-0316-8.[19] Szirmai, J. Decomposition method related to saturated hyper-ball packings, Ars Math. Contemp. , (2019), 349–358, DOI:10.26493/1855-3974.1485.0b1.[20] Szirmai, J. Upper bound of density for packing of congruent hy-perballs in hyperbolic 3 − space, Submitted manuscript , (2019).[21] Szirmai, J. Horoball packings to the totally asymptotic regularsimplex in the hyperbolic n -space, Aequat. Math. , (2013),471-482, DOI: 10.1007/s00010-012-0158-6.[22] Szirmai, J. Horoball packings and their densities by generalizedsimplicial density function in the hyperbolic space, Acta Math.Hung., (2012), 39–55, DOI: 10.1007/s10474-012-0205-8.[23] Szirmai, J. The p -gonal prism tilings and their optimal hyper-sphere packings in the hyperbolic 3-space, Acta Math. Hungar. (2006)
111 (1-2) , 65–76.[24] Szirmai, J. The regular prism tilings and their optimal hyperballpackings in the hyperbolic n -space, Publ. Math. Debrecen (2006)
69 (1-2) , 195–207. overings with horo- and hyperballs . . .
Kragujevac Journal of Math-ematics (2016) , 260-270.[26] Szirmai, J. The least dense hyperball covering to the regularprism tilings in the hyperbolic n -space, Ann. Mat. Pur. Appl. (2016) , 235-248, DOI: 10.1007/s10231-014-0460-0.[27] Szirmai, J. A candidate for the densest packing with equal ballsin Thurston geometries,
Beitr. Algebra Geom. (2014) , 441-452, DOI: 10.1007/s13366-013-0158-2.[28] Szirmai, J. Hyperball packings in hyperbolic 3-space,
Mat.Vesn. (2018), , 211-221.[29] Szirmai, J. Hyperball packings related to cube and octahedrontilings in hyperbolic space,
Contributions to Discrete Mathemat-ics (to appear) (2019), arXiv:1803.04948.[30] Szirmai, J. Congruent and non-congruent hyperball packingsrelated to doubly truncated Coxeter orthoschemes in hyper-bolic 3-space,
Acta Univ. Sapientiae Math. (to appear) (2019),arXiv:1811.03462.[31] Vermes, I. ¨Uber die Parkettierungsm¨oglichkeit des dreidimen-sionalen hyperbolischen Raumes durch kongruente Polyeder,
Studia Sci. Math. Hungar. (1972) , 267–278.[32] Vermes, I. Ausf¨ullungen der hyperbolischen Ebene durch kon-gruente Hyperzykelbereiche, Period. Math. Hungar. (1979) , 217–229.[33] Vermes, I. ¨Uber regul¨are ¨uberdeckungen der Bolyai-Lobatschewskischen Ebene durch kongruente Hyperzykel-bereiche,
Period. Math. Hungar. (1981) , 249–261. ∼∼