A Polytopal Generalization of Apollonian Packings and Descartes’ Theorem

A POLYTOPAL GENERALIZATION OF APOLLONIAN PACKINGS AND DESCARTES’ THEOREM

JORGE L. RAM´IREZ ALFONS´IN† AND IVAN´ RASSKIN

Abstract. In this paper we introduce and study the class of d-ball packings arising from edge-scribable polytopes. We are able to generalize Apollonian disk packings and the well-known Descartes’ theorem in different settings and in higher dimensions. After introducing the notion of Lorentzian curvature of a polytope we present an analogue of the Descartes’ theorem for all regular polytopes in any dimension. The latter yields to nice curvature relations which we use to construct integral Apollonian packings based on the Platonic solids. We show that there are integral Apollonian packings based on the tetrahedra, cube and dodecahedra containing the sequences of perfect squares. We also study the duality, unicity under Möbiustransformations as well as generalizations of the Apollonian groups. We show that these groups are hyperbolic Coxeter groups admitting an explicit matrix representation. An unexpected invariant, that we call Möbiusspectra, associated to Möbiusunique polytopes is also discussed.

1. Introduction In the last decades, Apollonian1 packings have drawn increasing attention to the scientiﬁc community. For instance, Apollonian sphere packings have been used to model the dynamics of granular systems and foams structures (e.g., [AM95; SHP06; Kwo+20]). However, in mathematics, many questions are still open about the behaviour of the curvatures of integral Apollonian packings [Fuc13].

Apollonian packings are constructed from an initial configuration of four pairwise tangent disks in R2, also known as a Descartes configuration. For each interstice made by three disks, there is a unique disk that can be inscribed. By adding the inscribed disk to each interstice we obtain a larger disk packing containing more interstices. We then repeat the process of inscribing new disks to each new interstice indefinitely as in Figure 1.

Figure 1. An Apollonian packing.

The name of Descartes conﬁguration comes from Ren´eDescartes who stated in a letter to the Princess Elizabeth of Bohemia in 1643 [Bos10] an algebraic relation equivalent to the following theorem.

2010 Mathematics Subject Classiﬁcation. 51M20, 52C26, 05B40. Key words and phrases. Apollonian packings, Curvatures, Apollonian group, Descartes’ theorem, Polytopes. † Partially supported by IEA-CNRS. 1The origin of the name goes back to a work due to Apollonius (circa 200 BCE). 1 Theorem 1.1 (Descartes). The curvatures of four pairwise tangent disks in the plane satisﬁes

2 2 2 2 2 (1) (κ1 + κ2 + κ3 + κ4) = 2(κ1 + κ2 + κ3 + κ4).

In 1936, Sir Frederick Soddy2 published in Nature a paper rediscovering Descartes’ theorem [Sod36]. In that paper, he noticed, by resolving the above equation for κ4, that√ if κ1, κ2, κ3 are the curvatures of three disks in a Descartes conﬁguration, such that κ1, κ2, κ3 and κ1κ2 + κ1κ3 + κ2κ3 are integers, then all the curvatures in the Apollonian packing containing the Descartes conﬁguration are integers. An Apollonian packing satisfying this property is said to be integral (see Figure 2).

81 49 25 64 49 81 25 36 49 36 25 81 49 64 25 49 81 81 49 25 64 49 81 25 36 49 36 25 81 49 64 25 49 81

76 16 84 84 16 76 76 16 84 84 16 76 9 52 52 9 9 52 52 9 81 81 81 81 28 72 72 28 28 72 72 28 57 57 4 57 57

64 64 64 64 33 33 33 33 88 88 12 88 88

73 73 73 73 24 40

60 84

84 60 40 24 73 73 73 73

88 88 12 88 88 33 33 33 33 64 64 64 64

57 157 57 1 57

28 72 72 28 28 72 72 28 81 81 81 81

52 52 52 52

76 76 76 76 9 84 84 94 9 84 84 9 25 16 25 36 36 25 16 25 25 16 25 36 36 25 16 25 81 49 64 49 81 49 81 49 64 49 81 81 49 64 49 81 49 81 49 64 49 81 -3 0 Figure 2. Two integral Apollonian packings with initial curvatures (−3, 5, 8) and (0, 0, 1).

Both Apollonian packings and Descartes’ theorem have been generalized in many ways. In [Sod36], Soddy extended Descartes’ theorem in dimension three. One year later, Thorold Gosset presented in [Gos37] a proof for a generalization of Descartes’ theorem in arbitrary dimensions.

Theorem 1.2 (Soddy-Gosset). The curvatures of d + 2 pairwise tangent d-balls in Rd satisﬁes

2 2 2 (2) (κ1 + ··· + κd+2) = d κ1 + ··· + κd+2 .

The Soddy-Gosset theorem was discovered and rediscovered many times [LMW01]. Another type of generalizations in the plane have been studied by modifying the initial set of Descartes configurations. By a similar process of construction, Guettler and Mallows defined in [GM08] octahedral Apollonian packings from disk packings whose tangency graph is the octahedral graph. In [Bol+18], the authors defined the Apollonian ring packings where Descartes configurations are replaced by disk packings where each disk is surrounded by n disks. For n = 3 and n = 4, they obtained the classical and octahedral Apollonian packings, respectively. In the case n = 5, they obtained an icosahedral Apollonian packing (see Figure 3).

Kontorovich and Nakamura studied disk packings arising from edge-scribed realizations of convex polyhedra [KN19]. They explored the close connection between the polyhedra structure behind some known generalizations of Apollonian packings. Eppstein, Kuperberg and Ziegler [EKZ03] used a similar approach to improve the upper bound of the average kissing number of 3-ball packings from edge-scribed 4-polytopes. Similarly, Chen studied in [Che16] the relation between Apollonian d-ball packings and stacked (d + 1)-polytopes.

2Chemistry Nobel Prize laureate in 1921 for discovering isotopes. 2 Figure 3. An octahedral (left) and an icosahedral (right) Apollonian packing.

On this direction, we deﬁne a class of d-ball packings that we call polytopal d-ball packings arising similarly from edge-scribed realizations of (d + 1)-polytopes. As we will see, the combinatorial structure information in these packings comes from a polytope, not only from the tangency graph, as usual when considering the circle packings.

These packings allow us to obtain attractive algebraic, geometric and arithmetic properties of Apol- lonian packings and polytopes. The main contributions of the paper are: (i) We propose new projections of regular polytopes that we call centered d-ball packings projections. (ii) An unexpected invariant that we call Möbiusspectra, associated to Möbiusunique polytopes, is introduced. (iii) We define some generalizations of the Apollonian groups for polytopal d-ball packings. When the corresponding polytope is one of of the Platonic solids, we show that these groups are hyperbolic Coxeter groups given by an explicit matrix representation. (iv) We prove that there are integral Apollonian packings based on the tetrahedron, cube and dodecahedron containing the sequences of perfect squares. (v) By introducing the notion of Lorentzian curvature of a polytope, we present a result analogue of Descartes’ theorem for regular polytopes in any dimension. (vi) We obtain sufficient conditions for an Apollonian packing based on the Platonic solids to be integral. 1.1. Organization of the paper. The paper is organized as follows: in Section 2 we discuss the background theories needed for the rest of the paper. First, we recall fundamental concepts about the Lorentzian and projective model of the space of d-balls and polytopes. Then, we present the construction of polytopal d-ball packings and we show the centered disk packing projections of the Platonic solids. After that, we discuss aspects concerning duality and unicity of polytopal d-ball packings. We end this section by introducing and discussing the Möbiusspectra and by putting forward a couple of questions concerning this invariant.

In Section 3, we give the generalization of the Apollonian groups for polytopal ball packings. Then, we present a matrix representation of these groups when the corresponding polytope is one of the Platonic solids. The matrix representations are used to prove (iv).

In Section 4, we present the result analogue to Descartes’ theorem for regular polytopes. We show that Soddy-Gosset’ generalization follows when we apply this result to the (d + 1)-simplices and we extend it to the family of (d + 1)-cubes, for every d ≥ 2.

Finally, in Section 5, we study the applications of the main theorem of Section 4 for polytopal disk packings coming from the Platonic solids. In particular, we derive the Guettler-Mallows’ generalization in the octahedral case. We use the results of this section to characterize the integrality of the Apollonian packings based on the Platonic solids. 3 2. Polytopal ball packings: background Let d ≥ 1 be an integer. We denote by Ed+1 and Ld+1,1 the Euclidean space of dimension d + 1 and the Lorentzian space of dimension d + 2, respectively. Recall that Ld+1,1 is the real vector space of dimension d + 2 endowed with the Lorentzian product d+1,1 hx, yi = x1y1 + ··· + xd+1yd+1 − xd+2yd+2, x, y ∈ L . d+1,1 A vector L is future-directed (resp. past-directed) if xd+2 > 0 (resp. xd+2 < 0). There is a well- known bijection between the space of d-balls of Rbd := Rd ∪{∞}, denoted by Balls(Rbd), and the one-sheet hyperboloid of the normalized space-like vectors of Ld+1,1 which we will denoted by S(Ld+1,1). We refer the reader to [Wil81] as well as [RR21, Section 2] and the reference therein for further details.

Let b ∈ Balls(Rbd) of center c and curvature κ ∈ R (resp. normal vector n pointing to the interior and d+1,1 signed distance to the origin δ ∈ R when b is a half-space). We denote by xb ∈ S(L ) the Lorentzian vector corresponding to b. The inversive coordinates of b deﬁned by Wilker [Wil81] correspond to the d+1,1 Cartesian coordinates of xb in the canonical basis of L . These coordinates can be given in terms of its center c and curvature κ and (resp. normal vector n and signed distance to the origin δ) as follows  κ (2c, kck2 − κ−2 − 1, kck2 − κ−2 + 1) if κ 6= 0,  2 (3) xb =  (n, δ, δ) if κ = 0

where k · k denotes the Euclidean norm of Rbd. The curvature of b can be deduced from its inversive coordinates by

(4) κ(b) = −hxN , xbi d+1,1 where xN = ed+1 + ed+2 and ei denotes the i-th canonical vector of L . The inversive product of two d-balls hb, b0i := hv(b), v(b0)i is a fundamental tool to encode arrangements of d-balls. Indeed, if the Lorentzian vectors of b and b0 are not both past-directed then  0  < −1 if b and b are disjoint  0  = −1 if b and b are externally tangent (5) hb, b0i = 0 if b and b0 are orthogonal  0  = 1 if b and b are internally tangent  > 1 if b and b0 are nested A d-ball arrangement B is a d-ball packing if any two d-balls are externally tangent or disjoint. For any d-ball arrangement B we denoted by Gram(B) the Gramian of the corresponding Lorentzian vectors in Ld+1,1. The Möbiusgroup Möb(Rbd) is the group generated by inversions on d-balls. It is isomorphic to the Orthochronous Lorentz group made by the linear maps of Ld+1,1 to itself preserving the Lorentz product and the time direction. The Orthochronous Lorentz group is isomorphic to the following group of matrices ↑ T T Od+1,1(R) = {M ∈ GLd+2(R) | M Qd+2M = Qd+2 and ed+2Med+2 > 0}. This connection gives a matrix representation of Möb(Rbd) by d ↑ Möb(Rb ) → O (R) (6) d+1,1 T sb 7→ Id+2 − 2xbxb Qd+2

where sb is the inversion on the d-ball b, Id+2 = diag(1,..., 1, 1) and Qd+2 = diag(1,..., 1, −1). The M¨obiusgroup sends d-balls to d-balls and preserves the inversive product [Wil81].

4 2.1. The projective model of the space of balls. The one-sheet hyperboloid S(Ld+1,1) can be regarded in the oriented projective space of Ld+1,1 d+1,1 d+1,1 (7) P+L = {x ∈ L \ 0}/∼ d+1,1 where x ∼ y if there is a real λ > 0 such that x = λy. P+L is equivalent to the Euclidean unit sphere Sd+1 ⊂ Ld+1,1 which, under the gnomonic projection, becomes the union of two aﬃne hyperplanes ↑ ↓ 0 d+1 Π = {xd+2 = 1} and Π = {xd+2 = −1} together with Π = {(x, 0) | x ∈ S }. The composition of the isomorphism Balls(Rbd) → S(Ld+1,1) with the projection d+1,1 ↑ 0 ↓ S(L ) → Π ∪ Π ∪ Π x if x = 0 x 7→ d+2 1 x otherwise |xd+2| gives an isomorphism between Balls(Rbd) and Π↑ ∪ Π0 ∪ Π↓. We call the latter the projective model of Balls(Rbd). By identifying Π↑ with Ed+1 we get a bijection between d-balls whose Lorentzian vector is future-directed and points of Ed+1 whose Euclidean norm is strictly greater than 1. Such a point of Ed+1 will be called an outer-sphere point. The reciprocal bijection between an outer-sphere point u ∈ Ed+1 and a d-ball b(u) ∈ Balls(Rbd) can be obtained geometrically by taking a light source which illuminates Sd from p. The spherical illuminated region is a spherical cap which becomes, under the stereographic projection from the north pole, the d-ball b(u). We say that u is the light source of b(u) and reciprocally we say that b(u) is the illuminated region of u.

Rcd b(u)

d+1 E u

Figure 4. A d-ball and its light source.

The inversive coordinates of a d-ball whose Lorentzian vector is future-directed can be computed from the coordinates of its light source by 1 (8) xb(u) = (u, 1). pkuk2 − 1 This equality implies that for any two d-balls whose Lorentzian vectors are future-directed, their inversive product is related to the Euclidean inner product of their corresponding light sources by the following equation 1 (9) hb(u), b(v)i = (u · v − 1) p(kuk2 − 1)(kvk2 − 1) where · denotes the Euclidean inner product. In the projective model of space of d-balls, the M¨obius group acts as the group of projective transformations preserving the unit sphere of Ed+1 [Zie04].

5 2.2. Polytopes. Let us quickly recall some basic polytope notions and deﬁnitions needed for the rest of the paper. We refer the reader to [Sch04] for further details. We consider here a (d + 1)-polytope as the convex hull of a ﬁnite collection of points in Ed+1. A 2-polytope and a 3-polytope are usually called polygon and polyhedron respectively. A polygon with n sides will be referred as a n-gon.

Let P be a (d+1)-polytope. For every 0 ≤ k ≤ d+1 we denote by Fk(P) the set of k-faces of P and by Sd+1 F(P) = {∅} ∪ k=0 Fk(P). A ﬂag of P is a chain of faces (f0, f1, . . . , fd, P) where for each k = 0, . . . , d, fk ⊂ fk+1. The elements of F0(P), F1(P), Fd−1(P) and Fd(P) are called vertices, edges, ridges and facets of P, respectively. The graph of P is the graph induced by the vertices and the edges of P. The face lattice (F(P), ⊂) encodes all the combinatorial information about P. Two polytopes P and P0 are combinatorially equivalent if there exists an isomorphism between their face lattices. If they are combinatorially equivalent, we say they have the same combinatorial type and P0 is said to be a realization of P.

The polar of a set X ⊂ Ed+1 is deﬁned as the set of points X∗ = {u ∈ Ed+1 | u · v ≤ 1 for all v ∈ X} where · denotes the Euclidean inner product. If P is a (d + 1)-polytope containing the origin in its interior then P∗ is also a (d + 1)-polytope containing the origin in its interior and holding the dual relation (P∗)∗ = P. There is a bijection between F(P) and F(P∗) which reverses incidences. Indeed, ∗ every facet f ∈ Fd(P) corresponds to a unique vertex vf ∈ F0(P ) and they are related by

(10) f = {u ∈ P | u · vf = 1}. The symmetric group of P is defined as the group of Euclidean isometries of Ed+1 preserving P. P is said to be regular if its symmetric group acts transitively on the set of flags P. In this case, the symmetric group of P can be generated from a flag (f0, . . . , fd, P) by considering the maximal simplex C (chamber) in the barycentric subdivision of P whose vertices are the barycenters of the faces in (f0, . . . , fd, P). Then C is the fundamental region of the symmetric group of P which is generated by the reflections r0, . . . , rd in the walls of C containing the barycenter of P, where rk is the reflection in the wall opposite to the vertex of C corresponding to the k-face in (f0, . . . , fd). We call the set {r0, . . . , rd} the fundamental generators of the symmetric group of P with respect to the flag (f0, . . . , fd, P). The symmetric group of P is a spherical Coxeter group with Coxeter graph where {p1, . . . , pd} is the Schläflisymbol of P. p1 pd For every 0 ≤ k ≤ d, a (d+1)-polytope P is said to be k-scribed if all its k-faces are tangent to the unit sphere of Ed+1.A k-scribed polytope is called inscribed, edge-scribed, ridge-scribed and circunscribed if k = 0, 1, d − 1, d respectively. A polytope is said to be k-scribable if it admits a realization which is k-scribed. Any regular (d + 1)-polytope is k-scribable for every 0 ≤ k ≤ d (by properly centering and rescaling by a suitable value). If P is a (d + 1)-polytope containing the origin in its interior then P is k-scribed if and only if P∗ is (d − k)-scribed (see [CP17] for more general results).

For any regular (d + 1)-polytope P, we denote by `P the half edge-length of a regular edge-scribed realization of P. The values of `P for every regular (d + 1)-polytope are presented in the following table adapted from [Cox73, p. 293].

d Regular (d + 1)-polytope P Schl¨aﬂisymbol `P π d = 1 p-gon {p} tan( p ) d+2 1/2 (d + 1)-simplex {3,..., 3} ( d ) | {z } d (d + 1)-cross polytope {3,..., 3, 4} 1 d ≥ 1 | {z } d−1 (d + 1)-cube {4, 3,..., 3} d−1/2 | {z } d−1 Icosahedron {3, 5} ϕ−1 d = 2 Dodecahedron {5, 3} ϕ−2 24-cell {3, 4, 3} 3−1/2 d = 3 600-cell {3, 3, 5} 5−1/4ϕ−3/2 120-cell {5, 3, 3} 3−1/2ϕ−3

The half edge-length of a regular edge-scribed realization of all the regular (d+1)- Table 1. √ 1+ 5 π polytopes for every d ≥ 1. The letter ϕ denotes the Golden Ratio 2 = 2 cos( 5 ). 6 In the case d = 2, we notice that the values of `P can be obtained from the Schl¨aﬂisymbol of P by v u 2 π 2 π usin ( q ) − cos ( p ) (11) `{p,q} = t 2 π . cos ( p )

2.3. The ball arrangement projection. Let P be an outer-sphere (d + 1)-polytope, i.e. with all its vertices outer-sphere. We deﬁne the ball arrangement projection of P, denoted by B(P), the collection of d-balls whose light sources are the vertices of P. Reciprocally, we say that P is the light-source polytope of the ball arrangement B(P). As Chen noticed in [Che16], if B(P) is a packing then then tangency graph of B(P) is a spanning subgraph of the graph of P.

Figure 5. (Left) An outer-sphere polyhedron with the spherical illuminated regions of its vertices; (right) the corresponding ball arrangement projection.

For any two vertices u and v joined by an edge e of P, the d-balls b(u) and b(v) are disjoint, externally tangent or they have intersecting interiors if and only if e cuts transversely, is tangent or avoids strictly Sd, respectively. Therefore, if P is an edge-scribed (d + 1)-polytope then B(P) is a d-ball packing. A d d-ball packing BP is deﬁned as polytopal if there exists µ ∈ M¨ob(Rb ) such that µ(BP ) = B(P). We notice that the tangency graph of BP and the graph of P are isomorphic. We say that P is the tangency polytope of BP . We observe that not all the d-ball packings are polytopal. An example of such packing is illustrated in Figure 6.

Figure 6. A non-polytopal disk packing (left), its light-source polyhedron (center) and same polyhedron with the unit sphere (right). The edge cutting the sphere does not correspond to any edge of the tangency graph of the disk packing.

We say that a ball arrangement projection of P has a k-face centered at the infinity if there is a k-face of P whose barycenter lies on the ray spanned by the North pole. For a regular (d + 1)-polytope P, we define the centered d-ball packing projections of P as the collection of the ball arrangement projections with a k-face centered at the infinity of a regular edge-scribed realization of P, up to Euclidean isometries. Table 2 illustrates the centered disk packing projections of the Platonic solids. 7 Edge-scribed realization Vertex centered at ∞ Edge centered at ∞ Face centered at ∞

n κ n κ √ n κ √ 1 1 p ` = 2 1 √ (1 − 3) 2 0 3 √ (1 − 1/3) {3,3} 2 √ 2 √ 3 √1 (1 + p1/3) 2 2 1 √1 (1 + 3) 2 2

n κ n κ n κ √ p `{3,4} = 1 1 1 − 2 2 0 3 1 − 2/3 p 4 1 √ 2 1 3 1 + 2/3 1 1 + 2 2 2

n κ √ √ n κ 1 2 − 3 n κ √ 2 0 √ ` = √1 3 2 − p1/3 √ 4 2 − 1 {4,3} 2 √ 4 2 √ 3 2 + p1/3 √ 4 2 + 1 √ √ 2 2 2 1 2 + 3

n κ n κ √ n κ 1 ϕ − ϕ + 2 2 0 3 ϕ − ϕ2p1/3 q ϕ+2 2 ϕ − 1 ` = 1 5 ϕ − 3 ϕ − ϕ−1p1/3 {3,5} ϕ 5 4 ϕ q ϕ+2 3 ϕ + ϕ−1p1/3 5 ϕ + 2 ϕ + 1 √ 5 2p 1 ϕ + ϕ + 2 2 2ϕ 3 ϕ + ϕ 1/3

n κ n κ √ n κ 2 2 0 q 1 ϕ − ϕ 3 2 2 1 2 p 4 ϕ − ϕ 5 ϕ − 5 (7 + 11ϕ) 3 ϕ − ϕ 5/3 2 2 ϕ − 1 2 q 1 ` = 1 2 p 5 ϕ − (3 − ϕ) {5,3} ϕ2 6 ϕ − ϕ 1/3 2 5 2 p 4 ϕ q 6 ϕ + ϕ 1/3 2 5 ϕ2 + 1 (3 − ϕ) p 2 ϕ + 1 5 3 ϕ2 + ϕ 5/3 q √ 4 ϕ2 + ϕ 2 1 2 5 ϕ + (7 + 11ϕ) 1 ϕ + ϕ 3 2 2ϕ2 5

Table 2. Centered disk packing projections of the Platonic solids together with a table showing the number of circles n with the same curvature κ.

8 2.4. Duality of polytopal ball packings. For d ≥ 2, let P be an edge-scribed (d + 1)-polytope. We can always send P by a projective transformation preserving the sphere into an edge-scribed (d + 1)- polytope P0 containing the origin in its interior. Therefore, for any polytopal d-ball packing BP there is a Möbiustransformation µ such that µ(BP ) = B(P0). We define the dual of BP as the ball arrangement ∗ −1 ∗ ∗ BP := µ (B(P0 )). The dual BP does not depend on the choice of P0. For any vertex v of P0 and ∗ for any vertex of P0 corresponding to a facet f of P0 containing v the corresponding d-balls bv ∈ BP ∗ and bf ∈ BP are orthogonal (this can be easily obtained by combining (5) with (9) and (10)). Since P ∗ ∗ is edge-scribed, P0 is edge-scribed and P0 is ridge-scribed. In the case d = 2, P0 is also edge-scribed ∗ ∗ and therefore BP and BP are both disk-packings. The union BP ∪ BP has been called a primal-dual circle representation of P [FR19]. Brightwell and Scheinerman proved in [BS93] the existence and the unicity up to Möbiustransformations of primal-dual circle representations for every polyhedron. It can be thought of as a stronger version of the Koebe-Andreev-Thurston Circle packing theorem [BS04].

Figure 7. (Left) An edge-scribed icosahedron I and its polar in blue; (center) the spherical illuminated regions of I and its polar; (right) a primal-dual circle representation of I.

2.5. Möbiusunicity. Two d-ball packings are Möbiusequivalent if one can be sent to the other by a Möbiustransformation. If such a Möbiustransformation is a Euclidean isometry then we say that the two packings are Euclidean congruent. We recall that a d-ball packing B is said to be standard if it i contains the half-spaces bi = {xd ≥ 1} and bj = {xd ≤ −1}, denoted by [B]j. For any tangent d-balls bi and bj of a d-ball packing B there is always a Möbiustransformation µ mapping B into a standard form i [B]j (see [RR21]). We call such a Möbiustransformation a standard transformation. Lemma 2.1. [RR21] Let B and B0 be two d-ball packings with same tangency graph G and let ij be an 0 i 0 i edge of G. Then B and B are Möbiusequivalent if and only if [B]j and [B ]j are Euclidean congruent. We present here another approach to detect Möbiusequivalence. We say that a d-ball arrangement has maximal rank if the rank of its Gramian is equal to d + 2 (it cannot be larger). In particular, polytopal d-ball packings have maximal rank. Proposition 2.1. Let B and B0 two d-ball packings with maximal rank. Then B is Möbiusequivalent to B0 if and only if Gram(B) = Gram(B0). Proof. The necessity is trivial since Möbius transformations preserve the inversive product. Now let 0 0 us suppose that Gram(B) = Gram(B ) is a matrix of rank d + 2. Let V = (v1, . . . , vn) and V = 0 0 0 (v1, . . . , vn) be the Lorentzian vectors of the d-balls of B and B respectively. Without loss of gen- erality we can consider that the vectors of V and V 0 are future-directed. By the rank condition, V d+1,1 0 contains a basis ∆ = {vi1 , . . . , vid+2 } of L . Since Gram(B) = Gram(B ) the corresponding collection ∆0 = {v0 , . . . , v0 } ⊂ B0 is also a basis. Let P and P0 be the light-source polytopes of the d-balls i1 id+2 corresponding to ∆ and ∆0 respectively. Since these are basis the polytopes P and P0 are (d + 1)- simplices of Ed+1 intersecting the unit sphere Sd. If P does not contain the origin then we can find a Möbiustransformation µe0 such that the corresponding projective transformation maps P into another d+1 (d + 1)-simplex P0 containing the origin of E . Such a Möbiustransformation can be easily found by d combining translations and scalings of Rb which are also Möbiustransformations. Let µ0 be the element in the Orthochronous Lorentz group corresponding to µe0 and let V0 := µ0(V ), ∆0 := µ0(∆). Let M0 and M1 be the (d + 2)-square matrices formed by the column-matrices of the coordinates of the vectors of 9 0 ∆0 and ∆ respectively with respect to the canonical basis. The matrices M0 and M1 are non-singular and therefore 0 T T Gram(∆0) = Gram(∆ ) ⇔ M0 Qd+2M0 = M1 Qd+2M1 T −1 T ⇔ Qd+2(M1 ) M0 Qd+2M0 = M1 where Qd+2 = diag(1,..., 1, −1) is the matrix of the Lorentz product. The matrix T −1 T A := Qd+2(M1 ) M0 Qd+2 d+1,1 0 defines a linear map of R to itself which maps ∆0 to ∆ . Moreover, one can check that T A Qd+2A = Qd+2. 0 ↑ T Since the origin belongs to P0 we have that Aed+2 ⊂ P ⊂ Π . Thus, ed+2Aed+2 > 0 and therefore ↑ 0 A ∈ Od+1,1(R). The coordinates for the remaining vectors of V0 and V with respect to the basis ∆0 and ∆0 respectively are completely determined by the entries of Gram(B). The equality of the Gramians 0 and linearity implies that the Lorentz transformation µ given by A maps V0 to V . Therefore, µµ0 0 corresponds to a Möbiustransformation which maps B to B . Remark 1. Proposition 2.1 does not hold for ball arrangements in general. Indeed, for any ball packing B whose Gramian is maximal, the ball arrangement −B, made by taking the d-balls corresponding the vectors −xb for every b ∈ B, is not a packing, and therefore B and −B are not Möbiusequivalent. However, Gram(B) = Gram(−B). We say that an edge-scribable (d + 1)-polytope is Möbiusunique if the ball arrangement projections of all its edge-scribed realizations are Möbiusequivalent. Lemma 2.1 and Proposition 2.1 are useful to study Möbiusunicity. Corollary 2.1. Let P and P0 be two edge-scribed realizations of a regular (d + 1)-polytope which is 0 0 0 Möbiusunique. If P and P share a facet f then P = P or P = sf (P) where sf is the projective transformation corresponding to the inversion on the sphere f ∩ Sd. 0 ∗ Proof. Let BP and BP0 the ball arrangement projections of P and P , respectively. Let bf ∈ BP and 0 ∗ bf ∈ BP0 the d-balls corresponding to the facet f. Let u and v two vertices of f connected by an edge. 0 After applying a standard transformation with respect to the edge uv to BP and BP we have that bf and 0 bf becomes two half-spaces whose boundary is a hyperplane H orthogonal to {xd ≥ 1} and {xd ≤ −1}, u respectively. All the d-balls of [BP ]v corresponding to vertices of P which are not contained in f lie on 0 u one side of H. Similarly for [BP ]v . If they lie on the same side, since the combinatorial type of P and 0 u 0 u P is regular and Möbiusunique, Lemma 2.1 implies that [BP ]v = [BP ]v . If they lie on different sides, 0 u 0 u same lemma and the regularity implies that [BP ]v = s([BP ]v ), where s is the reflection on H and the result follows. We notice that the Gramian of a ball arrangement projection of an edge-scribed (d + 1)-simplex is the (d + 2)-square matrix with 1 at the diagonal entries and −1 at the non-diagonal entries. Therefore, Proposition 2.1 implies the following. Corollary 2.2. For every d ≥ 1, the (d + 1)-simplex is Möbiusunique. It is clear that all polygons are edge-scribable but not all are Möbiusunique. Corollary 2.3. The only polygon which is Möbiusunique is the triangle.

n Proof. Let P be an edge-scribed n-gon with n > 3 and let [BP ]1 = (b1, b2, . . . , bn) be the standard 1-ball packing obtained by applying a standard transformation to the ball arrangement projection of P. Let −1 < x < 1 be the contact point of the two closed intervals b2 and b3. By replacing b2 and b3 by 0 0 two closed intervals b2 and b3 obtained by moving x a suitable distance we can construct another 1-ball 0 n n n 0 n packing [BP ]1 which is not Euclidean congruent to [BP ]1 . By Lemma 2.1, [BP ]1 and [BP ]1 are not Möbiusequivalent and therefore P is not Möbius unique. Möbiusunicity in dimension 3 is much less restricted. Indeed, the Brightwell and Scheinerman’ theorem [BS93] stating the existence and unicity of primal-dual circle representations can be restated as follows. Theorem 2.1 (Brightwell and Scheinerman). Every polyhedron is edge-scribable and Möbiusunique. 10 In dimension 4, there are combinatorial polytopes which are not edge-scribable (e.g. [Sch87; EKZ03; Che16]). However, as far as we are aware, it is not known if there are edge-scribable 4-polytopes which are not Möbiusunique. Conjecture 1. For every d ≥ 2 all edge-scribable (d + 1)-polytopes are Möbiusunique. As shown above Conjecture 1 holds for the (d + 1)-simplex. It turns out that it also holds for the (d + 1)-cube with d ≥ 2. In order to prove this we need the following. d+1 Proposition 2.2. Let d ≥ 2 and let BCd+1 be a polytopal d-ball packing where C is an edge-scribed realization of the (d + 1)-cube. Then, for every two vertices u, v of Cd+1, we have

hbu, bvi = 1 − 2dG(u, v)(12) d+1 where dG(u, v) is the distance between u and v in the graph of C . Proof. We proceed by induction on d ≥ 2.

Let C3 be the regular edge-scribed cube where the coordinates of the vertices are given by the 8 signed permutations of √1 (±1, ±1, ±1). By using Equations (8) and (9), we obtain that the proposition holds 2 for C3. Since, by Theorem 2.1, the 3-cube is M¨obius unique, then the proposition holds for any other edge-scribed realization of the 3-cube.

Let us now suppose that the proposition holds for any edge-scribed realization of the (d + 1)-cube d+2 d+2 for some d ≥ 2. Let BCd+2 be a polytopal (d + 2)-ball packing where C ⊂ E is an edge-scribed d+2 realization of the (d + 2)-cube. Let u, v be two vertices of C . We notice that 0 ≤ dG(u, v) ≤ d + 2. d+1 d+2 Letu ¯ the unique vertex in C such that dG(u, u¯) = d + 2. If v 6= u then there is a facet f of C containing u and v. Since Cd+2 is edge-scribed then f intersects Sd+1 in a d-sphere S. We may find a Möbiustransformation µ ∈ Möb(Sd+1) which sending S to the equator of Sd+1. After identifying d+2 d+1 the hyperplane {xd+2 = 0} ⊂ E with E we obtain that µ(f) becomes an edge-scribed realization d+1 d+1 Ce of the (d + 1)-cube. The identification {xd+2 = 0}' E preserves the inversive product of the illuminated regions of the outer-sphere points lying in {xd+2 = 0}. Moreover, the distance between u and v in the graph of Cd+2 is equal to the distance in the graph of Ced+1. By the invariance of the inversive product under Möbiustransformations and the induction hypothesis we have that (12) holds for any d+2 two (d + 2)-balls of BCd+2 corresponding to two vertices of C at distance strictly less than d + 2. It remains thus to show that (12) holds for bu and bu¯.

d+2 Let v1, . . . , vd+2 be the neighbours of u in the graph of C . We consider the (d+1)-ball arrangement

d+2 ∆ = (bu, bv1 , . . . , bvd+2 , bu¯) ⊂ BC . By the induction hypothesis we have  1 −1 · · · −1 λ   −1 1 − 2d    (13) Gram(∆) =  . .   . Ad+2 .     −1 1 − 2d  λ 1 − 2d ··· 1 − 2d 1

where An denotes the n-square matrix with 1’s in the diagonal entries and −3 everywhere else and λ := hbu, bu¯i. It can be computed that det(Gram(∆)) = −4d+1(d(2d − 3λ + 3) − 2λ + 2)(2d + λ + 3). Since ∆ corresponds to a collection of d + 4 vectors of Ld+2,1 the Gramian of ∆ must be singular. Therefore, by the above equality we have that 2d2 det(Gram(∆)) = 0 ⇔ λ = −3 − 2d or λ = + 1 > 0 for every d ≥ 2. 3d + 2

Since bu and bu¯ are disjoint then, by (5), λ < −1 and therefore we have that

hbu, bu¯i = −3 − 2d = 1 − 2(d + 2) = 1 − 2dG(u, u¯). Corollary 2.4. The (d + 1)-cube is M¨obiusunique for every d ≥ 2.

Proof. The result follows by combining Propositions 2.1 and 2.2. 11 2.6. The Möbius spectra. Spectral techniques, based on the eigenvalues and the eigenvectors of the adjacency or the Laplace matrices of graphs are strong and successful tools to investigate different graph’s properties. On this spirit, we define the Möbiusspectra of an edge-scribable Möbiusunique (d + 1)- polytope P as the multiset of the eigenvalues of the Gramian of B(P0) where P0 is any edge-scribed realization of P.

Eigenvalues Multiplicities Tetrahedron −2, 2 1, 3 Octahedron −6, 0, 4 1, 2, 3 Cube −16, 0, 8 1, 4, 3 Icosahedron −12ϕ2, 0, 4ϕ + 8 1, 8, 3 Dodecahedron −20ϕ4, 0, 20ϕ2 1, 16, 3

Table 3. The Möbiusspectra of the Platonic solids. The well-known theorem of Steinitz states that the graph of a polyhedron is a 3-connected simple planar graph. Such graphs are called polyhedral graphs. Since all the polyhedra are edge-scribable and Möbiusunique, the Möbiusspectra can be defined for any polyhedral graph. Möbiusunicity implies that the Möbiusspectra does not depend on the edge-scribed realization. One may wonder if Möbius spectra defines a complete invariant for edge-scribable Möbiusunique polytopes and, in particular, for polyhedral graphs. Question 1. Are there two combinatorially different edge-scribable Möbiusunique (d + 1)-polytopes with same Möbiusspectra? In particular, are there two non isomorphic polyhedral graphs with same Möbius spectra?

3. The Apollonian groups of polytopal ball packings In this section we introduce a generalization of the Apollonian group introduced by Graham et al. in [Gra+05; Gra+06a; Gra+06b] and Lagarias et al. in [LMW01] for any polytopal d-ball packing.

d For d ≥ 2, let BP a polytopal d-ball packing. We consider the following subgroups of M¨ob(Rb ): • The symmetric group of BP is the group d Sym(BP ) := hµ ∈ M¨ob(Rb ) | µ(BP ) = BP i. ∗ We notice that Sym(BP ) = Sym(BP ). • The Apollonian group of BP is the group ∗ A(BP ) := hS(BP )i ∗ ∗ where S(BP ) denotes the set of inversions on the d-balls of BP . • The symmetrized Apollonian group of BP is the group

SA(BP ) := hSym(BP ) ∪ A(BP )i ∗ It can be checked that for every r ∈ Sym(BP ) and every b ∈ BP we have

rsbr = sr(b) ∈ A(BP )(14)

where sb denotes the inversion of b. And consequentely, we have that SA(BP ) = A(BP )oSym(BP ). • The super symmetrized Apollonian group of BP is the group ∗ SSA(BP ) := hA(BP ) ∪ Sym(BP ) ∪ A(BP )i. Remark 2. In [Gra+05; Gra+06a; Gra+06b; KN19] the authors defined the dual Apollonian group and ∗ the super Apollonian group which, with our notation, correspond to the groups A(BP ) and hA(BP ) ∪ ∗ A(BP )i, respectively. For the purposes of this paper these groups are not needed. Neverthless, both groups are subgroups of SSA(BP ). 0 For any other polytopal d-ball packing BP Möbiusequivalent to BP the four groups defined above 0 for BP are congruent to their analogues in BP . Therefore, if P is Möbiusunique, the four groups can be defined for P independently of the choice of BP . We define the Platonic Apollonian groups as the Apollonian groups of the Platonic solids.

12 Since inversions of M¨ob(Rbd) can be seen as hyperbolic reﬂections of the (d+1)-dimensional hyperbolic space, Apollonian groups are hyperbolic Coxeter groups. The Apollonian cluster of BP , denoted by Ω(BP ), is the union of the orbits of the action of A(BP ) on BP . If Ω(BP ) is a packing we shall call it the Apollonian packing of BP . For d = 2, Ω(BP ) is always a packing for any polytopal disk packing. This is not true in higher dimensions. For instance, if P is an edge-scribed (d + 1)-simplex with d ≥ 4 then Ω(BP ) is not packing (see [Gra+06b]).

The four groups presented above give different actions on the Apollonian cluster. For instance, each disk of a polytopal disk packing BP has a different orbit under the action of A(BP ). This can be used to present colorings of the disks in the Apollonian packing of BP . Indeed, if BP admits a proper coloring then is not hard to see that the coloring can be extended to Ω(BP ) by the action of the A(BP ). We present in Figure 8 three Apollonian packings with a minimal coloration. In the first case each orbit has a different color.

Figure 8. A tetrahedral (left), octahedral (center) and cubical (right) Apollonian packing with a minimal coloration. An interesting issue about these colorings is that the white disks, including the disk with negative curvature, are induced by optical illusion from the other colored disks.

∗ If P is regular then Sym(BP ) acts transitively on the d-balls of BP , BP and their corresponding Apollonian clusters. Let Σ(BP ) be a set of generators of Sym(BP ). By equation (14), we have that for ∗ any sf ∈ S(BP ) and any sv ∈ S(BP ) the group SA(BP ) is generated by Σ(BP ) ∪ {sf } and SSA(BP ) is generated by Σ(BP ) ∪ {sv, sf }.

We observe that the transitivity of the action of the symmetrized Apollonian group for regular polytopes could be useful to ﬁnd special sequences of curvatures in the Apollonian clusters. We refer the reader to [Fuc13] for an excellent survey on problems related to ﬁnding sequences of prime curvatures.

Theorem 3.1. There is a tetrahedral, cubical and dodecahedral Apollonian packing where the set of curvatures of the disks contains all the perfect squares.

1 Proof. For every p = 3, 4, 5, let [BP ]2 be the standard polytopal disk packing where P is the regular polyhedron of Schläflisymbol {p, 3}. Let (v, e, f) be the flag of P where v is the vertex corresponding 0 to the half-space bv = {y ≤ −1}, e is the edge of P with ends v and v where bv0 = {y ≤ −1}, and f is a face of P containing e. Let us apply a translation and a reflection if needed so bf becomes the half-space 2 π {x ≥ 0}. We then rescale by a factor of λ := 4 cos ( p ). Let Bp,3, for p = 3, 4, 5, be the transformed 1 disk packing obtained from [BP ]2 (see Figure 9). The rescaling factor has been chosen to obtain that the disks with minimal non-zero curvature of Bp,3 have curvature equals to one.

13 rf sf bv0 rf sf bv0 rf sf bv0

rv rv rv

re bv bv bv re

Figure 9. The disk packings Bp,3 for p = 3 (left), p = 4 (center) and p = 5 (right). In dashed the generators of Sym(Bp,3).

Since P is regular then SA(Bp,3) is generated by {rv, re, rf , sf } where {rv, re, rf } are the elements in Sym(Bp,3) corresponding to the fundamental generators of the symmetric group of P with respect to the ﬂag (v, e, f) and sf is the inversion on the bf . Therefore, for p = 3, 4, 5, we have the following:

• bv is the half-space {y ≤ −λ} and bv0 is the half-space {y ≥ λ}, • rv is the reflection on the line {y = 0}, √ • re is the inversion on the circle centered√ at (0, −λ) and radius 2 λ, • rf is the reflection on the line {x = − λ}, • sf is the reflection on the line {x = 0}. For every p = 3, 4, 5 and for every integer n ≥ 0 we define the element n µp,n := re(rf sf ) re ∈ SA(Bp,3) and the disk bn := µp,n(bv0 ) ∈ Ω(Bp,3).

Let us compute the curvature of bn. The inversive coordinates of bv and the matrices representing re, rf and sf obtained by (6) are given by

  0 1 0 0 0 λ 1 2 1 2 0 1 − 2 4 λ − 4λ − 1 − 4 λ − 4λ + 1 1  2     2 2 2  bv 7→ Bλ =   re 7→ Eλ =  1 2 (λ −4λ−1) 1−(λ−4) λ , λ 0 4 λ − 4λ − 1 1 − 8λ − 8λ  2 2  λ 1 2 1−(λ−4)2λ2 (λ −4λ+1) 0 4 λ − 4λ + 1 8λ 1 + 8λ √ √  −1 0 2 λ −2 λ −1 0 0 0 0 1 0 0 0 1 0 0 rf 7→ Fλ =  √ , sf 7→ S =  , 2√λ 0 1 − 2λ 2λ   0 0 1 0 2 λ 0 −2λ 2λ + 1 0 0 0 1 By induction on n we obtain √ √  1 0 2 λn −2 λn  n  √0 1 0 0  (FλS) =  2 2  −2√λn 0 1 − 2λn 2λn  −2 λn 0 −2λn2 1 + 2λn2 and therefore, by (4), we have

κ(bn) = κ(µp,n(bv)) n = KEλ(FλS) EλBλ = n2 where K = diag(0, 0, −1, 1). The right image of Figures 2, 12 and 15 illustrates the corresponding tetrahedral, cubical and dodecahedral Apollonian packings containing the sequence of perfect squares.

14 By deﬁnition, we have that for any polytopal d-ball packing A(BP ) < SA(BP ) < SSA(BP ). Therefore we can ﬁnd representations of A(BP ) as a subgroup of SSA(BP ). Even if SSA(BP ) is a larger group than A(BP ), it may be generated with a smaller set of generators.

Theorem 3.2. Let BP a polytopal disk packing where P is a Platonic Solid with Schl¨aﬂisymbol {p, q}. The group SSA(B ) is the hyperbolic Coxeter Group with Coxeter graph . Moreover, P ∞ p q ∞

SSA(BP ) is generated by the following ﬁve matrices:

−1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0  0 1 0 0 0 −1 −2 2 0 −1 0 0 0 1/2 1 −1/2 S =   S∗ =  , V =  , E =    0 0 1 0 0 −2 −1 2 0 0 1 0 0 1 −1 1  0 0 0 1 0 −2 −2 3 0 0 0 1 0 1/2 −1 3/2

 π π  −1 0 4 cos( q ) −4 cos( q )  0 1 0 0  Fq =  π 2 π 2 π . 4 cos( q ) 0 1 − 8 cos ( q ) 8 cos ( q )  π 2 π 2 π 4 cos( q ) 0 −8 cos ( q ) 1 + 8 cos ( q ) 1 Proof. We follow a similar strategy as in the proof of Theorem 3.1. For every q = 3, 4, 5, let [BP ]2 be the standard polytopal disk packing where P is the regular polyhedron of Schläflisymbol {3, q}. We choose the flag (v, e, f) satisfying the same conditions as in the proof of Theorem 3.1. Let us now apply a translation and a reflection if needed so bf becomes the half-space {x ≥ 0}. Let B3,q be the disk packing obtained after the transformations. Since P is regular, SSA(B3,q) is generated by {sv, rv, re, rf , sf } where {rv, re, rf } are the elements in Sym(B3,q) corresponding to the fundamental generators of the symmetric ∗ group of P with respect to the flag (v, e, f), sf is the inversion on the disk in B3,q corresponding to f and sv is the inversion on the disk in B3,q corresponding to v. For each q = 3, 4, 5 we have that

• sv is the reflection on the line {y = 1}, • rv is the reflection on the line {y = 0}, • re is the inversion on the circle centered at (1, 1) and radius 2, π • rf is the reflection on the line {x = −2 cos( q )}, • sf is the reflection on the line {x = 0}.

re re re

sv sv sv

rv rv rv

rf sf rf sf rf sf

Figure 10. The disk packings B3,q with their dual (in blue) for q = 3 (left), q = 4 (center) and q = 5 (right). In dashed the generators of the Symmetric Group of B3,q. The central disk is a unit disk centered at the origin.

By using the inversive coordinates and (6) we obtain a faithful linear representation of SSA(B3,q) as ↑ a discrete subgroup of O3,1(R) where the generators are mapped to: ∗ sv 7→ S rv 7→ V re 7→ E rf 7→ Fq sf 7→ S The relations of finite order in the Coxeter graph can be checked by straightforward computations ∗ n n on the matrices. For every n ∈ N the last entry in the diagonal of (S V) and (FqS) is, respectively, 2 2 π 2 1 + 2n and 1 + 8 cos ( q )n and therefore every product has infinite order for every q = 3, 4, 5. Since ∗ SSA(B3,q) = SSA(B3,q) then the results also hold for the cube and the dodecahedron. Indeed, the ∗ generators of SSA(B3,q) are given by swapping rv with rf and sv with sf . 15 The previous theorem allows to exhibit a matrix representation of the Apollonian groups of the ↑ √ Platonic solids in O3,1(K) where K is one of these 3 number fields: Z, Z[ 2] and Z[ϕ]. We illustrates this for the tetrahedron, octahedron and cube. ↑ Corollary 3.1. The tetrahedral Apollonian group is isomorphic to hT1, T2, T3, T4i < O3,1(Z) where

−1 0 0 0 −1 0 4 −4 0 1 0 0 0 1 0 0 T =   = ST =   = F T F 4  0 0 1 0 3  4 0 −7 8  3 4 3 0 0 0 1 4 0 −8 9 −1 2 0 −2 −1 −2 0 −2 2 −1 0 2 −2 −1 0 −2 T =   = ET ET =   = VT V 2  0 0 1 0  3 1  0 0 1 0  2 2 −2 0 3 2 2 0 3 Corollary 3.2. The octahedral Apollonian group is isomorphic to ↑ √ hC123, C123, C123, C123, C123, C123, C123, C123i < O3,1(Z[ 2]) where √ √ −1 0 0 0  −1 0 4 2 −4 2  0 1 0 0  √0 1 0 0  C123 = = SC123 = = F4C123F4  0 0 1 0 4√2 0 −15 16  0 0 0 1 4 2 0 −16 17 √ √ √ √     −√1 2 2 0 −2 2 −√1 −2 2 0 −2 2 2 2 −3 0 4  −2 2 −3 0 −4  C123 = = EC123EC123 = = VC123V  √0 0 1 0   √0 0 1 0  2 2 −4 0 5 2 2 4 0 5 √ √ √ √ √ √     −√17 6 2 12 2 −18 2 −17√ −6 2 12 2 −18 2 6 2 −3 −8 12 −6 2 −3 8 −12 C =  √  = F C F C =  √  = VC V 123   4 123 4 123   123 12√2 −8 −15 24 12√2 8 −15 24 18 2 −12 −24 37 18 2 12 −24 37 √ √ √  −17 0 0 −12 2  −49 0 20 2 −40 2  0 1 0 0   0√ 1 0 0  C123 = = EC123EC123 = = F4C123F4  0√ 0 1 0  20√2 0 −15 32  12 2 0 0 17 40 2 0 −32 65

↑ √ Corollary 3.3. The cubical Apollonian group is isomorphic to hO1, O2, O3, O3, O2, O1i < O3,1(Z[ 2]) where

1 0 0 0 1 0 0 0  0 −1 −2 2 0 −1 2 −2 O =   = S∗ O =   = VO V 1 0 −2 −1 2 2 0 2 −1 2  1 0 −2 −2 3 0 2 −2 3 √ √ 1 0 0 0  −15 0 12 2 −16 2 0 1 0 0  0√ 1 0 0  O3 = = EO2EO3 = = F4O3F4 0 0 −1 0 12√2 0 −17 24  0 0 0 1 16 2 0 −24 33 √ √ √ √ √ √     −√15 4 2 4 2 −12 2 −15√ −4 2 4 2 −12 2 4 2 −1 −2 6 −4 2 −1 2 −6 O =  √  = EO EO =  √  = VO V 2   3 1   2 4 √2 −2 −1 6 4 √2 2 −1 6 12 2 −6 −6 19 12 2 6 −6 19

16 4. A polytopal Descartes theorem for regular polytopes. We shall deﬁne the notion of Lorentzian curvature of a polytope needed for the proof of our main contribution of this section. We ﬁrst extend the notion of curvature to any vector x ∈ Ld+1,1 (not only for normalized space-like vectors as in (4)) by

(15) κ(x) = −hxN , xi

d+1,1 d+1 where xN = ed+1 + ed+2 and ei denotes the i-th canonical vector of L . Let P ⊂ E be an d+1,1 outer-sphere polytope. We deﬁne the Lorentzian barycenter of P the vector xP ∈ L given by 1 X (16) xP := xb(v) |F0(P)| v∈F0(P)

where b(v) is the illuminated region of v. We define the Lorentzian curvature of P as κP := κ(xP ). By linearity we have that 1 X κP = κ(b(v))(17) |F0(P)| v∈F0(P) We notice that every face f of P is also an outer-sphere polytope. We can thus define the Lorentzian barycenter and Lorentzian curvature of f analogously as above. Theorem 4.1. For d ≥ 1, let P be a regular edge-scribed realization of a regular (d + 1)-polytope with Schläflisymbol {p1, . . . , pd}. The Lorentzian curvatures of any flag (f0, f1, . . . , fd, fd+1 = P) satisfies

d 2 X (κfi − κfi+1 ) (18) κ2 = L (d + 1) P P L (i + 1) − L (i) i=0 P P where  −1 i = 0  (19) LP (i) := 0 i = 1  `−2 if 2 ≤ i ≤ d + 1 {p1,...,pi−1}

where `{p1,...,pi−1} is the half edge-length of a regular edge-scribed realization of fi. The following lemmas are needed for the proof of Theorem 4.1.

d+1,1 Lemma 4.1. Let ∆ = (x1,..., xd+2) be a collection of d + 2 vectors in L . Then ∆ is a basis if and only if Gram(∆) is non-singular. Moreover, the vector (κ1, . . . , κd+2) where κi := κ(xi) satisﬁes   κ1 −1  .  (κ1, . . . , κd+2) Gram(∆)  .  = 0(20) κd+2 Proof. Let M be the matrix whose columns are the coodinates of the vectors of ∆ with respect to d+1,1 T the canonical basis of L , N = (0,..., 0, 1, 1) and Qd+2 = diag(1,..., 1, −1). We have that T (κ1, . . . , κd+2) = (κ(x1), . . . , κ(xd+2)) = −M Qd+2N. Therefore,

  κ1 −1  .  T −1 T −1 T (κ1, . . . , κd+2) Gram(∆)  .  = (−N Qd+2M)(M Qd+2(M ) )(−M Qd+2N) κd+2 T = N Qd+2N = 0

Remark 3. As Boyd noticed in [Boy74], the Soddy-Gosset theorem (Theorem 1.2) can be obtained directly by applying the above lemma to the Lorentzian vectors of a packing of d + 2 pairwise tangent d-balls. 17 Let a1, . . . , an be a sequence of real numbers. We deﬁne the corner matrix C(a1, . . . , an) as the matrix with the following shape   a1 a2 a3 ··· an−1 an  a2 a2 a3 ··· an−1 an    a3 a3 a3 ··· an−1 an (21) C(a , . . . , a ) :=   1 n  ......   ......    an−1 an−1 an−1 ··· an−1 an an an an ··· an an

If ai 6= ai+1 for each i = 1, . . . , n − 1 then C(a1, . . . , an) is non-singular. It can be easily checked that its inverse is equal to  1 −1 0 ··· 0 0 0  a1−a2 a1−a2    .   −1 1 1 −1 ..   a −a a −a + a −a a −a 0 0 0   1 2 1 2 2 3 2 3     . .   −1 1 1 .. ..   0 a −a a −a + a −a 0 0   2 3 2 3 3 4       ......   ......       . .   0 0 .. .. 1 + 1 −1 0   an−3−an−2 an−2−an−1 an−2−an−1       .   . −1 1 1 −1   0 0 0 . +   an−2−an−1 an−2−an−1 an−1−an an−1−an    0 0 0 ··· 0 −1 1 + 1 an−1−an an−1−an an

Moreover, if Xn = (X1,...,Xn) then it can also be checked that n−1 2 2 X (Xi − Xi+1) X (22) X C(a , . . . , a )−1XT = + n n 1 n n a − a a i=1 i i+1 n Lemma 4.2. Let P be a regular edge-scribed (d + 1)-polytope with d ≥ 1. The Lorentzian vectors of 1 the vertices of P are contained in the hyperplane {xd+2 = } where `P is the half edge-length of P. `P Furthermore, for every f ∈ F(P) we have hxf − xP , xP i = 0. Proof. Since P is regular and edge-scribed the barycenter of P is at the origin of Ed+1. Therefore all its p 2 vertices have the same Euclidean norm and the length of each half edge of P is equal to `P = kvk − 1. By Equation (8), the Lorentzian vectors of the vertices of P are contained in the hyperplane 1 1 d+1,1 H = {xd+2 = √ = } ⊂ kvk2−1 `P L implying that for every face f of P the Lorentzian vector xf is also in H. Since the barycenter of P is d+1 1 the origin of then xP = ed+2 implying the second part of the Lemma. E `P

Proof of Theorem 4.1. Let (f0, f1, . . . , fd, fd+1 = P) be a ﬂag of P and let ∆ = (xf0 ,..., xfd , xP ) ⊂ Ld+1,1. We shall show that

(23) for any d ≥ 1, Gram(∆) = C(−LP (0), −LP (1),..., −LP (d + 1)). The desired equality will then follows from this as explained below. Since Gramians and corner matrices are symmetric then equation (23) is equivalent to

(24) for any d ≥ 1, Gram(∆)i,j = −LP (j − 1) for every 1 ≤ i ≤ j ≤ d + 2 where Gram(∆)i,j denotes the (i, j)-entry of Gram(∆). We shall consider three cases according to the value of j in (24): 0 (i) j = 1, 2. Let v = f0, e = f1 and v be the other vertex of e. Then,

• Gram(∆)1,1 = hxv, xvi == hxb(v), xb(v)i = 1 = −LP (0). 1 1 • Gram(∆)1,2 = hxv, xei = hxv, 2 (xv + xv0 )i = 2 (1 − 1) = 0 = −LP (1). 18 1 1 1 • Gram(∆)2,2 = hxe, xei = h 2 (xv + xv0 ), 2 (xv + xv0 )i = 4 (1 − 2 + 1) = 0 = −LP (1). (ii) j = d + 2. Since P is a regular edge-scribed realization, the ﬁrst part of the Lemma 4.2 implies −1 d+1,1 that xP = ` ed+2 where ed+2 is the (d + 2)-th vector of the canonical basis of . Therefore, {p1,...,pd} L −2 • Gram(∆)d+2,d+2 = hxP , xP i = −` = −LP (d + 1). {p1,...,pd} The second part of Lemma 4.2 implies that for each i = 1, . . . , d + 2 we have

• Gram(∆)i,d+2 = hxfi−1 , xP i = hxP , xP i = −LP (d + 1). (iii) 3 ≤ j ≤ d + 2. If d = 1, then (i) and (ii) cover all the possible values for j. Let d > 1. First, we shall prove that (24) holds for j = d + 1. Since P is edge-scribed, the facet fd intersects the unit sphere Sd ⊂ Ed+1 in a (d − 1)-sphere S. Let µ ∈ Möb(Rbd) given by the composition of the following two Möbius transformations: d d - A rotation of S which sends the barycenter of fd to the line spanned by the North pole of S . - The Möbiustransformation corresponding to a rescaling of Rbd, which sends the rotated S to the equator of Sd. d+1 d By identifying the hyperplane {xd+1 = 0} ⊂ E with E we have that µ(fd) becomes an edge-scribed realization fed of the regular d-polytope with Schläflisymbol {p1, . . . , pd−1}. Moreover, since P is regular fed is also regular. We can then apply again Lemma 4.2 to obtain hx , x i = −`−2 for every 1 ≤ i ≤ d + 1 fei−1 fed {p1,...,pd−1} d where fei is the i-polytope obtained from µ(fi) after the identification {xd+1 = 0}' E . This identification preserves the inversive product of the illuminated regions of the outer-sphere points lying in {xd+1 = 0}. Since the inversive product is also invariant under Möbiustransformations then, for each i = 1, . . . , d + 1, we have:

• Gram(∆)i,d+1 = hxf , xf i = hx , x i = −LP (d). i−1 d fei−1 fed

Then, to prove (24) for j = d we apply the same arguments as above by considering this time fed−1 as a facet of the regular edge-scribed d-polytope fed. By carrying on these arguments for the rest of the values 3 ≤ j ≤ d + 2 we obtain (24) which gives (23).

−2 −2 Finally, since the sequence (−LP (0), −LP (1),..., −LP (d + 1)) = (1, 0, −` ,..., −` ) is {p1} {p1,...,pd} strictly decreasing then Gram(∆) is non-singular. Therefore, by applying the Lemma 4.1 to ∆ we obtain −1 T 0 = (κf0 , . . . , κfd+1 ) Gram(∆) (κf0 , . . . , κfd+1 ) −1 T = (κf0 , . . . , κfd+1 )C(−LP (0),..., −LP (d + 1)) (κf0 , . . . , κfd+1 ) d+1 2 2 X (κfi − κfi−1 ) κ = − P L (i) − L (i − 1) L (d + 1) i=1 P P P as desired. We notice that Equation (18) is invariant under M¨obiustransformations. By combinining the previous theorem with Corollaries 2.2 and 2.4, and also with the values of the half edge-length of a regular edge- scribed (d + 1)-simplex and (d + 1)-cube given in Table 1 we obtain the following two corollaries: d+1 Corollary 4.1. For any ﬂag (f0, f1, . . . , fd, fd+1 = T ) of an edge-scribed realization of a (d + 1)- simplex with d ≥ 1 the following holds d 2 d X i + 2 2 (25) κ d+1 = (κ − κ ) . T d + 2 2 fi fi+1 i=0

The latter can be used to obtain the curvatures in a d-ball packing BT where T is the (d + 1)-simplex. 1 Indeed, if we replace in the previous equation κfi = i+1 (κ1 + ··· + κi+1) for every 0 ≤ i ≤ d + 1 we obtain the Soddy-Gosset theorem. d+1 Corollary 4.2. For any flag (f0, f1, . . . , fd, fd+1 = C ) of an edge-scribed realization of a (d + 1)-cube with d ≥ 2 the following holds d 2 X 2 (26) κCd+1 = d (κfi − κfi+1 ) . i=0 19 5. Integrality of the Platonic Apollonian packings In this last section, we shall discuss and present numerous consequences of Theorem 4.1 for polytopal disk packings based on the Platonic solids. Polytopal 3-ball packings based on the regular 4-polytopes will be explored in a future work [RR]. Theorem 5.1. Let P and edge-scribed realization of a Platonic solid with Schläflisymbol {p, q}. The Lorentzian curvatures of any flag (v, e, f, P) satisfies

cos2( π ) sin2( π ) sin2( π ) (27) κ2 = p (κ − κ )2 + p (κ − κ )2 + p (κ − κ )2. P 2 π 2 π v e 2 π 2 π e f cos2( π ) f P sin ( q ) − cos ( p ) sin ( q ) − cos ( p ) q Moreover, the following relations hold: - Relation between two consecutive vertices:

κ + κ 0 (28) v v = κ 2 e 0 0 where v is the vertex in F0(P) \{v} satisfying v ⊂ e. - Relation between two consecutive edges:

κ + κ 0 (29) e e = cos( π )2κ + sin( π )2κ 2 p v p f 0 0 where e is the edge in F1(P) \{e} satisfying v ⊂ e ⊂ f. - Relation between two consecutive faces:

2 π 2 π 2 π κ + κ 0 cos ( ) sin ( ) − cos ( ) (30) f f = q κ + q p κ 2 2 π e 2 π P sin ( p ) sin ( p ) 0 0 where f is the face in F2(P) \{f} satisfying e ⊂ f ⊂ P. - Relation between two consecutive polyhedra: 2 π κ + κ 0 sin ( ) (31) P P = p κ 2 2 π 2 π f sin ( p ) − cos ( q ) 0 2 where P = sf (P) where sf is the projective transformation preserving S which corresponds to ∗ the inversion on the disk bf ∈ BP . Proof. By Theorem 2.1, every polyhedron is M¨obiusunique. Since Equation (18) is invariant under M¨obiustransformations then Theorem 4.1 holds for every edge-scribed realization not necessarily regular. By replacing `{p} and `{p,q} by their expression in terms of p and q given in Table 1 and Equation (11), we obtain cos2( π ) L (2) = cot2( π ) and L (3) = p P p P 2 π 2 π sin ( q ) − cos ( p )

and this gives (27). Equation (28) can be obtained directly by the definition of κe. By resolving the quadratic equation (27) for κe, κf , and then for κP , we obtain two solutions in each of the three cases. In the first two cases, one of the solutions is the Lorentzian curvature of e and f, respectively, and the other 0 0 solution is the Lorentzian curvature of e = re(e) and f = rf (f), where re and rf are the fundamental generators of the symmetric group of P with respect to the flag (v, e, f, P). In the last case, one of the solution is the Lorentzian curvature of P and the other solution, by Corollary 2.1, is the Lorentzian 0 curvature of P = sf (P). Equations (29), (30) and (31) follow by adding both solutions in each of the three cases.

Corollary 5.1. Let BP a polytopal disk packing where P is an edge-scribed regular polyhedron. All the curvatures of the Apollonian packing of BP can be obtained from the curvatures of three consecutive tangent disks of BP corresponding to three vertices of P lying in a same face.

Proof. Let {p, q} the Schl¨aﬂisymbol of P and let κi−1, κi and κi+1 the curvatures of three consecutive tangent disks bi−1, bi, bi+1 ∈ BP . Let f be the face of P containing the the vertices vi−1, vi, vi+1 corre- 0 sponding to three disks and let e and e be the edges with vertices vi−1, vi and vi, vi+1, respectively. By 20 1 0 1 replacing κe = 2 (κi + κi+1) and κ(e ) = 2 (κi + κi−1) in Equation (29) we obtain 1 1 (32) κf = 2 π (κi+1 + κi−1) + (1 − 2 π )κi 4 sin ( p ) 2 sin ( p ) This equation holds for every three consecutive vertices of f and, therefore, we can obtain the curvatures of the remaining vertices in f in terms of κi−1, κi and κi+1. By replacing in (27) κv by κi, κe by 1 2 (κi + κi+1) and κf by the right-hand-side of (32) and then resolving the quadratic equation for κP we obtain 2π 1 π q 2 π 2 − cos( p )κi + 2 (κi+1 + κi−1) ± cos( q ) (1 − 4 cos ( p ))κi + κiκi+1 + κiκi−1 + κi+1κi−1 (33) κ ± = . P 2 π 2 π 2(sin ( q ) − cos ( p ))

Corollary 2.1 implies that one of the two solutions is κP and the other is the Lorentzian curvature of 0 ∗ P = sf (P), where sf is the inversion on the boundary of bf ∈ BP . Once that κP is fixed, we can obtain the curvatures of the disks corresponding to the vertices of P by applying recursively (33). If, instead, we fix κ(P0), we obtain the curvatures of the disks corresponding to the vertices of P0. By changing faces and polyhedra we can obtain the curvatures for all the disks in Ω(BP ) in terms of κi, κi+1 and κi−1. 5.1. Integrality conditions. 5.1.1. Octahedral Apollonian packings. From Corollary 4.2 we have that the Lorentzian curvatures of every flag (v, e, f, O) of an edge-scribed octahedron O satisfies 3 (34) κ2 = (κ − κ )2 + 3(κ − κ )2 + (κ − κ )2 O v e e f 2 f O 1 1 By replacing κv = κ1, κe = 2 (κ1 + κ2) and κf = 3 (κ1 + κ2 + κ3) in the previous equation we obtain the generalization of the Descartes theorem given by Guettler and Mallows where κO plays the role of w (see [GM08]). The antipodal relation can be obtained from the relation (30). From Equation (33) we obtain the Lorentzain curvatures of two octahedra sharing a triangle f in terms of the curvatures of the vertices of f are given by p (35) κO± = κ1 + κ2 + κ3 ± 2(κ1κ2 + κ2κ3 + κ1κ3). The above equality leads to the condition of an octahedral Apollonian packings to be integral.

Proposition 5.1 (Guettler-Mallows). Let κ1, κ2, κ3 be the curvatures of three pairwise tangent disks of p an octahedral disk packing BO. If κ1, κ2, κ3 and 2(κ1κ2 + κ1κ3 + κ2κ3) are integers then Ω(BO) is integral.

293 365 365 293 398 229 229 398 294 173 173 294 206 125 300 300 125 206 134 85 212 212 85 134

356 356 381 140 140 381

309 309 260 53 53 260 340 78 78 340 84 398 398 84 180 180 189 382357 357382 189 220 29 302 302 29 220

358 358 141 364 364 141 373 222 222 373 116 38 262 262 38 116 341 156 237 237 156 341 356 356 238 325 325 238 294 294 0 300 44 44 300

198189 189198 158 158

340 340 308 308 166 166

221236 236221

50 49 81 49 50 81 49 81 49 50 72 49 81 81 49 72 50 49 81 49 81 50 49 81 49 50 389 68 68 389 32 82 32 32 74 74 32 32 82 32 390 390 58 25 50 50 25 58 25 25 58 25 50 50 25 58 268 268 18 18 18 18 374 374 34 34 13 365 365 13 69 84 348 348 84 69 26 26 26 26 228 349 278 110 110 278 349 228 244 244 9 88 88 88 88 9 9 88 88 88 88 9 173 126 126 173 380 380 90 89 8 8 89 90 90 89 8 8 89 90 117 117 325 325 388 388 64 72 72 64 64 72 72 64 150156 317 317 156150 65 65 56 56 65 65 65 65 56 56 65 65 157 157 10 58 58 10 10 58 58 10

277 277 41 41 41 41 366350 350366 40 40 40 40 269 269 397 397 388 20 20 388 32 32 32 32 93 93 133 133 133 133 66 66 66 66 180 373 373 180 164 124 284 94 94 284 124 164 14 118 334 358 4 358 334 118 14 78 245 245 78 292396 396292 246172 172246 342 342 349 302 302 349 380 380 45 388 388 45 229182 182229 230 220 220 230 268 268 221 52 52 221

317286 286317 80 80 80 80 270 270 270 270 42 42 2 42 42 148 148 36 205 205 36 57 57 333 333 56 56 56 56 206 397 397 206 398 398 382 254 254 382 62 197 389 389 197 62 101108 69 262 262 69 108101 16 16 64 64 49 49 70 70 70 70 17 64 64 17

181188 262269 269262 188181 16 16 204 204 56 56 56 56 116 116 57 57

285292 292285 42 42 42 42

174 174 80 80 80 80

244 244

326 326

214 214 278285 12 285278 76 76 66 66 66 66 292 292 77 221 221 77 70 286 286 70 32 32 32 32 236 236 40 40 40 40 237 237 41 41 41 41 172 172 318 318 11 55 58 58 58 58 356 356 56 56 56 56 270 270 65 10 65 65 10 65 65 10 65 65 10 65 64 72 72 64 64 72 72 64 196 196 332 332 325 325 90 89 89 90 90 89 89 90 382 382 134 134 88 88 88 88 2 88 88 88 88 220 220 220 220 213 316 316 213 317 317 26 26 26 26 84 132 102 102 132 84 34 34 374 124 124 374 8 50 50 8 8 50 50 8 9 58 58 9 9 58 58 9 74 74 125 342 125 125 342 125 18 82 18 18 82 18 364 364 25 32 32 25 32 25 25 32 25 32 32 25 50 49 81 49 50 81 49 81 49 50 72 49 81 81 49 72 50 49 81 49 81 50 49 81 49 50 286238 388 158166 166158 388 238286 301 381 381 301 46 68 181 181 68 46

293356 356293 308 61 28 28 61 308 173 173 252 229 229 252 230 230 188 188 230 230 189 222 166 166 222 189 389 20 20 389

269 380 132 132 380 269 358 358 109 262 262 109 110 332317 317332 110 388 388 365 365 276 21 6 21 276 76 76 390 390

276 276 206 206 253 68 68 253 156 156 349 53 53 349 334 334 148 101 268 268 101 148 260 165 165 260 245 245 -2 341 341 0 Figure 11. Two integral octahedral Apollonian packings with initial curvatures (−2, 4, 5) and (0, 0, 1).

21 5.1.2. Cubical Apollonian packings. Cubical√ Apollonian packings has been studied in [Sta15] as a particular case of the Schmidt arrangement Q[ 2]. It also appeared at the end of [Bol+18]. From Corollary 4.2 we have that the Lorentzian curvatures of every ﬂag (v, e, f, C) of an edge-scribed cube C satisﬁes

2 2 2 2 (36) κC = 2(κv − κe) + 2(κe − κf ) + 2(κf − κC) .

By combining (32) with the deﬁnition of κf we obtain a relationship of the curvatures involving the vertices of a square-face

(37) κi + κi+2 = κi+1 + κi+3 where the indices are given in cyclic order around the square-face. Moreover, by combining (37) with (30) we obtain the antipodal relation between any two vertices v andv ¯ at distance 3 in the graph of C by κ + κ (38) κ = v v¯ C 2 With Equations (37) and (38) we can compute the curvatures of the vertices of C from the curvatures of four vertices not lying all in a same face. The Lorentzian curvatures of two edge-scribed cubes C+ and C− sharing a square f in terms of the curvatures of the vertices of three consecutives vertices of f are q 2 (39) κC± = κi+1 + κi−1 ± −κi + κiκi+1 + κiκi−1 + κi−1κi+1 The above equality leads to an integrality condition for cubical Apollonian packings.

Proposition 5.2. Let κi−1, κi, κi+1 be the curvatures of three consecutive tangent disks in a cubical p 2 disk packing BC. If κi−1, κi, κi+1 and −κi + κiκi+1 + κiκi−1 + κi−1κi+1 are integers then Ω(BC) is integral. 0

462 462 404 404 350 470 470 350 412 412 300 358 358 300 254 308 308 254 212 469 469 212 477262 262477 174389 220 220 389174 397 397 317 182 182 317 489 140 325 325 140 489 253 148 148 253 261 261 377490110 110490377 464 464 197 118 118 197 84 205 205 84 281368 433 433 368281 346433 433346 149 92 92 149 157 157 424 424 489 489 476 62 402337 337402 62 476 201266 266201 422 485 485 422 313 493 493 313 469 248 70 70 248 469 109 109 117 322 322 117 478 478 486 369 369 486 413 257 257 413 421 304 304 421

326373 44 44 373326 466 466 184 184 137 349357 52 52 357349 137 217 382390 390382 217 424 424 284 462 462 284 457 170 240 240 170 457 317 77 77 317 485 85 273 193 193 273 85 485 340348 226 226 348340 420 420 453 453 293301 301293

389422 30 38 38 30 422389 381 381 245 245253 348 348 253245 245 212 488 488 212 337 276 276 337 304 268 178 178 268 304 413 145 145 413 122 436 201 201 436 122 458 5 458 145 168 481 318 318 481 168 145 89 61 341 341 61 89 498 53 53 498 406 406 274 465 465 274 112 373 238246 246238 373 112 182 297 297 182 205 205213 213205 205 494 494 309 309 332 453461 461453 332

470 470 493 493 493 493

293270 270293 28 401378 378401 28 136113 113136 413 413 413 413 405 360252 252360 405 430 430 377 445 445 377 181 488 153 269 269 153 488 181 173 130 462 462 130 173 198 497 497 198 412 412 432 190 389 389 190 432 20 449 449 20

413 413 413 413 390 45 430 430 45 390 268 228 228 268 245 405 245 320 320 245 405 245 382 157 277 457 337 222 397 397 222 337 457 277 157 382 494 434 314 314 434 494 337 294 180 97 412 412 97 180 294 337 354 237 237 354 134 217 172 157 114 329 329 114 157 172 217 134 194 80 80 194 493 37 37 493 401 476 112 112 476 401 356 149 433 433 149 356 416 448 97 378 468 341 341 448 448 341 341 468 378 97 448 416 465 333 129 129 333 465 425 276 356 425 425 356 276 425 442 336 22 442 442 22 336 442 333 333 313 350 350 313 57 74 253 253 74 57 386 386 401 401 386 317 317 386 176 436 436 176 326 193 309 141 141 309 193 326 445 445 428 156 37 37 156 428 445 445 133 133 385368 368385 133 133 116 422 20 422 116 148 414 498481 481498 414 148 333 449 449 333 316 464 464 316 330 412 150 150 412 330 214 345 133 133 345 214 197 404 404 197 406293 293406 317 142 398285 285398 142 317 300 125 125 300 442 442 442 442 425 425 425 425 329 329 312 312

438453 29 29 453438 14 205220 220205 14 264 264 249 437 437 249 214 420 420 214 258 197 197 258 365 73 241 241 73 365 394 409 409 394 277 460 460 277 377 58 345 345 58 377 445 182 330 330 182 445 260428 277109153 328 328 153109277 428260 490 313 313 490 165 260 473 473 260 165 494 92 136 56 56 136 92 494 385 385 385 385 341 472 341 341 472 341 457 474 474 457 246290 145 430 430 145 290246 476 41 130 101 101 130 41 476 280369 12 369280 325 253 253 325 462 236 164 164 236 462 86 238 238 86 397245 245397 149 382 382 149 452 344 344 452 230 461 461 230 329 329 -3 446 446 0

Figure 12. Two cubical Apollonian packings with initial curvatures (−3, 5, 12) (left) and (0, 0, 1) (right).

22 5.1.3. Icosahedral Apollonian packings. Icosahedral Apollonian packings appeared in [Bol+18] as a particular case of Apollonian ring packings when n = 5. By applying Theorem 5.1 with p = 3 and q = 5 we obtain that the Lorentzian curvatures of every ﬂag (v, e, f, I) of an edge-scribed icosahedron I satisﬁes

2 2 2 2 2 −2 2 (40) κI = ϕ (κv − κe) + 3ϕ (κe − κf ) + 3ϕ (κf − κI ) For any two consecutive faces f and f 0 of I sharing an edge e we have

κ + κ 0 1 f f = (ϕ2κ + ϕ−2κ )(41) 2 3 e I The latter implies the following relation between the Lorentzian curvatures of the two vertices u ∈ f \ e and u0 ∈ f 0 \ e by

κ + κ 0 (42) u u = ϕ−1κ + ϕ−2κ 2 e I By applying recursively this equation we can obtain the antipodal relation between two vertices v andv ¯ at distance 3 in the graph of I by κ + κ (43) κ = v v¯ I 2 As above, the Lorentzian curvatures of all the vertices of I can be easily computed from the curvatures of four vertices of I. In order to obtain the curvatures of the icosahedral Apollonian packing we can use the Equation (33), which gives the Lorentzian curvatures of two edge-scribed icosahedra I+ and I− sharing a triangle f, in terms of the curvatures of the vertices of f by

2 3√ (44) κI± = ϕ (κ1 + κ2 + κ3) ± ϕ κ1κ2 + κ1κ3 + κ2κ3. The above equality leads to a suﬃcient condition for the curvatures of a dodecahedral Apollonian packing to be in Z[ϕ]. We say that such a packing is ϕ-integral.

Proposition 5.3. Let κ1, κ2,√ κ3 be the curvatures of three mutually tangent disks of an icosahedral disk packing BI . If κ1, κ2, κ3 and κ1κ2 + κ1κ3 + κ2κ3 are in Z[ϕ] then Ω(BI ) is ϕ-integral.

11 φ+6 18 φ- 8

22 φ+ 2 6 φ+ 4 11 φ- 6 12 φ+7

10 φ+6 14 φ+ 8 10 φ+6

18 φ+ 10 18 φ+ 10 12 φ+7 12 φ+7 12 φ- 1 0 13 φ+ 12 14 φ 3 φ+2 6 φ- 4 φ 8 φ+ 3 10 φ+6 6 φ+ 2

12 φ+7 11 φ+6 6 φ+ 4 6 φ+ 4 9 φ+ 4 9 φ+ 4 4 φ+3 4 φ+3 12 φ+3 11 φ+ 2 8 φ- 1

18 φ+ 10 6 φ+ 6 6 φ- 2 13 φ+ 8 6 φ 4 φ+3 20 φ- 1 22 φ+ 2 2φ

8 φ+ 3 17 φ- 4 5 φ+4 3 φ- 2 6 φ+ 2 7 φ+2

17 φ+ 4 13 φ

15 φ+ 10 4 φ- 1 14 φ+2

14 φ+ 6 16 φ+ 11 2 φ+ 2 9 φ+ 4

14 φ+ 8 12 φ+3 14 φ+ 8 15 φ+ 6 -1 0 Figure 13. Two icosahedral Apollonian packings with initial curvatures (−1, ϕ, 2ϕ) (left) and (0, 0, 1) (right).

By comparing Proposition 5.3 with the integrality condition of Apollonian packings given in the introduction, we obtain that every triple of integers which produces an integral tetrahedral Apollonian packing produces also a ϕ-integral icosahedral Apollonian packing. We show in Figure 14 the integral tetrahedral and the ϕ-integral icosahedral Apollonian packings given by the curvatures (−4, 8, 9). 23 369 369 297 297 233 233 177 177 129 129 276 89 89 276

312 312 164 164 329 57 57 329

200 200

392 392 209 33 33 209 84 84 272 305 305 272 161 161 264 332 120 120 332 264 393 393 257 257

348 348 273 8 273 185 185 168 17 17 168 72 72 89 288 188 161 161 188 288 89 356 284 284 356 225 225

353 353 128 272 272 128 252 252 393 393 36 329 137 137 329 36 396 396 369 369 185 56 185 224 224 65 140 65 260 321 360 360 321 104 104

153 153

212 212

281 281

360 360 308 176 99 80 209 209 396 396

305 305 377 377 296 296 368 368 225 144 144 225 164 392 248 248 392 164 113 113 353 72 72 353

336 20 336 209 41 41 209

264 105 105 264 201 201 -4 329 329 -4

Figure 14. An integral tetrahedral (left) and a ϕ-integral icosahedral (right) Apollonian packings both with initial curvatures (−4, 8, 9).

5.1.4. Dodecahedral Apollonian packings. By applying Theorem 5.1 with p = 5 and q = 3 we obtain that the Lorentzian curvatures of every ﬂag (v, e, f, D) of an edge-scribed dodecahedron D satisﬁes

2 4 2 2 −2 2 (45) κD = ϕ (κv − κe) + (2 + ϕ)(κe − κf ) + (ϕ + 1)(κf − κD) .

By combining (32) with the deﬁnition of κf we obtain that the curvatures of the vertices of a pentagonal face are related by

(46) ϕ(κi+1 − κi) = κi+2 − κi−1

where the indices are given in cyclic order. Moreover, by combining the relations between consecutive edges and consecutives faces we can express the Lorentzian curvature of D in terms of the curvature of a vertex v and its three neighbours u1, u2 and u3 by

ϕ2 1 + 3ϕ (47) κ = (κ + κ + κ ) − κ . D 2 u1 u2 u3 2 v

The previous two equations allow to compute the curvatures of all the vertices of D from the curvatures of a vertex and its three neighbours.

Equation (33) gives the Lorentzian curvatures of the two edge-scribed dodecahedra D+ and D− sharing a pentagon f, in terms of the curvatures of the vertices of three consecutives vertices of f, by

q 2 2 (48) κD± = ϕκi + ϕ κi+1 + κi−1 ± −ϕκi + κiκi+1 + κiκi−1 + κi−1κi+1 .

Proposition 5.4. Let κi−1, κi, κi+1 be the curvatures of three consecutive tangent disks in a dodecahedral p 2 2 disk packing BD. If κi−1, κi, κi+1 and −ϕ κi + κiκi+1 + κiκi−1 + κi−1κi+1 are in Z[ϕ] then Ω(BD) is ϕ-integral. 24 0

-1 0

Figure 15. Two dodecahedral Apollonian packings with initial curvatures (−1, ϕ + 1, 2ϕ) (left) and (0, 0, 1) (right).

References [AM95] S. V. Anishchik and N. N. Medvedev. “Three-Dimensional Apollonian Packing as a Model for Dense Granular Systems”. In: Phys. Rev. Lett. 75 (23 1995), pp. 4314–4317. doi: 10. 1103/PhysRevLett.75.4314. url: https://link.aps.org/doi/10.1103/PhysRevLett. 75.4314. [BS04] A. Bobenko and B. Springborn. “Variational principles for circle patterns and Koebe’s theorem”. In: Transactions of the American Mathematical Society 356 (Feb. 2004). doi: 10.1090/S0002-9947-03-03239-2. [Bol+18] A. Bolt et al. “Apollonian Ring Packings”. In: May 2018, pp. 283–296. isbn: 9781107153981. doi: 10.1017/9781316650295.018. [Bos10] E. J. Bos. “Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)”. In: Historia Mathematica 37.3 (2010). Contexts, emergence and issues of Cartesian geometry: In honour of Henk Bos’s 70th birthday, pp. 485–502. issn: 0315-0860. doi: https://doi.org/10. 1016/j.hm.2009.11.004. url: https://www.sciencedirect.com/science/article/ pii/S0315086009001256. [Boy74] D. W. Boyd. “A new class of inﬁnite sphere packings.” In: Paciﬁc Journal of Mathematics 50.2 (1974), pp. 383–398. doi: pjm/1102913226. url: https://doi.org/. [BS93] G. R. Brightwell and E. R. Scheinerman. “Representations of Planar Graphs”. In: SIAM Journal on Discrete Mathematics 6.2 (1993), pp. 214–229. doi: 10.1137/0406017. eprint: https://doi.org/10.1137/0406017. url: https://doi.org/10.1137/0406017. [Che16] H. Chen. “Apollonian Ball Packings and Stacked Polytopes”. In: Discrete & Computational Geometry 55.4 (2016), pp. 801–826. issn: 1432-0444. doi: 10.1007/s00454-016-9777-3. url: https://doi.org/10.1007/s00454-016-9777-3. [CP17] H. Chen and A. Padrol. “Scribability problems for polytopes”. In: European Journal of Combinatorics 64 (2017), pp. 1–26. issn: 0195-6698. doi: https://doi.org/10.1016/ j.ejc.2017.02.006. url: https://www.sciencedirect.com/science/article/pii/ S0195669817300136. [Cox73] H.S.M. Coxeter. Regular Polytopes. Dover books on advanced mathematics. Dover Publica- tions, 1973. isbn: 9780486614809. [EKZ03] D. Eppstein, G. Kuperberg, and G.M. Ziegler. “Fat 4-polytopes and fatter 3-spheres”. In: Discrete geometry: In honor of W. Kuperberg’s 60th birthday (2003), pp. 239–265. [FR19] S. Felsner and G. Rote. “On Primal-Dual Circle Representations”. In: SOSA. 2019.

25 [Fuc13] E. Fuchs. “Counting problems in Apollonian packings”. In: Bulletin of the American Math- ematical Society 50 (2013), pp. 229–266. [Gos37] T. Gosset. “The Hexlet”. In: Nature 139 (1937), p. 62. [Gra+05] R. Graham et al. “Apollonian Circle Packings: Geometry and Group Theory I. The Apol- lonian Group”. In: Discrete & Computational Geometry 34.4 (2005), pp. 547–585. doi: 10.1007/s00454-005-1196-9. url: https://doi.org/10.1007/s00454-005-1196-9. [Gra+06a] R. Graham et al. “Apollonian Circle Packings: Geometry and Group Theory II. Super- Apollonian Group and Integral Packings”. In: Discrete & Computational Geometry 35 (Sept. 2006), pp. 1–36. doi: 10.1007/s00454-005-1195-x. [Gra+06b] R. Graham et al. “Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions”. In: Discrete & Computational Geometry 35 (2006). doi: 10.1007/s00454- 005-1197-8. [GM08] G. Guettler and C. Mallows. “A generalization of Apollonian packing of circles”. In: Journal of Combinatorics 1 (Jan. 2008). doi: 10.4310/JOC.2010.v1.n1.a1. [KN19] A. Kontorovich and K. Nakamura. “Geometry and arithmetic of crystallographic sphere packings”. In: Proceedings of the National Academy of Sciences 116.2 (2019), pp. 436–441. issn: 0027-8424. doi: 10 . 1073 / pnas . 1721104116. eprint: https : / / www . pnas . org / content/116/2/436.full.pdf. url: https://www.pnas.org/content/116/2/436. [Kwo+20] S. Kwok et al. “Apollonian packing in polydisperse emulsions”. In: Soft Matter 16 (10 2020), pp. 2426–2430. doi: 10.1039/C9SM01772K. url: http://dx.doi.org/10.1039/ C9SM01772K. [LMW01] J. Lagarias, C. Mallows, and A. Wilks. “Beyond the Descartes Circle Theorem”. In: The American Mathematical Monthly 109 (2001). doi: 10.2307/2695498. [RR21] J. L. Ram´ırezAlfons´ınand I. Rasskin. “Ball packings for links”. In: European Journal of Combinatorics 96 (2021), p. 103351. issn: 0195-6698. doi: https://doi.org/10.1016/ j.ejc.2021.103351. url: https://www.sciencedirect.com/science/article/pii/ S0195669821000433. [RR] J. L. Ram´ırezAlfons´ınand I. Rasskin. “The regular polytopal 3-ball packings”. In: (), In preparation. [SHP06] S. Sauerbrei, E. C. Hass, and P. Plath. “The Apollonian decay of beer foam bubble size distribution and the lattices of Young diagrams and their correlated mixing functions”. In: Discrete Dynamics in Nature and Society 2006 (May 2006). doi: 10.1155/DDNS/2006/ 79717. [Sch87] E. Schulte. “Analogues of Steinitz’s theorem about non-inscribable polytopes”. In: Colloq. Math. Soc. J´anosBolyai 48 (Jan. 1987). [Sch04] E. Schulte. “Symmetry of Polytopes and Polyhedra”. In: Handbook of Discrete and Compu- tational Geometry, 2nd Ed. 2004. [Sod36] F. Soddy. “The Kiss Precise”. In: Nature 137.3477 (1936), pp. 1021–1021. issn: 1476-4687. doi: 10.1038/1371021a0. url: https://doi.org/10.1038/1371021a0. [Sta15] K. Stange. “The Apollonian structure of Bianchi groups”. In: Transactions of the American Mathematical Society 370 (May 2015). doi: 10.1090/tran/7111. [Wil81] J. B. Wilker. “Inversive Geometry”. In: (1981). Ed. by Chandler Davis, Branko Gr¨unbaum, and F. A. Sherk, pp. 379–442. [Zie04] G. Ziegler. “Convex Polytopes: Extremal Constructions and f-Vector Shapes”. In: 13 (Dec. 2004).

IMAG Univ. Montpellier, CNRS, France Email address: [email protected]