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A POLYTOPAL GENERALIZATION OF APOLLONIAN PACKINGS AND
DESCARTES’ THEOREM
JORGE L. RAMIREZ ALFONSIN† 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 apolytope 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 Mobius transformations as well as generalizations of the Apollonian groups. We show that these
groups are hyperbolic Coxeter groups admitting an explicit matrix representation. An unexpectedinvariant, that we call Mobius spectra, associated to Mobius unique 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 andfoams structures (e.g., [AM95; SHP06; Kwo+20]). However, in mathematics, many questions are stillopen 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 diskthat can be inscribed. By adding the inscribed disk to each interstice we obtain a larger disk packingcontaining more interstices. We then repeat the process of inscribing new disks to each new intersticeindefinitely as in Figure 1.
Figure 1. An Apollonian packing.
The name of Descartes configuration comes from Rene Descartes who stated in a letter to the PrincessElizabeth 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
(κ1 + κ2 + κ3 + κ4)2 = 2(κ21 + κ2
2 + κ23 + κ2
4).(1)
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 curvaturesof 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).
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0Figure 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
(κ1 + · · ·+ κd+2)2
= d(κ2
1 + · · ·+ κ2d+2
).(2)
The Soddy-Gosset theorem was discovered and rediscovered many times [LMW01]. Another type ofgeneralizations in the plane have been studied by modifying the initial set of Descartes configurations. Bya similar process of construction, Guettler and Mallows defined in [GM08] octahedral Apollonian packingsfrom disk packings whose tangency graph is the octahedral graph. In [Bol+18], the authors defined theApollonian ring packings where Descartes configurations are replaced by disk packings where each diskis surrounded by n disks. For n = 3 and n = 4, they obtained the classical and octahedral Apollonianpackings, 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 convexpolyhedra [KN19]. They explored the close connection between the polyhedra structure behind someknown generalizations of Apollonian packings. Eppstein, Kuperberg and Ziegler [EKZ03] used a similarapproach to improve the upper bound of the average kissing number of 3-ball packings from edge-scribed4-polytopes. Similarly, Chen studied in [Che16] the relation between Apollonian d-ball packings andstacked (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 arisingsimilarly from edge-scribed realizations of (d+ 1)-polytopes. As we will see, the combinatorial structureinformation in these packings comes from a polytope, not only from the tangency graph, as usual whenconsidering 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 Mobius spectra, associated to Mobius unique 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 hyperbolicCoxeter groups given by an explicit matrix representation.
(iv) We prove that there are integral Apollonian packings based on the tetrahedron, cube and do-decahedron containing the sequences of perfect squares.
(v) By introducing the notion of Lorentzian curvature of a polytope, we present a result analogue ofDescartes’ theorem for regular polytopes in any dimension.
(vi) We obtain sufficient conditions for an Apollonian packing based on the Platonic solids to beintegral.
1.1. Organization of the paper. The paper is organized as follows: in Section 2 we discuss thebackground theories needed for the rest of the paper. First, we recall fundamental concepts about theLorentzian and projective model of the space of d-balls and polytopes. Then, we present the construc-tion 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 thissection by introducing and discussing the Mobius spectra and by putting forward a couple of questionsconcerning this invariant.
In Section 3, we give the generalization of the Apollonian groups for polytopal ball packings. Then, wepresent a matrix representation of these groups when the corresponding polytope is one of the Platonicsolids. The matrix representations are used to prove (iv).
In Section 4, we present the result analogue to Descartes’ theorem for regular polytopes. We show thatSoddy-Gosset’ generalization follows when we apply this result to the (d+ 1)-simplices and we extend itto 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 diskpackings coming from the Platonic solids. In particular, we derive the Guettler-Mallows’ generalizationin the octahedral case. We use the results of this section to characterize the integrality of the Apollonianpackings based on the Platonic solids.
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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 + 1and the Lorentzian space of dimension d+ 2, respectively. Recall that Ld+1,1 is the real vector space ofdimension d+ 2 endowed with the Lorentzian product
〈x,y〉 = x1y1 + · · ·+ xd+1yd+1 − xd+2yd+2, x,y ∈ Ld+1,1.
A vector Ld+1,1 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 Rd := Rd∪∞, denoted by Balls(Rd), and the one-sheethyperboloid of the normalized space-like vectors of Ld+1,1 which we will denoted by S(Ld+1,1). We referthe reader to [Wil81] as well as [RR21, Section 2] and the reference therein for further details.
Let b ∈ Balls(Rd) of center c and curvature κ ∈ R (resp. normal vector n pointing to the interior andsigned distance to the origin δ ∈ R when b is a half-space). We denote by xb ∈ S(Ld+1,1) the Lorentzianvector corresponding to b. The inversive coordinates of b defined by Wilker [Wil81] correspond to theCartesian coordinates of xb in the canonical basis of Ld+1,1. These coordinates can be given in terms ofits center c and curvature κ and (resp. normal vector n and signed distance to the origin δ) as follows
xb =
κ
2(2c, ‖c‖2 − κ−2 − 1, ‖c‖2 − κ−2 + 1) if κ 6= 0,
(n, δ, δ) if κ = 0
(3)
where ‖ · ‖ denotes the Euclidean norm of Rd. The curvature of b can be deduced from its inversivecoordinates by
κ(b) = −〈xN ,xb〉(4)
where xN = ed+1 + ed+2 and ei denotes the i-th canonical vector of Ld+1,1. The inversive product oftwo d-balls 〈b, b′〉 := 〈v(b), v(b′)〉 is a fundamental tool to encode arrangements of d-balls. Indeed, if theLorentzian vectors of b and b′ are not both past-directed then
〈b, b′〉
< −1 if b and b′ are disjoint= −1 if b and b′ are externally tangent= 0 if b and b′ are orthogonal= 1 if b and b′ are internally tangent> 1 if b and b′ are nested
(5)
A d-ball arrangement B is a d-ball packing if any two d-balls are externally tangent or disjoint. Forany d-ball arrangement B we denoted by Gram(B) the Gramian of the corresponding Lorentzian vectors
in Ld+1,1. The Mobius group Mob(Rd) is the group generated by inversions on d-balls. It is isomorphicto the Orthochronous Lorentz group made by the linear maps of Ld+1,1 to itself preserving the Lorentzproduct and the time direction. The Orthochronous Lorentz group is isomorphic to the following groupof matrices
O↑d+1,1(R) = M ∈ GLd+2(R) |MTQd+2M = Qd+2 and eTd+2Med+2 > 0.
This connection gives a matrix representation of Mob(Rd) by
Mob(Rd)→ O↑d+1,1(R)
sb 7→ Id+2 − 2xbxTb Qd+2
(6)
where sb is the inversion on the d-ball b, Id+2 = diag(1, . . . , 1, 1) and Qd+2 = diag(1, . . . , 1,−1). TheMobius group sends d-balls to d-balls and preserves the inversive product [Wil81].
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2.1. The projective model of the space of balls. The one-sheet hyperboloid S(Ld+1,1) can beregarded in the oriented projective space of Ld+1,1
P+Ld+1,1 = x ∈ Ld+1,1 \ 0/∼(7)
where x ∼ y if there is a real λ > 0 such that x = λy. P+Ld+1,1 is equivalent to the Euclidean unitsphere Sd+1 ⊂ Ld+1,1 which, under the gnomonic projection, becomes the union of two affine hyperplanesΠ↑ = xd+2 = 1 and Π↓ = xd+2 = −1 together with Π0 = (x, 0) | x ∈ Sd+1. The composition of
the isomorphism Balls(Rd)→ S(Ld+1,1) with the projection
S(Ld+1,1)→ Π↑ ∪Π0 ∪Π↓
x 7→
x if xd+2 = 01
|xd+2|x otherwise
gives an isomorphism between Balls(Rd) and Π↑ ∪ Π0 ∪ Π↓. We call the latter the projective model of
Balls(Rd). By identifying Π↑ with Ed+1 we get a bijection between d-balls whose Lorentzian vector isfuture-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(Rd) can be obtained geometrically by taking a light source which illuminatesSd from p. The spherical illuminated region is a spherical cap which becomes, under the stereographicprojection from the north pole, the d-ball b(u). We say that u is the light source of b(u) and reciprocallywe say that b(u) is the illuminated region of u.
Rd
Sd
Ed+1
b(u)
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 fromthe coordinates of its light source by
xb(u) =1√
‖u‖2 − 1(u, 1).(8)
This equality implies that for any two d-balls whose Lorentzian vectors are future-directed, their inversiveproduct is related to the Euclidean inner product of their corresponding light sources by the followingequation
(9) 〈b(u), b(v)〉 =1√
(‖u‖2 − 1)(‖v‖2 − 1)(u · v − 1)
where · denotes the Euclidean inner product. In the projective model of space of d-balls, the Mobiusgroup 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 restof the paper. We refer the reader to [Sch04] for further details. We consider here a (d+ 1)-polytope asthe convex hull of a finite collection of points in Ed+1. A 2-polytope and a 3-polytope are usually calledpolygon 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
F(P) = ∅ ∪⋃d+1k=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 andfacets of P, respectively. The graph of P is the graph induced by the vertices and the edges of P. Theface lattice (F(P),⊂) encodes all the combinatorial information about P. Two polytopes P and P ′ arecombinatorially equivalent if there exists an isomorphism between their face lattices. If they are combi-natorially equivalent, we say they have the same combinatorial type and P ′ 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 ∈ Xwhere · denotes the Euclidean inner product. If P is a (d + 1)-polytope containing the origin in itsinterior then P∗ is also a (d + 1)-polytope containing the origin in its interior and holding the dualrelation (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
f = u ∈ P | u · vf = 1.(10)
The symmetric group of P is defined as the group of Euclidean isometries of Ed+1 preserving P. P issaid to be regular if its symmetric group acts transitively on the set of flags P. In this case, the symmetricgroup 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). ThenC is the fundamental region of the symmetric group of P which is generated by the reflections r0, . . . , rd inthe walls of C containing the barycenter of P, where rk is the reflection in the wall opposite to the vertexof C corresponding to the k-face in (f0, . . . , fd). We call the set r0, . . . , rd the fundamental generatorsof the symmetric group of P with respect to the flag (f0, . . . , fd,P). The symmetric group of P is aspherical Coxeter group with Coxeter graph
p1 pdwhere p1, . . . , pd is the Schlafli symbol of P.
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 unitsphere of Ed+1. A k-scribed polytope is called inscribed, edge-scribed, ridge-scribed and circunscribedif k = 0, 1, d − 1, d respectively. A polytope is said to be k-scribable if it admits a realization which isk-scribed. Any regular (d + 1)-polytope is k-scribable for every 0 ≤ k ≤ d (by properly centering andrescaling by a suitable value). If P is a (d + 1)-polytope containing the origin in its interior then P isk-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-scribedrealization of P. The values of `P for every regular (d+ 1)-polytope are presented in the following tableadapted from [Cox73, p. 293].
d Regular (d+ 1)-polytope P Schlafli symbol `Pd = 1 p-gon p tan(π
p)
d ≥ 1
(d+ 1)-simplex 3, . . . , 3︸ ︷︷ ︸d
( d+2d
)1/2
(d+ 1)-cross polytope 3, . . . , 3︸ ︷︷ ︸d−1
, 4 1
(d+ 1)-cube 4, 3, . . . , 3︸ ︷︷ ︸d−1
d−1/2
d = 2Icosahedron 3, 5 ϕ−1
Dodecahedron 5, 3 ϕ−2
d = 324-cell 3, 4, 3 3−1/2
600-cell 3, 3, 5 5−1/4ϕ−3/2
120-cell 5, 3, 3 3−1/2ϕ−3
Table 1. The half edge-length of a regular edge-scribed realization of all the regular (d+1)-
polytopes for every d ≥ 1. The letter ϕ denotes the Golden Ratio 1+√
52 = 2 cos(π5 ).
6
In the case d = 2, we notice that the values of `P can be obtained from the Schlafli symbol of P by
`p,q =
√√√√ sin2(πq )− cos2(πp )
cos2(πp ).(11)
2.3. The ball arrangement projection. Let P be an outer-sphere (d + 1)-polytope, i.e. with all itsvertices outer-sphere. We define the ball arrangement projection of P, denoted by B(P), the collection ofd-balls whose light sources are the vertices of P. Reciprocally, we say that P is the light-source polytopeof the ball arrangement B(P). As Chen noticed in [Che16], if B(P) is a packing then then tangencygraph 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 itsvertices; (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, externallytangent or they have intersecting interiors if and only if e cuts transversely, is tangent or avoids strictlySd, respectively. Therefore, if P is an edge-scribed (d + 1)-polytope then B(P) is a d-ball packing. A
d-ball packing BP is defined as polytopal if there exists µ ∈ Mob(Rd) such that µ(BP) = B(P). Wenotice that the tangency graph of BP and the graph of P are isomorphic. We say that P is the tangencypolytope of BP . We observe that not all the d-ball packings are polytopal. An example of such packingis illustrated in Figure 6.
Figure 6. A non-polytopal disk packing (left), its light-source polyhedron (center) andsame polyhedron with the unit sphere (right). The edge cutting the sphere does not corre-spond 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-faceof P whose barycenter lies on the ray spanned by the North pole. For a regular (d+ 1)-polytope P, wedefine the centered d-ball packing projections of P as the collection of the ball arrangement projectionswith 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 ∞
`3,3 =√
2
n κ
1 1√2
(1−√
3)
3 1√2
(1 +√
1/3)
n κ
2 0
2√
2
n κ
3 1√2
(1−√
1/3)
1 1√2
(1 +√
3)
`3,4 = 1n κ
1 1−√
24 1
1 1 +√
2
n κ
2 02 1
2 2
n κ
3 1−√
2/3
3 1 +√
2/3
`4,3 = 1√2
n κ
1√
2−√
3
3√
2−√
1/3
3√
2 +√
1/3
1√
2 +√
3
n κ
2 0
4√
2
2 2√
2
n κ
4√
2− 1
4√
2 + 1
`3,5 = 1ϕ
n κ
1 ϕ−√ϕ+ 2
5 ϕ−√ϕ+25
5 ϕ+√ϕ+25
1 ϕ+√ϕ+ 2
n κ
2 0
2 ϕ− 14 ϕ
2 ϕ+ 1
2 2ϕ
n κ
3 ϕ− ϕ2√
1/3
3 ϕ− ϕ−1√
1/3
3 ϕ+ ϕ−1√
1/3
3 ϕ+ ϕ2√
1/3
`5,3 = 1ϕ2
n κ
1 ϕ2 − ϕ√
3
3 ϕ2 − ϕ√
5/3
6 ϕ2 − ϕ√
1/3
6 ϕ2 + ϕ√
1/3
3 ϕ2 + ϕ√
5/3
1 ϕ2 + ϕ√
3
n κ
2 0
4 ϕ2 − ϕ2 ϕ2 − 1
4 ϕ2
2 ϕ2 + 14 ϕ2 + ϕ2 2ϕ2
n κ
5 ϕ2 −√
15
(7 + 11ϕ)
5 ϕ2 −√
15
(3− ϕ)
5 ϕ2 +√
15
(3− ϕ)
5 ϕ2 +√
15
(7 + 11ϕ)
Table 2. Centered disk packing projections of the Platonic solids together with a table showingthe 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. Wecan 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 isa Mobius transformation µ such that µ(BP) = B(P0). We define the dual of BP as the ball arrangementB∗P := µ−1(B(P∗0 )). The dual B∗P does not depend on the choice of P0. For any vertex v of P0 andfor any vertex of P∗0 corresponding to a facet f of P0 containing v the corresponding d-balls bv ∈ BPand bf ∈ B∗P are orthogonal (this can be easily obtained by combining (5) with (9) and (10)). Since Pis edge-scribed, P0 is edge-scribed and P∗0 is ridge-scribed. In the case d = 2, P∗0 is also edge-scribedand therefore BP and B∗P are both disk-packings. The union BP ∪ B∗P has been called a primal-dualcircle representation of P [FR19]. Brightwell and Scheinerman proved in [BS93] the existence and theunicity up to Mobius transformations of primal-dual circle representations for every polyhedron. It canbe 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 sphericalilluminated regions of I and its polar; (right) a primal-dual circle representation of I.
2.5. Mobius unicity. Two d-ball packings are Mobius equivalent if one can be sent to the other by aMobius transformation. If such a Mobius transformation is a Euclidean isometry then we say that thetwo packings are Euclidean congruent. We recall that a d-ball packing B is said to be standard if itcontains the half-spaces bi = xd ≥ 1 and bj = xd ≤ −1, denoted by [B]ij . For any tangent d-balls biand bj of a d-ball packing B there is always a Mobius transformation µ mapping B into a standard form[B]ij (see [RR21]). We call such a Mobius transformation a standard transformation.
Lemma 2.1. [RR21] Let B and B′ be two d-ball packings with same tangency graph G and let ij be anedge of G. Then B and B′ are Mobius equivalent if and only if [B]ij and [B′]ij are Euclidean congruent.
We present here another approach to detect Mobius equivalence. We say that a d-ball arrangementhas 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 B′ two d-ball packings with maximal rank. Then B is Mobius equivalent toB′ if and only if Gram(B) = Gram(B′).
Proof. The necessity is trivial since Mobius transformations preserve the inversive product. Now letus suppose that Gram(B) = Gram(B′) is a matrix of rank d + 2. Let V = (v1, . . . , vn) and V ′ =(v′1, . . . , v
′n) 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 ′ are future-directed. By the rank condition, Vcontains a basis ∆ = vi1 , . . . , vid+2
of Ld+1,1. Since Gram(B) = Gram(B′) the corresponding collection∆′ = v′i1 , . . . , v
′id+2 ⊂ B′ is also a basis. Let P and P ′ be the light-source polytopes of the d-balls
corresponding to ∆ and ∆′ respectively. Since these are basis the polytopes P and P ′ are (d + 1)-simplices of Ed+1 intersecting the unit sphere Sd. If P does not contain the origin then we can find aMobius transformation µ0 such that the corresponding projective transformation maps P into another(d+ 1)-simplex P0 containing the origin of Ed+1. Such a Mobius transformation can be easily found by
combining translations and scalings of Rd which are also Mobius transformations. Let µ0 be the elementin the Orthochronous Lorentz group corresponding to µ0 and let V0 := µ0(V ), ∆0 := µ0(∆). Let M0 andM1 be the (d + 2)-square matrices formed by the column-matrices of the coordinates of the vectors of
9
∆0 and ∆′ respectively with respect to the canonical basis. The matrices M0 and M1 are non-singularand therefore
Gram(∆0) = Gram(∆′)⇔M0TQd+2M0 = M1
TQd+2M1
⇔ Qd+2(M1T )−1M0
TQd+2M0 = M1
where Qd+2 = diag(1, . . . , 1,−1) is the matrix of the Lorentz product. The matrix
A := Qd+2(M1T )−1M0
TQd+2
defines a linear map of Rd+1,1 to itself which maps ∆0 to ∆′. Moreover, one can check that
ATQd+2A = Qd+2.
Since the origin belongs to P0 we have that Aed+2 ⊂ P ′ ⊂ Π↑. Thus, eTd+2Aed+2 > 0 and therefore
A ∈ O↑d+1,1(R). The coordinates for the remaining vectors of V0 and V ′ with respect to the basis ∆0
and ∆′ respectively are completely determined by the entries of Gram(B). The equality of the Gramiansand linearity implies that the Lorentz transformation µ given by A maps V0 to V ′. Therefore, µµ0
corresponds to a Mobius transformation which maps B to B′.
Remark 1. Proposition 2.1 does not hold for ball arrangements in general. Indeed, for any ball packingB whose Gramian is maximal, the ball arrangement −B, made by taking the d-balls corresponding thevectors −xb for every b ∈ B, is not a packing, and therefore B and −B are not Mobius equivalent.However, Gram(B) = Gram(−B).
We say that an edge-scribable (d+ 1)-polytope is Mobius unique if the ball arrangement projectionsof all its edge-scribed realizations are Mobius equivalent. Lemma 2.1 and Proposition 2.1 are useful tostudy Mobius unicity.
Corollary 2.1. Let P and P ′ be two edge-scribed realizations of a regular (d + 1)-polytope which isMobius unique. If P and P ′ share a facet f then P ′ = P or P ′ = sf (P) where sf is the projectivetransformation corresponding to the inversion on the sphere f ∩ Sd.
Proof. Let BP and BP′ the ball arrangement projections of P and P ′, respectively. Let bf ∈ B∗P andb′f ∈ B∗P′ the d-balls corresponding to the facet f . Let u and v two vertices of f connected by an edge.
After applying a standard transformation with respect to the edge uv to BP and B′P we have that bf andb′f becomes two half-spaces whose boundary is a hyperplane H orthogonal to xd ≥ 1 and xd ≤ −1,respectively. All the d-balls of [BP ]uv corresponding to vertices of P which are not contained in f lie onone side of H. Similarly for [B′P ]uv . If they lie on the same side, since the combinatorial type of P andP ′ is regular and Mobius unique, Lemma 2.1 implies that [BP ]uv = [B′P ]uv . If they lie on different sides,same lemma and the regularity implies that [B′P ]uv = s([B′P ]uv ), where s is the reflection on H and theresult follows.
We notice that the Gramian of a ball arrangement projection of an edge-scribed (d + 1)-simplex isthe (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 Mobius unique.
It is clear that all polygons are edge-scribable but not all are Mobius unique.
Corollary 2.3. The only polygon which is Mobius unique is the triangle.
Proof. Let P be an edge-scribed n-gon with n > 3 and let [BP ]n1 = (b1, b2, . . . , bn) be the standard 1-ballpacking 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 bytwo closed intervals b′2 and b′3 obtained by moving x a suitable distance we can construct another 1-ballpacking [B′P ]n1 which is not Euclidean congruent to [BP ]n1 . By Lemma 2.1, [BP ]n1 and [B′P ]n1 are notMobius equivalent and therefore P is not Mobius unique.
Mobius unicity 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 asfollows.
Theorem 2.1 (Brightwell and Scheinerman). Every polyhedron is edge-scribable and Mobius unique.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 whichare not Mobius unique.
Conjecture 1. For every d ≥ 2 all edge-scribable (d+ 1)-polytopes are Mobius unique.
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.
Proposition 2.2. Let d ≥ 2 and let BCd+1 be a polytopal d-ball packing where Cd+1 is an edge-scribedrealization of the (d+ 1)-cube. Then, for every two vertices u, v of Cd+1, we have
〈bu, bv〉 = 1− 2dG(u, v)(12)
where dG(u, v) is the distance between u and v in the graph of Cd+1.
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 signedpermutations of 1√
2(±1,±1,±1). By using Equations (8) and (9), we obtain that the proposition holds
for C3. Since, by Theorem 2.1, the 3-cube is Mobius unique, then the proposition holds for any otheredge-scribed realization of the 3-cube.
Let us now suppose that the proposition holds for any edge-scribed realization of the (d + 1)-cubefor some d ≥ 2. Let BCd+2 be a polytopal (d + 2)-ball packing where Cd+2 ⊂ Ed+2 is an edge-scribedrealization of the (d + 2)-cube. Let u, v be two vertices of Cd+2. We notice that 0 ≤ dG(u, v) ≤ d + 2.Let u the unique vertex in Cd+1 such that dG(u, u) = d + 2. If v 6= u then there is a facet f of Cd+2
containing u and v. Since Cd+2 is edge-scribed then f intersects Sd+1 in a d-sphere S. We may finda Mobius transformation µ ∈ Mob(Sd+1) which sending S to the equator of Sd+1. After identifyingthe hyperplane xd+2 = 0 ⊂ Ed+2 with Ed+1 we obtain that µ(f) becomes an edge-scribed realization
Cd+1 of the (d + 1)-cube. The identification xd+2 = 0 ' Ed+1 preserves the inversive product of theilluminated 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 Cd+1. By the invariance of the inversiveproduct under Mobius transformations and the induction hypothesis we have that (12) holds for anytwo (d + 2)-balls of BCd+2 corresponding to two vertices of Cd+2 at distance strictly less than d + 2. Itremains thus to show that (12) holds for bu and bu.
Let v1, . . . , vd+2 be the neighbours of u in the graph of Cd+2. We consider the (d+1)-ball arrangement∆ = (bu, bv1 , . . . , bvd+2
, bu) ⊂ BCd+2 . By the induction hypothesis we have
Gram(∆) =
1 −1 · · · −1 λ−1 1− 2d... Ad+2
...−1 1− 2dλ 1− 2d · · · 1− 2d 1
(13)
where An denotes the n-square matrix with 1’s in the diagonal entries and −3 everywhere else andλ := 〈bu, bu〉. 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
det(Gram(∆)) = 0⇔ λ = −3− 2d or λ =2d2
3d+ 2+ 1 > 0 for every d ≥ 2.
Since bu and bu are disjoint then, by (5), λ < −1 and therefore we have that
〈bu, bu〉 = −3− 2d = 1− 2(d+ 2) = 1− 2dG(u, u).
Corollary 2.4. The (d+ 1)-cube is Mobius unique for every d ≥ 2.
Proof. The result follows by combining Propositions 2.1 and 2.2. 11
2.6. The Mobius spectra. Spectral techniques, based on the eigenvalues and the eigenvectors of theadjacency or the Laplace matrices of graphs are strong and successful tools to investigate different graph’sproperties. On this spirit, we define the Mobius spectra of an edge-scribable Mobius unique (d + 1)-polytope P as the multiset of the eigenvalues of the Gramian of B(P ′) where P ′ is any edge-scribedrealization 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 Mobius spectra of the Platonic solids.
The well-known theorem of Steinitz states that the graph of a polyhedron is a 3-connected simpleplanar graph. Such graphs are called polyhedral graphs. Since all the polyhedra are edge-scribable andMobius unique, the Mobius spectra can be defined for any polyhedral graph. Mobius unicity impliesthat the Mobius spectra does not depend on the edge-scribed realization. One may wonder if Mobiusspectra defines a complete invariant for edge-scribable Mobius unique polytopes and, in particular, forpolyhedral graphs.
Question 1. Are there two combinatorially different edge-scribable Mobius unique (d+1)-polytopes withsame Mobius spectra? In particular, are there two non isomorphic polyhedral graphs with same Mobiusspectra?
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.
For d ≥ 2, let BP a polytopal d-ball packing. We consider the following subgroups of Mob(Rd):• The symmetric group of BP is the group
Sym(BP) := 〈µ ∈ Mob(Rd) | µ(BP) = BP〉.We notice that Sym(BP) = Sym(B∗P).
• The Apollonian group of BP is the group
A(BP) := 〈S(B∗P)〉where S(B∗P) denotes the set of inversions on the d-balls of B∗P .
• The symmetrized Apollonian group of BP is the group
SA(BP) := 〈Sym(BP) ∪ A(BP)〉It can be checked that for every r ∈ Sym(BP) and every b ∈ B∗P 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) := 〈A(B∗P) ∪ Sym(BP) ∪ A(BP)〉.
Remark 2. In [Gra+05; Gra+06a; Gra+06b; KN19] the authors defined the dual Apollonian group andthe super Apollonian group which, with our notation, correspond to the groups A(B∗P) and 〈A(BP) ∪A(B∗P)〉, respectively. For the purposes of this paper these groups are not needed. Neverthless, bothgroups are subgroups of SSA(BP).
For any other polytopal d-ball packing B′P Mobius equivalent to BP the four groups defined abovefor BP are congruent to their analogues in B′P . Therefore, if P is Mobius unique, the four groups canbe defined for P independently of the choice of BP . We define the Platonic Apollonian groups as theApollonian groups of the Platonic solids.
12
Since inversions of Mob(Rd) can be seen as hyperbolic reflections of the (d+1)-dimensional hyperbolicspace, Apollonian groups are hyperbolic Coxeter groups. The Apollonian cluster of BP , denoted byΩ(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 theApollonian packing of BP . For d = 2, Ω(BP) is always a packing for any polytopal disk packing. Thisis not true in higher dimensions. For instance, if P is an edge-scribed (d + 1)-simplex with d ≥ 4 thenΩ(BP) is not packing (see [Gra+06b]).
The four groups presented above give different actions on the Apollonian cluster. For instance, eachdisk of a polytopal disk packing BP has a different orbit under the action of A(BP). This can be used topresent colorings of the disks in the Apollonian packing of BP . Indeed, if BP admits a proper coloringthen is not hard to see that the coloring can be extended to Ω(BP) by the action of the A(BP). Wepresent in Figure 8 three Apollonian packings with a minimal coloration. In the first case each orbit hasa different color.
Figure 8. A tetrahedral (left), octahedral (center) and cubical (right) Apollonian packingwith a minimal coloration. An interesting issue about these colorings is that the whitedisks, including the disk with negative curvature, are induced by optical illusion from theother colored disks.
If P is regular then Sym(BP) acts transitively on the d-balls of BP , B∗P and their correspondingApollonian clusters. Let Σ(BP) be a set of generators of Sym(BP). By equation (14), we have that forany sf ∈ S(BP) and any sv ∈ S(B∗P) the group SA(BP) is generated by Σ(BP) ∪ sf and SSA(BP) isgenerated by Σ(BP) ∪ sv, sf.
We observe that the transitivity of the action of the symmetrized Apollonian group for regular poly-topes could be useful to find special sequences of curvatures in the Apollonian clusters. We refer thereader 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 ofcurvatures of the disks contains all the perfect squares.
Proof. For every p = 3, 4, 5, let [BP ]12 be the standard polytopal disk packing where P is the regularpolyhedron of Schlafli symbol p, 3. Let (v, e, f) be the flag of P where v is the vertex correspondingto the half-space bv = y ≤ −1, e is the edge of P with ends v and v′ where bv′ = y ≤ −1, and f is aface of P containing e. Let us apply a translation and a reflection if needed so bf becomes the half-spacex ≥ 0. We then rescale by a factor of λ := 4 cos2(πp ). Let Bp,3, for p = 3, 4, 5, be the transformed
disk packing obtained from [BP ]12 (see Figure 9). The rescaling factor has been chosen to obtain that thedisks with minimal non-zero curvature of Bp,3 have curvature equals to one.
13
rf
rv
bv′
bv
sf
re
rf
rv
bv′
bv
sf
re
rf
rv
bv′
bv
sf
re
Figure 9. The disk packings Bp,3 for p = 3 (left), p = 4 (center) and p = 5 (right). Indashed 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 inSym(Bp,3) corresponding to the fundamental generators of the symmetric group of P with respect to theflag (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 bv′ 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
µp,n := re(rfsf )nre ∈ SA(Bp,3) and the disk bn := µp,n(bv′) ∈ Ω(Bp,3).
Let us compute the curvature of bn. The inversive coordinates of bv and the matrices representing re, rfand sf obtained by (6) are given by
bv 7→ Bλ =
01λλ
re 7→ Eλ =
1 0 0 00 1− λ
214
(λ2 − 4λ− 1
)− 1
4
(λ2 − 4λ+ 1
)0 1
4
(λ2 − 4λ− 1
)1− (λ2−4λ−1)2
8λ− 1−(λ−4)2λ2
8λ
0 14
(λ2 − 4λ+ 1
) 1−(λ−4)2λ2
8λ1 +
(λ2−4λ+1)2
8λ
,
rf 7→ Fλ =
−1 0 2√λ −2
√λ
0 1 0 0
2√λ 0 1− 2λ 2λ
2√λ 0 −2λ 2λ+ 1
, sf 7→ S =
−1 0 0 00 1 0 00 0 1 00 0 0 1
,By induction on n we obtain
(FλS)n =
1 0 2
√λn −2
√λn
0 1 0 0
−2√λn 0 1− 2λn2 2λn2
−2√λn 0 −2λn2 1 + 2λn2
and therefore, by (4), we have
κ(bn) = κ(µp,n(bv))
= KEλ(FλS)nEλBλ
= n2
where K = diag(0, 0,−1, 1).
The right image of Figures 2, 12 and 15 illustrates the corresponding tetrahedral, cubical and dodec-ahedral 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). Thereforewe can find representations of A(BP) as a subgroup of SSA(BP). Even if SSA(BP) is a larger group thanA(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 Schlafli symbol p, q.The group SSA(BP) is the hyperbolic Coxeter Group with Coxeter graph ∞ p q ∞ . Moreover,
SSA(BP) is generated by the following five matrices:
S =
−1 0 0 00 1 0 00 0 1 00 0 0 1
S∗ =
1 0 0 00 −1 −2 20 −2 −1 20 −2 −2 3
, V =
1 0 0 00 −1 0 00 0 1 00 0 0 1
, E =
1 0 0 00 1/2 1 −1/20 1 −1 10 1/2 −1 3/2
Fq =
−1 0 4 cos(π
q) −4 cos(π
q)
0 1 0 04 cos(π
q) 0 1− 8 cos2(π
q) 8 cos2(π
q)
4 cos(πq
) 0 −8 cos2(πq
) 1 + 8 cos2(πq
)
.Proof. We follow a similar strategy as in the proof of Theorem 3.1. For every q = 3, 4, 5, let [BP ]12 bethe standard polytopal disk packing where P is the regular polyhedron of Schlafli symbol 3, q. Wechoose the flag (v, e, f) satisfying the same conditions as in the proof of Theorem 3.1. Let us now apply atranslation and a reflection if needed so bf becomes the half-space x ≥ 0. Let B3,q be the disk packingobtained after the transformations. Since P is regular, SSA(B3,q) is generated by sv, rv, re, rf , sf whererv, re, rf are the elements in Sym(B3,q) corresponding to the fundamental generators of the symmetricgroup of P with respect to the flag (v, e, f), sf is the inversion on the disk in B∗3,q corresponding to fand 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,• rf is the reflection on the line x = −2 cos(πq ),• sf is the reflection on the line x = 0.
sv
sf
rv
rf
re
sv
sf
rv
rf
re
sv
sf
rv
rf
re
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 centraldisk 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 O↑3,1(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 computationson the matrices. For every n ∈ N the last entry in the diagonal of (S∗V)n and (FqS)n is, respectively,1 + 2n2 and 1 + 8 cos2(πq )n2 and therefore every product has infinite order for every q = 3, 4, 5. Since
SSA(B3,q) = SSA(B∗3,q) then the results also hold for the cube and the dodecahedron. Indeed, thegenerators of SSA(B∗3,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 O↑3,1(K) where K is one of these 3 number fields: Z, Z[√
2] and Z[ϕ]. We illustratesthis for the tetrahedron, octahedron and cube.
Corollary 3.1. The tetrahedral Apollonian group is isomorphic to 〈T1,T2,T3,T4〉 < O↑3,1(Z) where
T4 =
−1 0 0 00 1 0 00 0 1 00 0 0 1
= S T3 =
−1 0 4 −40 1 0 04 0 −7 84 0 −8 9
= F3T4F3
T2 =
−1 2 0 −22 −1 0 20 0 1 02 −2 0 3
= ET3E T1 =
−1 −2 0 −2−2 −1 0 −20 0 1 02 2 0 3
= VT2V
Corollary 3.2. The octahedral Apollonian group is isomorphic to
〈C123,C123,C123,C123,C123,C123,C123,C123〉 < O↑3,1(Z[√
2]) where
C123 =
−1 0 0 00 1 0 00 0 1 00 0 0 1
= S C123 =
−1 0 4√
2 −4√
20 1 0 0
4√
2 0 −15 16
4√
2 0 −16 17
= F4C123F4
C123 =
−1 2√
2 0 −2√
2
2√
2 −3 0 40 0 1 0
2√
2 −4 0 5
= EC123E C123 =
−1 −2√
2 0 −2√
2
−2√
2 −3 0 −40 0 1 0
2√
2 4 0 5
= VC123V
C123 =
−17 6√
2 12√
2 −18√
2
6√
2 −3 −8 12
12√
2 −8 −15 24
18√
2 −12 −24 37
= F4C123F4 C123 =
−17 −6√
2 12√
2 −18√
2
−6√
2 −3 8 −12
12√
2 8 −15 24
18√
2 12 −24 37
= VC123V
C123 =
−17 0 0 −12√
20 1 0 00 0 1 0
12√
2 0 0 17
= EC123E C123 =
−49 0 20√
2 −40√
20 1 0 0
20√
2 0 −15 32
40√
2 0 −32 65
= F4C123F4
Corollary 3.3. The cubical Apollonian group is isomorphic to 〈O1,O2,O3,O3,O2,O1〉 < O↑3,1(Z[√
2])where
O1 =
1 0 0 00 −1 −2 20 −2 −1 20 −2 −2 3
= S∗ O2 =
1 0 0 00 −1 2 −20 2 −1 20 2 −2 3
= VO1V
O3 =
1 0 0 00 1 0 00 0 −1 00 0 0 1
= EO2E O3 =
−15 0 12√
2 −16√
20 1 0 0
12√
2 0 −17 24
16√
2 0 −24 33
= F4O3F4
O2 =
−15 4√
2 4√
2 −12√
2
4√
2 −1 −2 6
4√
2 −2 −1 6
12√
2 −6 −6 19
= EO3E O1 =
−15 −4√
2 4√
2 −12√
2
−4√
2 −1 2 −6
4√
2 2 −1 6
12√
2 6 −6 19
= VO2V
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 maincontribution of this section. We first extend the notion of curvature to any vector x ∈ Ld+1,1 (not onlyfor normalized space-like vectors as in (4)) by
κ(x) = −〈xN ,x〉(15)
where xN = ed+1 + ed+2 and ei denotes the i-th canonical vector of Ld+1,1. Let P ⊂ Ed+1 be anouter-sphere polytope. We define the Lorentzian barycenter of P the vector xP ∈ Ld+1,1 given by
xP :=1
|F0(P)|∑
v∈F0(P)
xb(v)(16)
where b(v) is the illuminated region of v. We define the Lorentzian curvature of P as κP := κ(xP). Bylinearity we have that
κP =1
|F0(P)|∑
v∈F0(P)
κ(b(v))(17)
We notice that every face f of P is also an outer-sphere polytope. We can thus define the Lorentzianbarycenter 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 withSchlafli symbol p1, . . . , pd. The Lorentzian curvatures of any flag (f0, f1, . . . , fd, fd+1 = P) satisfies
κ2P = LP(d+ 1)
d∑i=0
(κfi − κfi+1)2
LP(i+ 1)− LP(i)(18)
where
LP(i) :=
−1 i = 00 i = 1
`−2p1,...,pi−1 if 2 ≤ i ≤ d+ 1
(19)
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.
Lemma 4.1. Let ∆ = (x1, . . . ,xd+2) be a collection of d+ 2 vectors in Ld+1,1. Then ∆ is a basis if andonly if Gram(∆) is non-singular. Moreover, the vector (κ1, . . . , κd+2) where κi := κ(xi) satisfies
(κ1, . . . , κd+2) Gram(∆)−1
κ1
...κd+2
= 0(20)
Proof. Let M be the matrix whose columns are the coodinates of the vectors of ∆ with respect tothe canonical basis of Ld+1,1, N = (0, . . . , 0, 1, 1)T and Qd+2 = diag(1, . . . , 1,−1). We have that(κ1, . . . , κd+2) = (κ(x1), . . . , κ(xd+2)) = −MTQd+2N. Therefore,
(κ1, . . . , κd+2) Gram(∆)−1
κ1
...κd+2
= (−NTQd+2M)(M−1Qd+2(MT )−1)(−MTQd+2N)
= NTQd+2N = 0
Remark 3. As Boyd noticed in [Boy74], the Soddy-Gosset theorem (Theorem 1.2) can be obtaineddirectly by applying the above lemma to the Lorentzian vectors of a packing of d + 2 pairwise tangentd-balls.
17
Let a1, . . . , an be a sequence of real numbers. We define the corner matrix C(a1, . . . , an) as the matrixwith the following shape
C(a1, . . . , an) :=
a1 a2 a3 · · · an−1 ana2 a2 a3 · · · an−1 ana3 a3 a3 · · · an−1 an...
......
. . ....
...an−1 an−1 an−1 · · · an−1 anan an an · · · an an
(21)
If ai 6= ai+1 for each i = 1, . . . , n− 1 then C(a1, . . . , an) is non-singular. It can be easily checked that itsinverse is equal to
1a1−a2
−1a1−a2
0 · · · 0 0 0
−1a1−a2
1a1−a2
+ 1a2−a3
−1a2−a3
. . . 0 0 0
0 −1a2−a3
1a2−a3
+ 1a3−a4
. . .. . . 0 0
.... . .
. . .. . .
. . .. . .
...
0 0. . .
. . . 1an−3−an−2
+ 1an−2−an−1
−1an−2−an−1
0
0 0 0. . . −1
an−2−an−1
1an−2−an−1
+ 1an−1−an
−1an−1−an
0 0 0 · · · 0 −1an−1−an
1an−1−an
+ 1an
Moreover, if Xn = (X1, . . . , Xn) then it can also be checked that
XnC(a1, . . . , an)−1XTn =
n−1∑i=1
((Xi −Xi+1)2
ai − ai+1
)+X2n
an(22)
Lemma 4.2. Let P be a regular edge-scribed (d + 1)-polytope with d ≥ 1. The Lorentzian vectors ofthe vertices of P are contained in the hyperplane xd+2 = 1
`P where `P is the half edge-length of P.
Furthermore, for every f ∈ F(P) we have 〈xf − xP ,xP〉 = 0.
Proof. Since P is regular and edge-scribed the barycenter of P is at the origin of Ed+1. Therefore all itsvertices have the same Euclidean norm and the length of each half edge of P is equal to `P =
√‖v‖2 − 1.
By Equation (8), the Lorentzian vectors of the vertices of P are contained in the hyperplane
H = xd+2 = 1√‖v‖2−1
= 1`P ⊂ Ld+1,1
implying that for every face f of P the Lorentzian vector xf is also in H. Since the barycenter of P isthe origin of Ed+1 then xP = 1
`Ped+2 implying the second part of the Lemma.
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
for any d ≥ 1, Gram(∆) = C(−LP(0),−LP(1), . . . ,−LP(d+ 1)).(23)
The desired equality will then follows from this as explained below.Since Gramians and corner matrices are symmetric then equation (23) is equivalent to
for any d ≥ 1, Gram(∆)i,j = −LP(j − 1) for every 1 ≤ i ≤ j ≤ d+ 2(24)
where Gram(∆)i,j denotes the (i, j)-entry of Gram(∆). We shall consider three cases according to thevalue of j in (24):
(i) j = 1, 2. Let v = f0, e = f1 and v′ be the other vertex of e. Then,
• Gram(∆)1,1 = 〈xv,xv〉 == 〈xb(v),xb(v)〉 = 1 = −LP(0).
• Gram(∆)1,2 = 〈xv,xe〉 = 〈xv, 12 (xv + xv′)〉 = 1
2 (1− 1) = 0 = −LP(1).18
• Gram(∆)2,2 = 〈xe,xe〉 = 〈 12 (xv + xv′),12 (xv + xv′)〉 = 1
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 impliesthat xP = `−1
p1,...,pded+2 where ed+2 is the (d+ 2)-th vector of the canonical basis of Ld+1,1. Therefore,
• Gram(∆)d+2,d+2 = 〈xP ,xP〉 = −`−2p1,...,pd = −LP(d+ 1).
The second part of Lemma 4.2 implies that for each i = 1, . . . , d+ 2 we have
• Gram(∆)i,d+2 = 〈xfi−1,xP〉 = 〈xP ,xP〉 = −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, weshall 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 ∈ Mob(Rd) given by the composition of the following two Mobiustransformations:
- A rotation of Sd which sends the barycenter of fd to the line spanned by the North pole of Sd.- The Mobius transformation corresponding to a rescaling of Rd, which sends the rotated S to the
equator of Sd.By identifying the hyperplane xd+1 = 0 ⊂ Ed+1 with Ed we have that µ(fd) becomes an edge-scribed
realization fd of the regular d-polytope with Schlafli symbol p1, . . . , pd−1. Moreover, since P is regular
fd is also regular. We can then apply again Lemma 4.2 to obtain
〈xfi−1,xfd〉 = −`−2
p1,...,pd−1 for every 1 ≤ i ≤ d+ 1
where fi is the i-polytope obtained from µ(fi) after the identification xd+1 = 0 ' Ed. This iden-tification preserves the inversive product of the illuminated regions of the outer-sphere points lying inxd+1 = 0. Since the inversive product is also invariant under Mobius transformations then, for eachi = 1, . . . , d+ 1, we have:
• Gram(∆)i,d+1 = 〈xfi−1 ,xfd〉 = 〈xfi−1,xfd〉 = −LP(d).
Then, to prove (24) for j = d we apply the same arguments as above by considering this time fd−1
as a facet of the regular edge-scribed d-polytope fd. By carrying on these arguments for the rest of thevalues 3 ≤ j ≤ d+ 2 we obtain (24) which gives (23).
Finally, since the sequence (−LP(0),−LP(1), . . . ,−LP(d + 1)) = (1, 0,−`−2p1, . . . ,−`
−2p1,...,pd) is
strictly decreasing then Gram(∆) is non-singular. Therefore, by applying the Lemma 4.1 to ∆ weobtain
0 = (κf0 , . . . , κfd+1) Gram(∆)−1(κf0 , . . . , κfd+1
)T
= (κf0 , . . . , κfd+1)C(−LP(0), . . . ,−LP(d+ 1))−1(κf0 , . . . , κfd+1
)T
=
d+1∑i=1
((κfi − κfi−1
)2
LP(i)− LP(i− 1)
)− κ2
PLP(d+ 1)
as desired.
We notice that Equation (18) is invariant under Mobius transformations. By combinining the previoustheorem 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:
Corollary 4.1. For any flag (f0, f1, . . . , fd, fd+1 = T d+1) of an edge-scribed realization of a (d + 1)-simplex with d ≥ 1 the following holds
κ2T d+1 =
d
d+ 2
d∑i=0
(i+ 2
2
)(κfi − κfi+1
)2.(25)
The latter can be used to obtain the curvatures in a d-ball packing BT where T is the (d+1)-simplex.Indeed, if we replace in the previous equation κfi = 1
i+1 (κ1 + · · · + κi+1) for every 0 ≤ i ≤ d + 1 weobtain the Soddy-Gosset theorem.
Corollary 4.2. For any flag (f0, f1, . . . , fd, fd+1 = Cd+1) of an edge-scribed realization of a (d+ 1)-cubewith d ≥ 2 the following holds
κ2Cd+1 = d
d∑i=0
(κfi − κfi+1)2.(26)
19
5. Integrality of the Platonic Apollonian packings
In this last section, we shall discuss and present numerous consequences of Theorem 4.1 for polytopaldisk packings based on the Platonic solids. Polytopal 3-ball packings based on the regular 4-polytopeswill be explored in a future work [RR].
Theorem 5.1. Let P and edge-scribed realization of a Platonic solid with Schlafli symbol p, q. TheLorentzian curvatures of any flag (v, e, f,P) satisfies
κ2P =
cos2(πp )
sin2(πq )− cos2(πp )(κv − κe)2 +
sin2(πp )
sin2(πq )− cos2(πp )(κe − κf )2 +
sin2(πp )
cos2(πq )(κf − κP)2.(27)
Moreover, the following relations hold:
- Relation between two consecutive vertices:
κv + κv′
2= κe(28)
where v′ is the vertex in F0(P) \ v satisfying v′ ⊂ e.- Relation between two consecutive edges:
κe + κe′
2= cos(πp )2κv + sin(πp )2κf(29)
where e′ is the edge in F1(P) \ e satisfying v ⊂ e′ ⊂ f .- Relation between two consecutive faces:
κf + κf ′
2=
cos2(πq )
sin2(πp )κe +
sin2(πq )− cos2(πp )
sin2(πp )κP(30)
where f ′ is the face in F2(P) \ f satisfying e ⊂ f ′ ⊂ P.- Relation between two consecutive polyhedra:
κP + κP′
2=
sin2(πp )
sin2(πp )− cos2(πq )κf(31)
where P ′ = sf (P) where sf is the projective transformation preserving S2 which corresponds tothe inversion on the disk bf ∈ B∗P .
Proof. By Theorem 2.1, every polyhedron is Mobius unique. Since Equation (18) is invariant underMobius transformations 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
LP(2) = cot2(πp ) and LP(3) =cos2(πp )
sin2(πq )− cos2(πp )
and this gives (27). Equation (28) can be obtained directly by the definition of κe. By resolving thequadratic equation (27) for κe, κf , and then for κP , we obtain two solutions in each of the three cases. Inthe first two cases, one of the solutions is the Lorentzian curvature of e and f , respectively, and the othersolution is the Lorentzian curvature of e′ = re(e) and f ′ = rf (f), where re and rf are the fundamentalgenerators of the symmetric group of P with respect to the flag (v, e, f,P). In the last case, one of thesolution is the Lorentzian curvature of P and the other solution, by Corollary 2.1, is the Lorentziancurvature of P ′ = sf (P). Equations (29), (30) and (31) follow by adding both solutions in each of thethree cases.
Corollary 5.1. Let BP a polytopal disk packing where P is an edge-scribed regular polyhedron. All thecurvatures of the Apollonian packing of BP can be obtained from the curvatures of three consecutivetangent disks of BP corresponding to three vertices of P lying in a same face.
Proof. Let p, q the Schlafli symbol of P and let κi−1, κi and κi+1 the curvatures of three consecutivetangent disks bi−1, bi, bi+1 ∈ BP . Let f be the face of P containing the the vertices vi−1, vi, vi+1 corre-sponding to three disks and let e and e′ be the edges with vertices vi−1, vi and vi, vi+1, respectively. By
20
replacing κe = 12 (κi + κi+1) and κ(e′) = 1
2 (κi + κi−1) in Equation (29) we obtain
κf =1
4 sin2(πp )(κi+1 + κi−1) + (1− 1
2 sin2(πp ))κi(32)
This equation holds for every three consecutive vertices of f and, therefore, we can obtain the curvaturesof the remaining vertices in f in terms of κi−1, κi and κi+1. By replacing in (27) κv by κi, κe by12 (κi + κi+1) and κf by the right-hand-side of (32) and then resolving the quadratic equation for κP weobtain
κP± =− cos( 2π
p )κi + 12 (κi+1 + κi−1)± cos(πq )
√(1− 4 cos2(πp ))κ2
i + κiκi+1 + κiκi−1 + κi+1κi−1
2(sin2(πq )− cos2(πp )).(33)
Corollary 2.1 implies that one of the two solutions is κP and the other is the Lorentzian curvature ofP ′ = sf (P), where sf is the inversion on the boundary of bf ∈ B∗P . Once that κP is fixed, we can obtainthe curvatures of the disks corresponding to the vertices of P by applying recursively (33). If, instead,we fix κ(P ′), we obtain the curvatures of the disks corresponding to the vertices of P ′. By changing facesand 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 ofevery flag (v, e, f,O) of an edge-scribed octahedron O satisfies
κ2O = (κv − κe)2 + 3(κe − κf )2 +
3
2(κf − κO)2(34)
By replacing κv = κ1, κe = 12 (κ1 + κ2) and κf = 1
3 (κ1 + κ2 + κ3) in the previous equation we obtainthe generalization of the Descartes theorem given by Guettler and Mallows where κO plays the role ofw (see [GM08]). The antipodal relation can be obtained from the relation (30). From Equation (33) weobtain the Lorentzain curvatures of two octahedra sharing a triangle f in terms of the curvatures of thevertices of f are given by
κO± = κ1 + κ2 + κ3 ±√
2(κ1κ2 + κ2κ3 + κ1κ3).(35)
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
an octahedral disk packing BO. If κ1, κ2, κ3 and√
2(κ1κ2 + κ1κ3 + κ2κ3) are integers then Ω(BO) isintegral.
455
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0
0Figure 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 par-
ticular case of the Schmidt arrangement Q[√
2]. It also appeared at the end of [Bol+18]. From Corollary4.2 we have that the Lorentzian curvatures of every flag (v, e, f, C) of an edge-scribed cube C satisfies
κ2C = 2(κv − κe)2 + 2(κe − κf )2 + 2(κf − κC)2.(36)
By combining (32) with the definition of κf we obtain a relationship of the curvatures involving thevertices of a square-face
κi + κi+2 = κi+1 + κi+3(37)
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 and v at distance 3 in the graph of Cby
κC =κv + κv
2(38)
With Equations (37) and (38) we can compute the curvatures of the vertices of C from the curvaturesof four vertices not lying all in a same face. The Lorentzian curvatures of two edge-scribed cubes C+ andC− sharing a square f in terms of the curvatures of the vertices of three consecutives vertices of f are
κC± = κi+1 + κi−1 ±√−κ2
i + κiκi+1 + κiκi−1 + κi−1κi+1(39)
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
disk packing BC. If κi−1, κi, κi+1 and√−κ2
i + κiκi+1 + κiκi−1 + κi−1κi+1 are integers then Ω(BC) isintegral.
5
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257486
493
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304 421
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322478
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257486
493
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118368
489
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489182 317
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489182317
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220389
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262469
477
300308
300308
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358
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358
404
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404
412
462
470
462
470
-3
0
0Figure 12. Two cubical Apollonian packings with initial curvatures (−3, 5, 12) (left) and(0, 0, 1) (right).
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5.1.3. Icosahedral Apollonian packings. Icosahedral Apollonian packings appeared in [Bol+18] as a par-ticular case of Apollonian ring packings when n = 5. By applying Theorem 5.1 with p = 3 and q = 5 weobtain that the Lorentzian curvatures of every flag (v, e, f, I) of an edge-scribed icosahedron I satisfies
κ2I = ϕ2(κv − κe)2 + 3ϕ2(κe − κf )2 + 3ϕ−2(κf − κI)2(40)
For any two consecutive faces f and f ′ of I sharing an edge e we have
κf + κf ′
2=
1
3(ϕ2κe + ϕ−2κI)(41)
The latter implies the following relation between the Lorentzian curvatures of the two vertices u ∈ f \ eand u′ ∈ f ′ \ e by
κu + κu′
2= ϕ−1κe + ϕ−2κI(42)
By applying recursively this equation we can obtain the antipodal relation between two vertices v and vat distance 3 in the graph of I by
κI =κv + κv
2(43)
As above, the Lorentzian curvatures of all the vertices of I can be easily computed from the curvaturesof four vertices of I. In order to obtain the curvatures of the icosahedral Apollonian packing we canuse 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
κI± = ϕ2(κ1 + κ2 + κ3)± ϕ3√κ1κ2 + κ1κ3 + κ2κ3.(44)
The above equality leads to a sufficient condition for the curvatures of a dodecahedral Apollonianpacking 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 diskpacking BI . If κ1, κ2, κ3 and
√κ1κ2 + κ1κ3 + κ2κ3 are in Z[ϕ] then Ω(BI) is ϕ-integral.
φ
2φ4φ+3
5φ+4
2φ+24φ - 1
3φ - 26φ
8φ+3
7φ+26φ +2
8φ+3
9 φ +4
6φ +2
4φ+3
3φ+2
6 φ+4
12φ+711 φ+6
10φ+6
6 φ+4
6φ+6
4φ+3
6φ - 2
6φ - 4
8φ - 112φ+3
11φ +2
9 φ +4
14φ+8
15φ+6
16 φ +11
14φ+8
9 φ +4
12φ+3
14 φ+6
15 φ +10
14 φ+2
13 φ17φ+4
14φ+8
18φ +10
12φ+7
10φ+6
6 φ+4
12φ+7
18φ +10
13 φ+8
12φ+7
13 φ +12
10φ+6
12 φ - 1
11φ - 6
14φ
18φ +10
20φ - 1
17 φ - 4
22 φ +2
11 φ+6
22 φ +2
18 φ - 8
-1
0
0Figure 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 theintroduction, we obtain that every triple of integers which produces an integral tetrahedral Apollonianpacking produces also a ϕ-integral icosahedral Apollonian packing. We show in Figure 14 the integraltetrahedral and the ϕ-integral icosahedral Apollonian packings given by the curvatures (−4, 8, 9).
23
8
99
17
56
20
1772
36
33
137
80
72
140
41
36 137
41
33
161
128
120
188
65
84
185
89
57
252
176
161
144
128393
144
120
396
209
188
260
72
65
396
105
84
393
209
185
105
89
252
72
57
284
272
257
224
272
288
200
329
332
356
104
161
225
164
353
305
185
209
348
168
89
308
284
305
272
305
257
248
224
336
272
336
288
248
200
368
329
332
396
356
113
104
201
161
264
225
209
164
353
368
305
209
185
264
209
396
348
201
168
113
89
360
312
153
264
329
276
321
369
392
273
129
392
360
392
312
164
153
329
264
329
353
276
353
321
369
392
329
273
164
129
212
393
177
225
212
393
225
177
281
233
296
281
296
233
360
297
377
360
377
297369369
-4 -4Figure 14. An integral tetrahedral (left) and a ϕ-integral icosahedral (right) Apollonianpackings 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 thatthe Lorentzian curvatures of every flag (v, e, f,D) of an edge-scribed dodecahedron D satisfies
κ2D = ϕ4(κv − κe)2 + (2 + ϕ)(κe − κf )2 + (ϕ−2 + 1)(κf − κD)2.(45)
By combining (32) with the definition of κf we obtain that the curvatures of the vertices of a pentagonalface are related by
ϕ(κi+1 − κi) = κi+2 − κi−1(46)
where the indices are given in cyclic order. Moreover, by combining the relations between consecutiveedges and consecutives faces we can express the Lorentzian curvature of D in terms of the curvature ofa vertex v and its three neighbours u1, u2 and u3 by
κD =ϕ2
2(κu1 + κu2 + κu3)− 1 + 3ϕ
2κv.(47)
The previous two equations allow to compute the curvatures of all the vertices of D from the curvaturesof a vertex and its three neighbours.
Equation (33) gives the Lorentzian curvatures of the two edge-scribed dodecahedra D+ and D− sharinga pentagon f , in terms of the curvatures of the vertices of three consecutives vertices of f , by
κD± = ϕκi + ϕ2
(κi+1 + κi−1 ±
√−ϕκ2
i + κiκi+1 + κiκi−1 + κi−1κi+1
).(48)
Proposition 5.4. Let κi−1, κi, κi+1 be the curvatures of three consecutive tangent disks in a dodecahedral
disk packing BD. If κi−1, κi, κi+1 and√−ϕ2κ2
i + κiκi+1 + κiκi−1 + κi−1κi+1 are in Z[ϕ] then Ω(BD) isϕ-integral.
24
-1
0
0Figure 15. Two dodecahedral Apollonian packings with initial curvatures (−1, ϕ+ 1, 2ϕ)(left) and (0, 0, 1) (right).
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IMAG Univ. Montpellier, CNRS, FranceEmail address: [email protected]
IMAG Univ. Montpellier, CNRS, FranceEmail address: [email protected]
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