2.1. The pseudosphere in Lorentz space

Hyperspheres in ${\mathbb{R}}^n$ can be represented by $(n+2)$ -dimensional vectors. Boyd calls such coordinates polyspherical coordinates and attributes them to Clifford and Darboux from the late 19th century [Reference Boyd6]. The more modern interpretation is that they represent planes in ${\mathbb{H}}^{n+1}$ imbedded in an $(n+2)$ -dimensional Lorentz space, which in turn represent hyperspheres on the boundary $\partial {\mathbb{H}}^{n+1}$ . We refer the reader to [Reference Ratcliffe20] for more details.

Let us set $N=n+2$ .

Given a symmetric matrix J with signature $(1,N-1)$ , we define the Lorentz space, ${\mathbb{R}}^{1,N-1} $ to be the set of N-tuples over ${\mathbb{R}}$ equipped with the negative Lorentz product

\begin{equation*}\mathbf u\cdot \mathbf v=\mathbf u^TJ\mathbf v.\end{equation*}

The surface $\mathbf x\cdot \mathbf x=1$ is a hyperboloid of two sheets. Let us distinguish a vector D with $D\cdot D>0$ and select the sheet ${\mathcal H}$ by

\begin{equation*}{\mathcal H}: \qquad \mathbf x\cdot \mathbf x=1, \qquad \mathbf x\cdot D>0.\end{equation*}

We define a distance on ${\mathcal H}$ by

\begin{equation*}\cosh\!(|AB|)=A\cdot B.\end{equation*}

Then ${\mathcal H}$ equipped with this metric is a model of ${\mathbb{H}}^{N-1}$ , sometimes known as the vector model. Equivalently, one can define

\begin{equation*}V^+=\{\mathbf x\in {\mathbb{R}}^{1,N-1}\;:\; \mathbf x\cdot \mathbf x>0, \mathbf x\cdot D>0\}\end{equation*}

and ${\mathcal H}=V^+/{\mathbb{R}}^+$ , together with the metric defined by

\begin{equation*}\cosh\!(|AB|)=\frac{A\cdot B}{|A||B|},\end{equation*}

where $|\mathbf x|=\sqrt{\mathbf x\cdot \mathbf x}$ for $\mathbf x\in V$ . For $\mathbf x\cdot \mathbf x<0$ , we define $|\mathbf x|=i\sqrt{-\mathbf x\cdot \mathbf x}$ .

Hyperplanes in ${\mathcal H}$ are the intersection of ${\mathcal H}$ with hyperplanes $\mathbf n\cdot \mathbf x=0$ in ${\mathbb{R}}^{1,N-1}$ . Such a plane intersects ${\mathcal H}$ if and only if $\mathbf n\cdot \mathbf n<0$ . Let $H_{\mathbf n}$ represent both the plane $\mathbf n\cdot \mathbf x=0$ in ${\mathbb{R}}^{1,N-1}$ and its intersection with ${\mathcal H}$ . The direction of $\mathbf n$ distinguishes a half space

\begin{equation*}H_{\mathbf n}^+=\{\mathbf x\;:\; \mathbf n\cdot \mathbf x>0\},\end{equation*}

in either ${\mathbb{R}}^{1,N-1}$ or ${\mathcal H}$ .

The angle $\theta$ between two intersecting planes $H_{\mathbf n}$ and $H_{\mathbf m}$ in ${\mathcal H}$ is given by

(2.1) \begin{equation}|\mathbf n||\mathbf m|\cos \theta=\mathbf n\cdot \mathbf m,\end{equation}

where $\theta$ is the angle in $H_{\mathbf n}^+\cap H_{\mathbf m}^+$ . If $|\mathbf n\cdot \mathbf m|=|\,|\mathbf n||\mathbf m|\,|$ , then the planes are parallel (i.e. tangent at infinity). If $|\mathbf n\cdot \mathbf m|>|\,|\mathbf n||\mathbf m|\,|$ , then the planes are ultraparallel, and the quantity $\psi$ in $|\mathbf n||\mathbf m|\cosh \psi=|\mathbf n\cdot \mathbf m |$ is the shortest hyperbolic distance between the two planes.

The group of isometries of ${\mathcal H}$ is given by

\begin{equation*}{\mathcal O}^+({\mathbb{R}})=\{T\in M_{N\times N}\;:\; \hbox{$T\mathbf u\cdot T\mathbf v=\mathbf u\cdot \mathbf v$ for all $\mathbf u, \mathbf v\in {\mathbb{R}}^{1,N-1}$, and $T{\mathcal H}={\mathcal H}$.}\}.\end{equation*}

Reflection in the plane $H_{\mathbf n}$ is given by

\begin{equation*}R_{\mathbf n}(\mathbf x)={\mathbf x-2}\mathop{\textrm{proj}}\nolimits_{\mathbf n}\!({\mathbf x})={\mathbf x-2}\frac{{\mathbf n}\cdot {\mathbf x}}{{\mathbf n}\cdot {\mathbf n}}{\mathbf n}.\end{equation*}

The group of isometries is generated by the reflections.

Let $\partial {\mathcal H}$ represent the boundary of ${\mathcal H}$ , which is an $(N-2)$ -sphere. It is represented by ${\mathcal L}^+/\Bbb R^+$ where

Given an $E\in{\mathcal L}^+$ , let $\partial {\mathcal H}_E=\partial {\mathcal H} \setminus E{\mathbb{R}}^+$ . Then $\partial {\mathcal H}_E$ equipped with the metric $|\cdot |_E$ defined by

\begin{equation*}|AB|_E^2=\frac{2A\cdot B}{(A\cdot E)(B\cdot E)}\end{equation*}

is the $(N-2)$ -dimensional Euclidean space that is the boundary of the Poincaré upper half hyperspace model of ${\mathcal H}$ with E the point at infinity. In $\partial {\mathcal H}_E$ , the plane $H_{\mathbf n}$ is represented by an $(N-3)$ -sphere, which we denote with $H_{\mathbf n,E}$ (or just $H_{\mathbf n}$ if E is understood, or sometimes just $\mathbf n$ ).

The curvature (the inverse of the radius, together with a sign) of $H_{\mathbf n, E}$ is given by the formula

\begin{equation*}\frac{\mathbf n\cdot E}{||\mathbf n||}\end{equation*}

using the metric $|\cdot |_E$ [Reference Baragar4]. Here, $||\mathbf n||=-i|\mathbf n|=\sqrt{-\mathbf n\cdot \mathbf n}$ . By choosing a suitable orientation for $\mathbf n$ , we get the appropriate sign for the curvature.

2.2. Coxeter graphs

Coxeter graphs can represent lattices in ${\mathbb{R}}^n$ or ${\mathbb{R}}^{1,n-1}$ , or groups of isometries of ${\mathbb{S}}^{n-1}$ or ${\mathbb{H}}^{n-1}$ . Each node in a Coxeter graph represents a plane, which can be represented by its normal vector. Two nodes are not connected if their corresponding planes are perpendicular. A regular edge between two nodes indicates that those planes intersect at an angle of $\pi/3$ , so their normal vectors are at an angle of $2\pi/3$ (some ambiguity here). If the angle between two planes is $\pi/4$ , then we indicate that with an edge subscripted (or superscripted) with a 4, or sometimes a double edge. In general, an edge with a superscript of m (or of multiplicity $m-2$ ) means the order of the composition of reflections in the two planes represented by the two nodes is m. A thick line, like the one in (1.1), means the two planes are parallel, and this is sometimes indicated with a superscript of $\infty$ . A dotted edge means the two planes are ultraparallel.

The Coxeter graph represents the lattice $\mathbf v_1{\mathbb{Z}}\oplus \cdots \oplus \mathbf v_n{\mathbb{Z}}$ , where the $\mathbf v_i$ are the nodes of the graph. Note that not all graphs give lattices, as the vectors may be linearly dependent. When the nodes are linearly independent (and sometimes when they are not), we get the bilinear form defined by the incidence matrix $[\mathbf v_i\cdot \mathbf v_j]$ . We can normalize the vectors $\mathbf v_i$ (say) so that $\mathbf v_i\cdot \mathbf v_i=-1$ . Then the angles noted by the edges define $\mathbf v_i\cdot \mathbf v_j$ , where we take the angle between the normal vectors of the planes to be obtuse (so $\mathbf v_i\cdot\mathbf v_j\geq 0$ ).

The Coxeter graph can also represent a group, the Weyl group $\langle R_{\mathbf v_1},...,R_{\mathbf v_n}\rangle$ . When representing a group, it is often necessary for the vectors to be linearly dependent.

We will explain the notion of weights in the next subsection, which will give us an example to investigate.

2.3. The Apollonian circle packing

The Apollonian circle packing is a well- known object, and we assume the reader is already familiar with it. The goal of this section is to think of it in a way that more naturally generalizes to higher dimensions.

Let us consider the strip version of the Apollonian circle packing shown in Figure 1, and think of the picture as lying on the boundary $\partial {\mathbb{H}}^3$ of hyperbolic space. Then each circle (and the two lines) represents a plane in ${\mathbb{H}}^3$ , which in turn are represented by their normal vectors. Let us represent four of the hyperbolic planes with the vectors $\mathbf e_i$ for $i=1,...,4$ , as shown in Figure 1. We orient $\mathbf e_i$ so that $H_{\mathbf e_i}^+$ contains $H_{\mathbf e_j}$ for $i\neq j$ , and assign them norms of $-2$ , meaning $\mathbf e_i\cdot \mathbf e_i=-2$ . Then, by the tangency conditions, $\mathbf e_i\cdot \mathbf e_j=2$ for $i\neq j$ (see equation (2.1)). We define $J_4=[\mathbf e_i\cdot \mathbf e_j]$ , which has signature (1, 3). (It is easy to see that 4 and $-4$ are eigenvalues, the latter with a 3-dimensional eigenspace.) Thus, $J_4$ defines a Lorentz product in ${\mathbb{R}}^{1,3}$ .

Figure 1. The strip version of the Apollonian packing. The dotted curves $\alpha_i$ represent symmetries of the packing. The fundamental domain ${\mathcal F}_4$ for the group ${\mathcal O}^+_{\Lambda_4}$ is the region above the shaded square and above the plane $H_{\alpha_3}$ (represented by the dotted circle).

Note that $\mathbf e_i\cdot(\mathbf e_i+\mathbf e_j)=0$ , so $\mathbf e_i+\mathbf e_j$ is the point of tangency between the circles represented by $\mathbf e_i$ and $\mathbf e_j$ . Thus, our point at infinity is $E=\mathbf e_1+\mathbf e_2$ , and the circle represented by $\mathbf e_i$ is $H_{\mathbf e_i,E}$ .

There are obvious symmetries of the Apollonian packing, as shown in Figure 1. In the plane, the symmetries are reflection in the lines $\alpha_1$ , $\alpha_2$ , and $\alpha_4$ and inversion in the circle $\alpha_3$ . The Apollonian packing is the orbit of $\mathbf e_1$ under the action of the group generated by these symmetries.

Thought of as actions in ${\mathbb{H}}^3$ , the symmetries are reflection in the planes $H_{\alpha_i}$ , so the packing is the ${\Gamma}_4$ -orbit of $H_{\mathbf e_1}$ , where

\begin{equation*}{\Gamma}_4=\langle R_{\alpha_1}, R_{\alpha_2}, R_{\alpha_3}, R_{\alpha_4}\rangle.\end{equation*}

It is fairly easy to solve for $\alpha_i$ in the basis $\mathbf e=\{\mathbf e_1,...,\mathbf e_4\}$ . For example, we note that $\alpha_1\cdot \mathbf e_i=0$ for $i\neq 4$ , since it is perpendicular to those planes. Solving, we get $\alpha_1=(1,1,1,-1)$ , up to scalars. The others are $\alpha_2=(0,0,-1,1)$ , $\alpha_3=(0,-1,1,0)$ , and $\alpha_4=(\!-1,1,0,0)$ .

We let $\Lambda_4=\mathbf e_1{\mathbb{Z}}\oplus ...\oplus \mathbf e_4{\mathbb{Z}}$ , which we call the Apollonian lattice. Let

\begin{equation*}{\mathcal O}^+_{\Lambda_4}=\{T\in {\mathcal O}^+({\mathbb{R}})\;:\; T\Lambda_4=\Lambda_4\}.\end{equation*}

It is straight forward to verify that $R_{\mathbf e_i}$ and $R_{\mathbf \alpha_i}\in {\mathcal O}^+_{\Lambda_4}$ . Let $G_4=\langle {\Gamma}_4, R_{\mathbf e_1}\rangle$ . A fundamental domain ${\mathcal F}_4$ for this group is the region bounded by the four planes $H_{\alpha_i}$ , the pane $H_{\mathbf e_1}$ , and with cusp E at the point at infinity:

\begin{equation*}{\mathcal F}_4=H^+_{\alpha_1}\cap \cdots \cap H^+_{\alpha_4} \cap H^+_{\mathbf e_1}.\end{equation*}

Since ${\mathcal F}_4$ has finite volume, we know $G_4$ has finite index in ${\mathcal O}^+_{\Lambda_4}$ , and it is not hard to verify that the two are equal.

Let us also use this example to learn a little about Coxeter graphs. The Coxeter graph for the planes that bound ${\mathcal F}_4$ is the following:

(2.2)

We note, from our earlier calculations, that $\alpha_i\cdot \alpha_i=-8$ . From the Coxeter graph, we therefore get $\alpha_1\cdot \alpha_2=8$ , $\alpha_2\cdot \alpha_3=4$ , $\alpha_3\cdot \alpha_4=4$ , and all other products $\alpha_i\cdot \alpha_j$ for $i<j$ are zero. This gives us the matrix

\begin{equation*}J_\alpha=[\alpha_i\cdot \alpha_j]=\left[\begin{array}{lccccccccc}-8&8&0&0 \\ 8&-8&4&0 \\ 0&4&-8&4 \\ 0&0&4&-8\end{array}\right].\end{equation*}

Our Lorentz product, in this basis, is $\mathbf x \cdot \mathbf y=\mathbf x^tJ_{\alpha}\mathbf y$ . Let $\Lambda_{\alpha}=\alpha_1{\mathbb{Z}}\oplus \cdots \oplus \alpha_4{\mathbb{Z}}$ , so $\Lambda_{\alpha}\subset \Lambda_{4}$ . Since $\det\!(J_\alpha)=4\det\!(J_4)$ , the index is two.

The fundamental domain ${\mathcal F}_4$ has five vertices: The point E at infinity, and the four vertices on $H_{\alpha_3}$ , the hemisphere above the dotted circle (see Figure 1). Let $w_i$ be the intersection of the three $\alpha_j$ with $j\neq i$ , so $w_i$ is a vertex of ${\mathcal F}_4$ for $i=1$ and 2, and $w_3$ is the point at infinity. The $w_i$ ’s satisfy $w_i\cdot \alpha_j=0$ for $i\neq j$ , so in the basis $\alpha=\{\alpha_1,...,\alpha_4\}$ , are the rows of $J_{\alpha}^{-1}$ , up to scalars. To express these points in the basis $\mathbf e$ , we multiply on the left by the change of basis matrix $Q=[\alpha_1|\alpha_2|\alpha_3|\alpha_4]$ whose columns are $\alpha_i$ ; the columns of the resulting matrix are the $w_i$ ’s. The $w_i$ ’s are called weights. The incidence matrix for the $w_i$ s is $[w_i\cdot w_j]=J_\alpha^{-1}J_\alpha (J_\alpha^{-1})^t=(J_\alpha^{-1})^t=J_\alpha^{-1}$ . The first and second diagonal elements of $J_\alpha^{-1}$ are positive, indicating that $w_1$ and $w_2$ lie in ${\mathcal H}$ . The third diagonal element is 0, indicating $w_3$ lies on $\partial {\mathcal H}$ , so is a cusp (it is the point E at infinity). The last diagonal element is negative, so $w_4$ represents a plane in ${\mathcal H}$ . That plane is perpendicular to $H_{\alpha_1}$ , $H_{\alpha_2}$ , and $H_{\alpha_3}$ , so is the plane $H_{\mathbf e_1}$ .

The other two vertices of ${\mathcal F}_4$ can be found in a similar way by replacing $\alpha_4$ with $\mathbf e_1$ . This is plain to see from the picture (see Figure 1). Combinatorially, the intersection of any three of the five faces should give us a potential vertex, but since $H_{\alpha_4}$ and $H_{\mathbf e_1}$ are parallel, any combination that includes those two will give at most their point of tangency.

2.4. Sphere packings

A sphere packing in ${\mathbb{R}}^n$ is a configuration of oriented $(n-1)$ -spheres that intersect tangentially or not at all. By oriented, we mean each sphere includes either the inside (a ball) or the outside. The trivial sphere packing is a sphere and its complement. We will not consider trivial sphere packings.

Maxwell calls ${\mathcal{P}}\subset {\mathbb{R}}^{1,N-1}$ a packing if for all $\mathbf n, \mathbf n'\in {\mathcal{P}}$ , there exists a positive constant k such that $\mathbf n\cdot \mathbf n=-k$ and $\mathbf n\cdot \mathbf n'\geq k$ [Reference Maxwell16]. Given a point E for the point at infinity, the packing ${\mathcal{P}}$ defines a sphere packing

\begin{equation*}{\mathcal{P}}_E=\bigcup_{\mathbf n\in {\mathcal{P}}}H_{\mathbf n,E}^-\subset \partial {\mathcal H}_E \cong {\mathbb{R}}^{n}.\end{equation*}

We call ${\mathcal{P}}_E$ a perspective of ${\mathcal{P}}$ .

For example, the Apollonian packing in ${\mathbb{R}}^{1,3}$ is ${\mathcal{P}}_4={\Gamma}_4(\mathbf e_1) $ (the ${\Gamma}_4$ -orbit of $\mathbf e_1$ ) and the strip packing of Figure 1 is ${\mathcal{P}}_{4,\mathbf e_1+\mathbf e_2}$ . An Apollonian packing derived from a different initial cluster of four mutually tangent circles is a different perspective of ${\mathcal{P}}_4$ .

We think of ${\mathcal{P}}$ as defining a cone

\begin{equation*}{\mathcal K}_{{\mathcal{P}}}=\bigcap_{\mathbf n\in {\mathcal{P}}}H_{\mathbf n}^+,\end{equation*}

in ${\mathbb{R}}^{1,N-1}$ , or a polyhedron

\begin{equation*}{\mathcal K}_{{\mathcal{P}}}\cap {\mathcal H},\end{equation*}

in ${\mathbb{H}}^{N-1}$ . The residual set of a sphere packing is the complement of the sphere packing in $\partial {\mathcal H}_E\cong {\mathbb{R}}^n$ and is the intersection of ${\mathcal K}_{{\mathcal{P}}}$ with $\partial {\mathcal H}_E$ . A sphere packing is maximal or dense if there is no space in the residual set where one can place another sphere of positive radius. It is complete if the residual set is of measure zero. A packing that is maximal (respectively complete) in one perspective is maximal (complete) in any perspective.

We call a packing of lattice type if ${\mathcal{P}}{\mathbb{Z}}$ forms a lattice in ${\mathbb{R}}^{1,N-1}$ [Reference Maxwell16]. We call a packing of general lattice type if there exists a lattice $\Lambda\subset {\mathbb{R}}^{1,N-1}$ so that a scalar multiple of $\mathbf n$ is in $\Lambda$ for all $\mathbf n\in {\mathcal{P}}$ , and ${\mathcal{P}}$ spans ${\mathbb{R}}^{1,N-1}$ .

A subgroup ${\Gamma} \leq {\mathcal O}_\Lambda^+$ is called geometrically finite if it has a convex fundamental domain with a finite number of faces. A packing is called crystallographic if there exists a geometrically finite group ${\Gamma}\leq {\mathcal O}_{\Lambda}^+$ and a finite set $S\subset \Lambda$ so that

\begin{equation*}{\mathcal{P}}=\{\gamma(\mathbf n)/|\gamma(\mathbf n)|\;:\; \gamma\in {\Gamma}, \mathbf n\in S\}.\end{equation*}

As the normality condition is not necessary, we will write ${\mathcal{P}}={\Gamma}(S)$ , the ${\Gamma}$ -orbit of S. The residual set for the packing is the limit set of ${\Gamma}$ .

Crystallographic packings are not always of lattice type, as was noted by Maxwell.

A crystallographic packing that is maximal is known to be complete. Furthermore, the Hausdorff dimension of the residual set is strictly less than $N-2$ [Reference Sullivan22].

The packings that are known as Boyd-Maxwell packings are crystallographic packings with the restriction that ${\Gamma}$ be a reflective group (i.e. is generated by a finite number of reflections). Kontorovich and Nakamura include this requirement in their definition of a crystallographic packing [Reference Kontorovich and Nakamura13].

We say a packing has the weak Apollonian property if it contains a cluster of $N=n+2$ mutually tangent spheres. We say it has the Apollonian property if every sphere is a member of a cluster of N mutually tangent spheres. We call a packing Apollonian if it is crystallographic, maximal, and has the Apollonian property.

Missing in this definition is a notion of efficiency. It is conjectured that the residual set for the Apollonian circle packing in two dimensions has minimal Hausdorff dimension among all nontrivial circle packings [Reference Melzak18], so an alternative definition would be that a packing is Apollonian if it is optimally efficient in that dimension. In ${\mathbb{R}}^7$ , there exists a weakly Apollonian packing that is more efficient than a known Apollonian packing (see Section 3.3), suggesting that there is still much to learn about the subject.

Remark 2. In [Reference Baragar4], we give a different definition for an Apollonian packing. We begin with a set $\{\mathbf e_1,...,\mathbf e_{N}\}$ with $\mathbf e_i\cdot \mathbf e_i=-1$ and $\mathbf e_i\cdot \mathbf e_j=1$ for $i\neq j$ (so a cluster of N mutually tangent spheres), and define the lattice

\begin{equation*}\Lambda_N=\mathbf e_1{\mathbb{Z}}\oplus\cdots\oplus \mathbf e_{N}{\mathbb{Z}}.\end{equation*}

We pick $D\in \Lambda_N$ with $D\cdot D>0$ and such that $D\cdot \mathbf n\neq 0$ for any $\mathbf n\in \Lambda_N$ with $\mathbf n\cdot \mathbf n=-1$ . (Such a D exists.) We define

\begin{equation*}{\mathcal E}_{-1}=\{\mathbf n\in \Lambda_N\;:\; \mathbf n\cdot \mathbf n=-1, \mathbf n\cdot D>0\}.\end{equation*}

We define the cone

\begin{equation*}{\mathcal K}_N=\bigcap_{\mathbf n\in {\mathcal E}_{-1}} H_{\mathbf n}^+,\end{equation*}

and the set

\begin{equation*}{\mathcal E}_{-1}^*=\{\mathbf n\in {\mathcal E}_{-1}\;:\; \hbox{$H_{\mathbf n}$ is a face of ${\mathcal K}_N$}\}.\end{equation*}

The set ${\mathcal E}_{-1}^*$ is what is called an Apollonian packing in that paper. It gives the Apollonian circle packing when $N=4$ , the Soddy sphere packing when $N=5$ , and yields Apollonian packings (as defined in this paper) in dimensions $n=4$ through 8 [Reference Baragar4, Reference Baragar5].

It is not clear that ${\mathcal E}_{-1}^*$ is a packing for all N, as it is a priori possible that spheres in the set properly intersect. There is a very nice modularity argument to show that this cannot happen. We first note that if $\mathbf n,\ \mathbf n'\in {\mathcal E}_{-1}^*$ , then $\mathbf n\cdot \mathbf n'\in {\mathbb{Z}}$ , so if the spheres intersect but not tangentially, then $\mathbf n\cdot \mathbf n'=0$ (so they intersect perpendicularly). Consider the difference $\mathbf m=\mathbf n-\mathbf n'$ . If $\mathbf n\cdot \mathbf n'=0$ , then $\mathbf m\cdot \mathbf m=-2$ . Let $\mathbf m=(m_1,...,m_N)$ so

\begin{equation*}\mathbf m\cdot \mathbf m=-\sum_{k=1}^N m_k^2+2\sum_{i\leq j} m_im_j.\end{equation*}

We consider this modulo four, which means we need only worry about the parity of each $m_k$ . Suppose there are K odd $m_k$ . Then

\begin{equation*}\mathbf m\cdot \mathbf m\equiv -K+2\left(\frac{K(K-1)}2\right)\equiv K(K-2) \pmod 4.\end{equation*}

Thus, $\mathbf m\cdot \mathbf m \neq 2 \pmod 4$ , so there are no spheres that intersect perpendicularly.

Using a result of Morrison [19], there exist K3 surfaces with intersection matrix $[J_{ij}]$ with $J_{ij}=2-4\delta_{ij}$ and $N\leq 10$ . The set ${\mathcal E}_{-2}^*$ gives the same packing as described in Remark 2. By [Reference Kovács14], the ample cone has no circular part for $N\geq 4$ , so these packings are maximal.

2.5. Enriques surfaces

For a nice introduction to Enriques surfaces, see [11]. Let X be an Enriques surface and let $\Lambda$ be its Picard group modulo torsion. Then $\Lambda$ is independent of the choice of X and is sometimes called the Enriques lattice $E_{10}$ . It is an even unimodular lattice of signature (1, 9) and can be decomposed as the orthogonal product of the $E_8$ lattice (with negative definite inner product) and the plane U equipped with the Lorentz product $\left[\begin{array}{lcl}0&1 \\ 1&0\end{array}\right]$ : $\Lambda=E_8\oplus U$ .

A nice representation of $E_{10}$ is given by the Coxeter graph $T_{237}$ whose nodes have norm $-2$ :

The subscript in $T_{237}$ means the graph is a tree with three branches of length 2, 3, and 7, as above. The lattice is $\Lambda=\alpha_0{\mathbb{Z}}\oplus \cdots \oplus \alpha_9{\mathbb{Z}}$ . Since $\alpha_i\cdot \alpha_i=-2$ , we know $\alpha_i\cdot \alpha_j=1$ if $\alpha_i\alpha_j$ is an edge of the graph, and 0 otherwise. Let $J=[\alpha_i\cdot \alpha_j]$ , so for $\mathbf x,\ \mathbf y\in \Lambda$ written in this basis, $\mathbf x\cdot \mathbf y=\mathbf x^tJ\mathbf y$ . Since J has integer entries and $-2$ ’s along the diagonal, the lattice is even, meaning $\mathbf x\cdot \mathbf x$ is even for all $\mathbf x\in \Lambda$ . One can verify $\det(J)=-1$ , so the lattice is unimodular.

The reflections

\begin{equation*}R_{\alpha_i}(\mathbf x)=\mathbf x -2\mathop{\textrm{proj}}\nolimits_{\alpha_i}(\mathbf x)=\mathbf x-2\frac{\mathbf x \cdot \alpha_i}{\alpha_i\cdot \alpha_i}\alpha_i=\mathbf x+(\mathbf x\cdot \alpha_i)\alpha_i,\end{equation*}

have integer entries, so are in ${\mathcal O}_\Lambda^+$ . The inverse of J (which appears in [11, p.11]) has non-negative entries along the diagonal, so the polytope bounded by the faces $H_{\alpha_i}$ has finite volume. Thus, the Weyl group

\begin{equation*}W_{237}=\langle R_{\alpha_0},...,R_{\alpha_{9}}\rangle,\end{equation*}

has finite index in ${\mathcal O}_{\Lambda}^+$ .

A generic Enriques surface has no nodal curves, meaning it has no smooth rational curves. The moduli space of Enriques surfaces is ten dimensional. If X contains a nodal curve $\nu$ , then we call it a nodal Enriques surface. By the adjunction formula, $\nu\cdot \nu=-2$ . Here we have abused notation by letting $\nu$ represent both the curve on X and its representation in $\Lambda$ . Since distinct irreducible curves on X have non-negative intersection, an element of $\Lambda$ represents at most one nodal curve on X. The moduli space of nodal Enriques surfaces is nine dimensional. The following is a rewording of a portion of [Reference Allcock1, Theorem 1.2]:

Theorem 2.1 (Coble, Looijenga, Cossec, Dolgachev, Allcock). Suppose X is a generic nodal Enriques surface with nodal curve $\nu$ . Let $\Lambda$ be its Picard group modulo torsion. Then there exist $\beta_0,..., \beta_9\in \Lambda$ so that $\beta_i\cdot\beta_i=-2$ and $\beta_0,...,\beta_9,\nu$ are the nodes of the Coxeter graph (1.1). Let

\begin{equation*}{\Gamma}_{\beta}=\langle R_{\beta_0},..., R_{\beta_9}\rangle \cong W_{246}.\end{equation*}

Then the image in $\Lambda$ of all nodal curves on X is the ${\Gamma}_{\beta}$ -orbit of $\nu$ .

As this version does not look much like Allcock’s result, let us show how it follows. By (5) (of [Reference Allcock1, Theorem 1.2]), $\mathop{\textrm{Aut}}\nolimits(X)$ acts transitively on the nodal curves on X. The image of $\mathop{\textrm{Aut}}\nolimits(X)$ in ${\mathcal O}_{\Lambda}^+$ is a subgroup of ${\Gamma}_{\beta}=W_{246}$ by (2), and by (3), contains a group identified as $\overline{W}_{246}(2)$ (see [Reference Allcock1] for the definition of this group). By (4), $\overline{W}_{246}(2)$ acts transitively on the faces of the ample cone (the interior of $\mathop{\textrm{nef}}\nolimits(X)$ ), so the image of the nodal curves on X includes all of them. By (1), $\mathop{\textrm{nef}}\nolimits(X)$ is the union of the ${\Gamma}_{\beta}$ -translates of the fundamental domain of the Coxeter group generated by the Coxeter graph (1.1) on page 2. The faces of the ample cone are therefore the ${\Gamma}_{\beta}$ image of $\nu$ , as all other faces of the fundamental domain correspond to reflections in $W_{246}$ . Hence, the nodal curves on X generate exactly the faces of the ample cone and is the ${\Gamma}_{\beta}$ -orbit of $\nu$ .

3.1. The strip version and the $\boldsymbol{E}_7$ lattice

To better understand this packing, we describe it in a couple of ways. Let us begin with a strip version, which is an analog of Figure 1. We can think of Figure 1 as an infinite set of circles of constant diameter that are each centered at lattice points of a one-dimensional lattice, then sandwiched between two lines, and filled in using the symmetries of the lattice together with inversion in $\alpha_3$ and reflection in $\alpha_4$ . We can describe ${\mathcal{P}}_\beta$ in a similar way:

The $E_7$ lattice has the Coxeter diagram $T_{234}$ :

It is common to assign each node the norm 2 so that the resulting bilinear form has integer entries, is even, and has determinant 2. Thus, the generating vectors have length $\sqrt 2$ , which is why the spheres above have radius $\sqrt 2/2$ . Adding a node to the appropriate branch gives us the group $W_{244}$ , which acts transitively on the $E_7$ lattice [Reference Vinberg24, Table 2]. Alternatively, the $E_7$ lattice can be thought of as the $W_{244}$ orbit of the point O, where O is the vertex of the canonical fundamental domain for $W_{244}$ that lies at the intersection of the planes represented by the $T_{234}$ subgraph. This is the representation we shall use.

Proof. Referring to $J_\beta^{-1}$ , we note that $w_8$ has norm zero, so is a point on $\partial {\mathcal H}$ . Let us consider the perspective ${\mathcal{P}}_{\beta,w_8}$ , which lies in $\partial {\mathcal H}_{w_8}\cong {\mathbb{R}}^8$ . Note that the spheres $s_0$ and $s_1$ are tangent at $w_8$ , so are parallel Euclidean hyperplanes in $\partial {\mathcal H}_{w_8}$ . These are our analogs of $\mathbf e_1$ and $\mathbf e_2$ , respectively, in Figure 1. Since $s_1=R_{\beta_9}\!(s_0)$ , the reflection $R_{\beta_9}$ is the analog of the symmetry represented by $\alpha_4$ , and we take V to be $H_{\beta_9}\cap \partial {\mathcal H}_{w_8}$ . Since $s_2=R_{\beta_8}(s_1)$ , the reflection $R_{\beta_8}$ is $\sigma$ , the analog of inversion in $\alpha_3$ . The sphere $s_2$ is the analog of $\mathbf e_3$ , and the points A and B are the points of tangency $s_0+s_2$ and $s_1+s_2$ , respectively. The rest of the generators are the analogs of $\alpha_1$ and $\alpha_2$ . They form the group ${\Gamma}_7=\langle R_{\beta_0},...,R_{\beta_7}\rangle \cong W_{244}$ . It remains to understand what the orbit ${\Gamma}_7(s_2)$ looks like.

Note that the generators of ${\Gamma}_7$ fix $w_8$ , so are Euclidean reflections in $\partial {\mathcal H}_{w_8}\cong {\mathbb{R}}^8$ . The planes through which they reflect are all perpendicular to $V=H_{\beta_9}\cap \partial {\mathcal H}_{w_8}\cong {\mathbb{R}}^7$ , so act as a group of reflections on V. Let O in V be the point of intersection of $H_{\beta_i}$ for $i\leq 6$ . Since the planes $H_{\beta_i}$ are perpendicular to $s_2$ for $i\leq 6$ (since $\beta_i\cdot s_2=0$ for $i\leq 6$ ), the point O must be the center of the sphere $s_2$ . Thus, the ${\Gamma}_7$ -orbit of $s_2$ is a set of congruent spheres that are centered on the lattice points of an $E_7$ lattice. Finally, since $s_3=R_{\beta_7}(s_2)$ , and $s_3$ is tangent to $s_2$ , the spheres must be of maximal radius so that they intersect tangentially or not at all. That is, scaling so that the generating vectors of the $E_7$ lattice have length $\sqrt 2$ , the spheres all have radius $\sqrt 2/2$ .

3.2. The $\boldsymbol{E}_8$ structure

The $E_8$ lattice has the Coxeter diagram $T_{235}$ , and the group $W_{236}$ acts transitively on it [Reference Vinberg24, Table 2]. The lattice can also be thought of as the $W_{236}$ orbit of the point O, where O is the vertex of the canonical fundamental domain for $W_{236}$ that lies at the intersection of the planes represented by the $T_{235}$ subgraph. Let us distinguish another vertex of the fundamental domain, the point P that lies at the intersection of the planes represented by the $T_{226}$ subgraph.

Proof. Referring to $J_\beta^{-1}$ , we see that $w_1$ is also on $\partial {\mathcal H}$ , as $w_1\cdot w_1=0$ . Let us consider the perspective ${\mathcal{P}}_{\beta, w_1}$ . The planes $H_{\beta_i}$ for $i\neq 1$ all go through $w_1$ (by the definition of $w_1$ ), so the corresponding reflections $R_{\beta_i}$ are Euclidean symmetries of ${\mathbb{R}}^8\cong \partial {\mathcal H}_{w_1}$ . Let us first consider the group ${\Gamma}_8=\langle R_{\beta_0}, R_{\beta_2},...,R_{\beta_9}\rangle\cong W_{236}$ .

We let O be the point on $\partial {\mathcal H}_{w_1}$ where the planes $H_{\beta_0}$ , $H_{\beta_2}$ , …, $H_{\beta_8}$ intersect, so ${\Gamma}_8(O)$ is an $E_8$ lattice in $\partial {\mathcal H}_{w_1}$ . Since these planes intersect $s_0=\nu$ perpendicularly, we know O is the center of $s_0$ . The ${\Gamma}_8$ -orbit of $s_0$ is therefore a set of spheres of constant radius centered on the vertices of an $E_8$ lattice. Since $R_{\beta_9}\!(s_0)=s_1$ is tangent to $s_0$ , these spheres are of maximal radius such that any pair intersect tangentially or not at all. They give the densest uniradial sphere packing of ${\mathbb{R}}^8$ [Reference Viazovska23].

It remains to understand the action of $R_{\beta_1}$ . The plane $H_{\beta_1}$ represents a sphere in $\partial {\mathcal H}_{w_1}$ . Since $\beta_1\cdot w_1=1=s_0\cdot w_1$ and $\beta_1\cdot \beta_1=-2=s_0\cdot s_0$ , the spheres have the same radii. Finally, $\beta_1\cdot \beta_i=0$ for $i=0$ , $3,...,9$ , so is centered at the point of intersection on V of the planes $H_{\beta_0}$ , $H_{\beta_3}$ , …, $H_{\beta_9}$ , which corresponds to the vertex P of the fundamental domain for $W_{236}$ . Thus, the described configuration is congruent to the perspective ${\mathcal{P}}_{\beta,w_1}$ .

3.3. Comparison with the packing in [Reference Baragar4]

Let

\begin{align*}\Lambda_{\beta}&=\beta_0{\mathbb{Z}}\oplus \cdots \oplus \beta_9{\mathbb{Z}}, \\\Lambda_{10}&=s_0{\mathbb{Z}}\oplus \cdots \oplus s_9{\mathbb{Z}}.\end{align*}

and let $J_{10}=[s_i\cdot s_j]$ . Then $\det\!(J_{10})=-2^{22}$ while $\det\!(J_\beta)=-4$ . It would appear that the packing ${\mathcal{P}}_{10}={\mathcal E}_{-2}^*$ generated by $\Lambda_{10}$ (see Remark 2) is very different than ${\mathcal{P}}_\beta$ . However, we can modify the underlying lattice for ${\mathcal{P}}_\beta$ in the following way. Consider the sublattice

\begin{equation*}\Lambda'_{\!\!\beta}=2\beta_0{\mathbb{Z}}\oplus \cdots \oplus 2\beta_8{\mathbb{Z}}\oplus \nu{\mathbb{Z}}.\end{equation*}

and its incidence matrix $J'_{\!\!\beta}$ . Then $\Lambda_{10}\subset \Lambda'_{\!\!\beta}\subset \Lambda_{\beta}$ and $\det\!(J'_{\!\!\beta})=-2^{20}$ . Thus, $\Lambda_{10}$ is a sublattice of $\Lambda'_{\!\!\beta}$ of index two. Though we have changed the underlying lattice for ${\mathcal{P}}_\beta$ , we have not changed its geometry. Furthermore, in this basis, ${\mathcal{P}}_\beta={\mathcal E}_{-2}^*$ . Thus, ${\mathcal K}_{{\mathcal{P}}_{10}}\supset {\mathcal K}_{{\mathcal{P}}_\beta}$ , so the residual set for the packing ${\mathcal{P}}_\beta$ is a subset of the residual set for ${\mathcal{P}}_{10}$ . Thus, ${\mathcal{P}}_\beta$ is a more efficient packing than ${\mathcal{P}}_{10}$ .