On \Sigma-classes in E^8. IV. The neighbourhood of F_{22}

1. Introduction

In d-dimensional Euclidean space $E^d$, a bounded, convex body of finite volume congruent copies of which can be juxtaposed by translations such that they fill space without gaps, and overlap only in border points, is denoted by Fedorov [10] as parallelohedron. A special kind of parallelohedron, the Dirichlet parallelohedron is obtained by applying Dirichlet’s famous construction [1] to a lattice of translations ${\Lambda }^d$. Voronoï [12,13] thoroughly investigated general properties of parallelohedra in spaces of arbitrary dimensions d. That is why they are often referred to as Voronoï parallelohedra.

Quadratic forms, translation lattices and parallelohedra play a predominant role in the geometry of numbers, but also in crystallography, where the lattice-like arrangement of atomic building blocks is a fundamental property of regular crystals. In crystallography, the discovery of quasicrystals, the structure of which can be viewed as projected from higher dimensional translation lattices, has greatly stimulated the investigation of lattices and parallelohedra in arbitrary dimensions.

In Euclidean space $E^8$, the collective of all lattices in the open cone ${\mathcal{C}}^{8\times 8}$ of positive definite quatratic forms is considered by studying its subdivision into $\Sigma$-subcones.

Voronoï proposed that each parallelohedron $\tilde{\mathsf{P}}$ is affinely equivalent to a Dirichlet parallelohedron $\mathsf{P}$, but he proved it for primitive parallelohedra only [12], §44. Primitive means that at each vertex of a parallelohedron $\mathsf{P}$ in its tiling exactly the minimal number $d+1$ of parallelohedra meet. These are the generic parallelohedra, for under small perturbations the combinatorial type remains fixed.

By using affinely invariant operations only, the zone contraction and zone extension, it becomes obvious that each combinatorial type of totally contracted Dirichlet parallelohedron is sufficient to prove that its complete contraction family can be represented by affine Dirichlet parallelohedra too. In dimensions $d\ge 0$ there exist totally contracted parallelohedra which are also referred to as minimal. But in dimensions $d\ge 6$ there exist parallelohedra which are as well minimal as maximal that is they allow no extension. This makes these parallelohedra particularly interesting.

In order to find ${\Sigma }_0$-classes in ${\mathbf{C}}^{8\times 8}$, the parallelohedra of the maximal finite irreducible subgroups of $G{L}_8\left(ZZ\right)$ given by Plesken and Pohst [11] are of importance. The five types viz.: $F_5$, $F_8$, $F_{15}$, $F_{22}$, and $F_{26}$, respectively, are as much minimal as maximal.

In previous papers the neighbourhood of the quadratic form ${\mathsf{Q}}_{F_5}$ [7], ${\mathsf{Q}}_{F_8}$ [8] ${\mathsf{Q}}_{F_{15}}$ [9] were investigated. In this note, the neighbourhood of ${\mathsf{Q}}_{F_{22}}$ will be looked at.

Table 1. The parallelohedra of the maximal finite irreducible subgroups of $G{L}_8\left(ZZ\right)$

No.	Group	Order	Comb. Type	Zones	Belts
$F_1$	$B_8$	$2^{8}8!$	16.256	8(8)	$4_{28}$
$F_2$	$D_8$	$2^{8}8!$	112.272	136(0)	$6_{224}$
$F_3$		$2·{1152}^2$	48.576	24(0)	$4_{114}{6}_{32}$
$F_4$	$D_8^{*}$	$2^{8}8!$	272.1120	56(0)	$4_{28}{6}_{512}$
$F_5$	$E_8$	$2^{14}{3}^5{5}^{2}7$	240.19449	8760(0)	$6_{1120}$
$F_6$		$3!·1152$	264.5304	36(0)	$4_{108}{6}_{972}$
$F_7$		${12}^{4}4!$	24.1296	12(12)	$4_{54}{6}_4$
$F_8$		$2·6^{4}4!$	348.3588	93(0)	$6_{1386}$
$F_9$		$2·{144}^2$	60.10404	18(18)	$4_2{256}_{48}$
$F_{10}$	$A_8$	$2·9!$	72.510	9(9)	$6_{84}$
$F_{11}$		$2·6^{4}4!$	186.5940	174(0)	$4_{54}{6}_{544}$
$F_{12}$		$2·6^{3}3!$	402.94254	1497(0)	$6_{2259}$
$F_{13}$	$A_8^{*}$	$2·9!$	510.362880	36(36)	$6_{3025}$
$F_{14}$		$2·{240}^2$	40.900	10(10)	$4_{100}{6}_{20}$
$F_{15}$		$2^7·{15}^2$	360.3840	180(0)	$6_{1700}$
$F_{16}$		$2·{240}^2$	60.14400	20(20)	$4_{225}{6}_{50}$
$F_{17}$		${(2·5!)}^2$	290.43310	1195(0)	$6_{1370}$
$F_{18}$		$2·3!·5!$	390.106200	450(0)	$6_{2160}$
$F_{19}$		$2·3!·5!$	180.6510	15(15)	$4_{90}{6}_{400}$
$F_{20}$		1152	296.13696	580(0)	$4_{66}{6}_{1168}$
$F_{21}$		1152	200.2992	304(0)	$4_{48}{6}_{564}$
$F_{22}$		$3·1152$	456.28632	264(0)	$6_{2260}$
$F_{23}$		672	398.90384	297(0)	$6_{2192}$
$F_{24}$		672	426.37638	21(21)	$4_{24}{6}_{1428}$
$F_{25}$		672	308.8028	1011(0)	$4_{21}{6}_{1386}$
$F_{26}$		672	398.127848	1036(0)	$6_{2227}$

Table 1. The parallelohedra of the maximal finite irreducible subgroups of $G{L}_8\left(ZZ\right)$

No.	Group	Order	Comb. Type	Zones	Belts
$F_1$	$B_8$	$2^{8}8!$	16.256	8(8)	$4_{28}$
$F_2$	$D_8$	$2^{8}8!$	112.272	136(0)	$6_{224}$
$F_3$		$2·{1152}^2$	48.576	24(0)	$4_{114}{6}_{32}$
$F_4$	$D_8^{*}$	$2^{8}8!$	272.1120	56(0)	$4_{28}{6}_{512}$
$F_5$	$E_8$	$2^{14}{3}^5{5}^{2}7$	240.19449	8760(0)	$6_{1120}$
$F_6$		$3!·1152$	264.5304	36(0)	$4_{108}{6}_{972}$
$F_7$		${12}^{4}4!$	24.1296	12(12)	$4_{54}{6}_4$
$F_8$		$2·6^{4}4!$	348.3588	93(0)	$6_{1386}$
$F_9$		$2·{144}^2$	60.10404	18(18)	$4_2{256}_{48}$
$F_{10}$	$A_8$	$2·9!$	72.510	9(9)	$6_{84}$
$F_{11}$		$2·6^{4}4!$	186.5940	174(0)	$4_{54}{6}_{544}$
$F_{12}$		$2·6^{3}3!$	402.94254	1497(0)	$6_{2259}$
$F_{13}$	$A_8^{*}$	$2·9!$	510.362880	36(36)	$6_{3025}$
$F_{14}$		$2·{240}^2$	40.900	10(10)	$4_{100}{6}_{20}$
$F_{15}$		$2^7·{15}^2$	360.3840	180(0)	$6_{1700}$
$F_{16}$		$2·{240}^2$	60.14400	20(20)	$4_{225}{6}_{50}$
$F_{17}$		${(2·5!)}^2$	290.43310	1195(0)	$6_{1370}$
$F_{18}$		$2·3!·5!$	390.106200	450(0)	$6_{2160}$
$F_{19}$		$2·3!·5!$	180.6510	15(15)	$4_{90}{6}_{400}$
$F_{20}$		1152	296.13696	580(0)	$4_{66}{6}_{1168}$
$F_{21}$		1152	200.2992	304(0)	$4_{48}{6}_{564}$
$F_{22}$		$3·1152$	456.28632	264(0)	$6_{2260}$
$F_{23}$		672	398.90384	297(0)	$6_{2192}$
$F_{24}$		672	426.37638	21(21)	$4_{24}{6}_{1428}$
$F_{25}$		672	308.8028	1011(0)	$4_{21}{6}_{1386}$
$F_{26}$		672	398.127848	1036(0)	$6_{2227}$

2. The Irreducible Subgroup $F_{22}$

Notations and methods developed in previous papers [3,4,5,6] will be applied.

Referred to an optimal basis the following Gram matrix is obtained.

$${\mathsf{Q}}_{F_{22}}:=\left(\begin{array}{cccccccc}\hfill 6& \hfill -3& \hfill 2& \hfill 3& \hfill 2& \hfill -1& \hfill 1& \hfill 0\\ \hfill -3& \hfill 6& \hfill -1& \hfill -3& \hfill 1& \hfill 0& \hfill -3& \hfill -3\\ \hfill 2& \hfill -1& \hfill 6& \hfill 3& \hfill 2& \hfill 1& \hfill -1& \hfill 2\\ \hfill 3& \hfill -3& \hfill 3& \hfill 6& \hfill -1& \hfill 1& \hfill 2& \hfill 3\\ \hfill 2& \hfill 1& \hfill 2& \hfill -1& \hfill 6& \hfill -3& \hfill -3& \hfill -2\\ \hfill -1& \hfill 0& \hfill 1& \hfill 1& \hfill -3& \hfill 6& \hfill 3& \hfill 3\\ \hfill 1& \hfill -3& \hfill -1& \hfill 2& \hfill -3& \hfill 3& \hfill 6& \hfill 3\\ \hfill 0& \hfill -3& \hfill 2& \hfill 3& \hfill -2& \hfill 3& \hfill 3& \hfill 6\end{array}\right).$$

The corresponding parallelohedron is computed by half-space intersections,

$$\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right):=\bigcap _{\forall \mathbf{t}\in {\Lambda }^d\setminus \left\{O\right\}}{\mathsf{H}}_{\mathbf{t}}.$$

(1)

$\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right)$ has 456 facets, 28’632 vertices, and 264 zones all of which are open. Therefore, $\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right)$ is minimal. The 228 pairs of facet vectors $\pm {\mathbf{f}}_i$ are listed in Table 2. The set of facet vectors is denoted by ${\mathcal{F}}_{F_{22}}$.

The condition for a zone vector ${\mathbf{z}}_k^{*}$ to be extendable is given by

$$L_i\left({\mathbf{z}}_k^{*}\right):={\mathbf{f}}_j{\mathbf{z}}_k^{*},i\in \{-1,0,1\},\forall {\mathbf{f}}_j\in {\mathcal{F}}_{F_{22}}$$

(2)

It is readily verified that for all zone vectors ${\mathbf{z}}_k^{*}$ with components $\mid z_l\mid \le 4$, $l=1,...,8$, the facet vectors ${\mathbf{f}}_j\in {\mathcal{F}}_{F_{22}}$ belong to layers ${\mathsf{L}}_i\left({\mathbf{z}}_k^{*}\right)$, $-n\le i\le n$, where $n\ge 2$. Therefore, $\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right)$ is as much minimal as maximal.

The 456 facet vectors of $\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right)$ belong to three equivalence classes of norm 6, 8, and 10, respectively. In order to obtain the group ${\mathcal{G}}_{F_{22}}$, the unified polytope scheme, described in [2], was used. Each identical scheme induces a permutation $\mathsf{S}$ of the facet vectors under their automorphism group. There exists an isomorphism between the class of identical schemes and the group of automorphisms of $\mathsf{P}\left({\mathsf{Q}}_{F_{22}}\right)$.

Below are given generating rotations of order 6 (${\mathsf{S}}_1$), 8 (${\mathsf{S}}_2$), and 12 (${\mathsf{S}}_3$), respectively, which generate the group ${\mathcal{G}}_{F_{22}}$ of order 3’456. The group contains only pure rotations.

$${\mathsf{S}}_1:=\left(\begin{array}{cccccccc}\hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill 1& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill 0& \hfill 0\\ \hfill -1& \hfill 1& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill -1\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill -1& \hfill 1& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill -1& \hfill -1\\ \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right),$$

$${\mathsf{S}}_2:=\left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill -1& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill -1& \hfill 1& \hfill -1& \hfill -1& \hfill 0& \hfill -1& \hfill -1& \hfill -1\\ \hfill -1& \hfill 1& \hfill -1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill -1& \hfill 0& \hfill -1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill 1& \hfill 0\end{array}\right),$$

$${\mathsf{S}}_3:=\left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill -1& \hfill -1& \hfill -1\\ \hfill 0& \hfill 1& \hfill 0& \hfill -1& \hfill 1& \hfill -1& \hfill -1& \hfill -1\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill -1& \hfill 1& \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill -1& \hfill 0& \hfill -1& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill -1& \hfill -1& \hfill 0\end{array}\right).$$

It holds:

$${\mathsf{S}}_i^t{\mathsf{Q}}_{F_{22}}{\mathsf{S}}_i={\mathsf{Q}}_{F_{22}},\forall {\mathsf{S}}_i\in {\mathcal{G}}_{F_{22}}.$$

(3)

3. The Neighbourhood of ${\mathsf{Q}}_{F_{22}}$

In the neighourhood of ${\mathsf{Q}}_{F_{22}}$, a form ${\mathsf{Q}}_1$ was found, whose parallelohedron is primitive and therefore, has the maximal number 510 of facets, as well as 362’304 vertices, and 761 zones all of which are open:

${\mathsf{Q}}_1:=$

$$\left(\begin{array}{cccccccc}\hfill 6.05228& \hfill -3.02808& \hfill 1.83912& \hfill 2.92092& \hfill 1.83868& \hfill -0.86624& \hfill 0.86536& \hfill -0.10748\\ \hfill -3.02808& \hfill 6.05256& \hfill -0.86504& \hfill -2.92108& \hfill 1.08004& \hfill -0.10812& \hfill -2.81348& \hfill -2.91876\\ \hfill 1.83912& \hfill -0.86504& \hfill 6.05272& \hfill 2.81244& \hfill 2.05364& \hfill 0.97484& \hfill -0.86644& \hfill 2.05432\\ \hfill 2.92092& \hfill -2.92108& \hfill 2.81244& \hfill 6.05292& \hfill -0.97332& \hfill 0.86608& \hfill 1.83976& \hfill 2.81120\\ \hfill 1.83868& \hfill 1.08004& \hfill 2.05364& \hfill -0.97332& \hfill 6.05352& \hfill -3.02916& \hfill -2.92188& \hfill -1.94620\\ \hfill -0.86624& \hfill -0.10812& \hfill 0.97484& \hfill 0.86608& \hfill -3.02916& \hfill 6.05368& \hfill 2.91984& \hfill 2.92224\\ \hfill 0.86536& \hfill -2.81348& \hfill -0.86644& \hfill 1.83976& \hfill -2.92188& \hfill 2.91984& \hfill 6.05396& \hfill 3.02740\\ \hfill -0.10748& \hfill -2.91876& \hfill 2.05432& \hfill 2.81120& \hfill -1.94620& \hfill 2.92224& \hfill 3.02740& \hfill 6.05412\end{array}\right).$$

The Gram matix ${\mathsf{Q}}_1$ defines the ${\Sigma }_0^1$-subcone. For each $\mathsf{Q}$ in the open ${\Sigma }_0^1$-subcone, the set of facet vectors ${\mathcal{F}}_{Q_1}$ is an invariant of ${\Sigma }_0^1$ and it is denoted by ${\mathcal{F}}_{{\Sigma }_0^1}$. The facet vectors are given in Table 4. Since $\mathsf{P}\left({\mathsf{Q}}_1\right)$ is generic it has the maximal number of facets. Verifying equation (2) for all zone vectors ${\mathbf{z}}_k^{*}$ with components $\mid z_l\mid \le 4$, it results that the facet vectors ${\mathbf{f}}_j\in {\mathcal{F}}_{{\Sigma }_0^1}$, lie in layers $L_i\left({\mathbf{z}}_k^{*}\right)$, $i\in \{-n\le i\le n\}$, where $n\ge 2$. Thus, $\mathsf{P}\left({\mathsf{Q}}_1\right)$ is as much minimal as maximal.

In order to calculate the ${\Sigma }_0^1$-subcone, all tripletts

$${\mathbf{f}}_i+{\mathbf{f}}_j+{\mathbf{f}}_k=0,{\mathbf{f}}_i,{\mathbf{f}}_j,{\mathbf{f}}_k\in {\mathcal{F}}_{{\Sigma }_0^1},$$

are determined. Their number is $N_b$=3’793 and the number of half-spaces ${\mathsf{H}}_i$ becomes $3N_b$. The closed subcone ${\Sigma }_0^1$ with apex in $\tilde{O}$ becomes

$${\Sigma }_0^1:=\bigcap _{i=1}^{3N_b}{\mathsf{H}}_i.$$

(4)

Because of the heigh complexity of the ${\Sigma }_0^1$-subcone, the direct computation of ${\Sigma }_0^1$ as well as of its $\Phi$-subcones is not practicable with small computers. Instead, existence charts may be defined which intersect the ${\Sigma }_0^1$-subcone.

Using condition

$${\mathbf{n}}_i^t\mathbf{q}>0,i=1,\cdots ,3N_b.$$

(5)

any further form ${\mathsf{Q}}_2\in {\Sigma }_0^1$ can be found viz.:

${\mathsf{Q}}_2:=$

$$\left(\begin{array}{cccccccc}\hfill 6.05104& \hfill -3.02844& \hfill 1.83868& \hfill 2.78892& \hfill 1.82188& \hfill -0.87588& \hfill 0.86456& \hfill -0.10868\\ \hfill -3.02844& \hfill 6.07416& \hfill -0.86660& \hfill -2.87348& \hfill 1.07004& \hfill -0.11692& \hfill -2.81192& \hfill -2.91592\\ \hfill 1.83868& \hfill -0.86660& \hfill 6.14472& \hfill 2.83496& \hfill 2.06568& \hfill 0.96604& \hfill -0.86968& \hfill 2.05112\\ \hfill 2.78892& \hfill -2.87348& \hfill 2.83496& \hfill 6.17292& \hfill -0.96612& \hfill 0.85528& \hfill 1.83936& \hfill 2.81088\\ \hfill 1.82188& \hfill 1.07004& \hfill 2.06568& \hfill -0.96612& \hfill 6.05596& \hfill -3.03876& \hfill -2.92208& \hfill -1.94636\\ \hfill -0.87588& \hfill -0.11692& \hfill 0.96604& \hfill 0.85528& \hfill -3.03876& \hfill 6.06728& \hfill 2.90464& \hfill 2.91460\\ \hfill 0.86456& \hfill -2.81192& \hfill -0.86968& \hfill 1.83936& \hfill -2.92208& \hfill 2.90464& \hfill 6.06596& \hfill 3.02140\\ \hfill -0.10868& \hfill -2.91592& \hfill 2.05112& \hfill 2.81088& \hfill -1.94636& \hfill 2.91460& \hfill 3.02140& \hfill 6.04612\end{array}\right).$$

The parallelohedron $\mathsf{P}\left({\mathsf{Q}}_2\right)$ is primitive and therefore, has the maximal number 510 of facets, as well as 362’160 vertices, and 778 zones all of which are open, and it proves that ${\mathsf{Q}}_{F_{22}}$ lies on the boundary of ${\Sigma }_0^1$.

The three forms

$${\mathsf{Q}}_{F_{22}},{\mathsf{Q}}_1,{\mathsf{Q}}_2\in {\Sigma }_0^1$$

determine a 2-section ${\Pi }^2$ through the ${\Sigma }_0^1$-subcone. with origin in ${\mathsf{Q}}_{F_{22}}$ and cartesian coordinates ${\overline{\mathbf{x}}}_1$, and ${\overline{\mathbf{x}}}_2$, respectively. Each $\mathsf{Q}\in {\Pi }^2$ is obtained by

$$\mathsf{Q}={\mathsf{Q}}_{F_{22}}+\sum _{i=1}^2{\lambda }_i{\overline{\mathbf{x}}}_i,{\lambda }_i\in IR.$$

The condition (5) is used to decide if $\mathsf{Q}$ is interior to a ${\Sigma }_0^k$-subcone. Considering that ${\mathcal{F}}_{{\Sigma }_0^k}$ is an invariant for ${\Sigma }_0^k$, the parallelohedron $\mathsf{P}\left(\mathsf{Q}\right)$ is easily computed.

In what follows, the surrounding of the ${\Sigma }_0^1$-subcone in the section ${\Pi }^2$ is investigated.

First, all walls ${\mathsf{W}}_k$ containing ${\mathsf{Q}}_{F_{22}}$ are determined along the four straight line segments

$$\begin{array}{c}0.1{\overline{\mathbf{x}}}_1+{\lambda }_2{\overline{\mathbf{x}}}_2\hfill \\ -0.1{\overline{\mathbf{x}}}_1+{\lambda }_2{\overline{\mathbf{x}}}_2\hfill \\ {\lambda }_1{\overline{\mathbf{x}}}_1+0.1{\overline{\mathbf{x}}}_2\hfill \\ {\lambda }_1{\overline{\mathbf{x}}}_1-0.1{\overline{\mathbf{x}}}_2\hfill \end{array}-0.1\le {\lambda }_1,{\lambda }_2\le 0.1,$$

forming a quadrangle arround the center ${\mathsf{Q}}_{F_{22}}$. Applying the method of nested intervals, the border point ${\mathsf{Q}}_k$ between two neighbouring ${\Sigma }_0^k$- and ${\Sigma }_0^{k+1}$-subcones is easily determind within an $ϵ$-limit. The value of $ϵ$ is chosen to be

$$ϵ=det\left(\mathsf{Q}\right){10}^{-10}.$$

The border point ${\mathsf{Q}}_k$ corresponds to a limiting parallelohedron $\mathsf{P}\left({\mathsf{Q}}_k\right)$ which is characterized by haveing some vertex $\mathbf{v}\in \mathsf{P}\left({\mathsf{Q}}_k\right)$ where at least d+1 facets meet. The wall ${\mathsf{W}}_k$ contains ${\mathsf{Q}}_k$ and the origin $\tilde{O}$. Thus, the wall normal ${\mathbf{n}}_k$ is easily calculated.

It proves that there exists a highly complex cluster ${\mathsf{C}}_{F_{22}}$ of ${\Sigma }_0^k$-subcones. In Figure 1 is shown the section through the ${\mathsf{C}}_{F_{22}}$-cluster. Only the border lines of the ${\Sigma }_0^k$-subcones are drawn. Each border line corresponds to a wall ${\mathsf{W}}_h$ having wall normal ${\mathbf{n}}_h$ of some ${\Sigma }_0^k$-subcone, and therefore, has a representation as a tensor product

$${\mathbf{n}}_h={\mathbf{f}}_a\otimes {\mathbf{f}}_b,{\mathbf{f}}_a,{\mathbf{f}}_b\in {\mathcal{F}}_{{\Sigma }_0^k}.$$

There exist two kinds of walls. Walls ${\mathsf{W}}^{\left(i\right)}$ that separate two adjacent ${\Sigma }_0$-subcones, and walls ${\mathsf{W}}^{\left(a\right)}$ that separate a ${\Sigma }_0$-subcone from a adjacent ${\Sigma }_1$-subcone. In order to determine all walls, it would require to compute a huge number of $\Phi$-subcones which at present is far out of reach.

The walls containing the center ${\mathsf{Q}}_{F_{22}}$ belong to three non-equivalent wall classes ${\mathcal{W}}_h$, $h=1,2,3$, under the group ${\mathcal{G}}_{F_{22}}$ which are shown in Table 6.

A representative wall normal ${\mathbf{n}}_j$ of ${\mathsf{W}}_j\in {\mathcal{W}}_h$ is given by a tensor product

$${\mathbf{n}}_j:=({\mathbf{f}}_a\otimes {\mathbf{f}}_b),{\mathbf{f}}_a,{\mathbf{f}}_b\in {\mathcal{F}}_{{\Sigma }_0^k}.$$

Although the walls ${\mathsf{W}}_i\in {\mathcal{W}}_h$ are equivalent, the ${\Sigma }_0^i$-subcones in the ${\mathsf{C}}_{F_{22}}$-cluster are not equivalent because the section ${\Pi }^2$ generally contains only one element of the orbit of some $\mathsf{Q}\in {\Sigma }_0^1$ under the group ${\mathcal{G}}_{F_{22}}$. Altogether there exist 684 walls containing ${\mathsf{Q}}_{F_{22}}$, whereof 102 intersect the ${\mathsf{C}}_{F_{22}}$-cluster.

For fixing the outer border of the ${\mathsf{C}}_{F_{22}}$-cluster further walls along suitable straight line segments were determined in order to find the walls which separate a ${\Sigma }_0^i$-subkone from a ${\Sigma }_1^j$-subcone. Thereby, the border lines avoiding the center ${\mathsf{Q}}_{F_{22}}$ were found which are shown in Figure 1. Remarkable in the Figure is the spike S of ${\Sigma }_1^j$-subcones entering into the ${\mathsf{C}}_{F_{22}}$-cluster. In the ${\Pi }^2$-section, the border lines do not necessarily extend over the whole cluster. When several border lines intersect then some border lines may end in the point of intersection. In Figure 1 it is seen that many of the walls ${\mathsf{W}}_i\in {\mathcal{W}}^{\left(i\right)}$ end in the center ${\mathsf{Q}}_{F_{22}}$. We don’t have further investigated this behaviour.

A rough estimate for the number of non-equivalent ${\Sigma }_0^k$-subcones within the ${\mathsf{C}}_{F_{22}}$-cluster in $E^8$ could amount up to ${10}^{10}$.