On the Σ classes in E6

Engel, P.

doi:10.1107/S2053273320009936

research papers

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 76| Part 5| September 2020| Pages 622-626

https://doi.org/10.1107/S2053273320009936

On the Σ classes in E⁶

P. Engel ^a ^*

^aJupiterstrasse 3, CH-3015 Berne, Switzerland
^*Correspondence e-mail: [email protected]

Edited by A. Altomare, Institute of Crystallography - CNR, Bari, Italy (Received 17 April 2020; accepted 20 July 2020; online 25 August 2020)

In E⁶, the cone of positive definite quadratic forms is subdivided into Σ_s subcones and its equivalence classes $[{\cal E}_{\Sigma _s}]$ are determined for s = 0–3, and 18–21.

Keywords: translation lattices; parallelohedra; cone of positive quadratic forms; Sigma classes.

1. Introduction

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. The classification of the combinatorial types of primitive parallelohedra $[{\sf P}]$ induces a structure on the cone of positive definite quadratic forms $[{\cal C}^{+}]$ .

In a series of papers, the shape of $[{\cal C}]$ and its subdivision into Φ and Σ subcones were discussed (Baburin & Engel, 2013 ; Engel, 2015 , 2019 ).

Ryshkov (1973 ) defined the S subcone which contains all parallelohedra that have the same set of facet vectors $[{\cal F}]$ , but without characterizing its boundary. The complete subcone was determined by Engel (2015) by the half-space intersection

$[\Sigma \left ( {\sf P} \right ): = \bigcap _{h = 1}^{3N_{b}} {\sf H}_{h}. \eqno(1)]$

The investigation of translation lattices becomes most attractive in E⁶ because many new phenomena appear for the first time in dimension 6.

In the report by Engel (2019), minimal and maximal Σ_s classes in E⁶, $[{\cal E}_{\Sigma _{0}}]$ and $[{\cal E}_{\Sigma _{21}}]$ , were investigated. The subscript `s' is an invariant of the class and denotes the number of closed zones of $[{\sf P}]$ [see Engel (2019), equations (12)–(13)]. This classification is continued for the Σ_s classes, s = 0, 1, 2, 3 and 18, 19, 20, 21.

The infinite family of Σ cones generate a face-to-face tiling of the cone $[{\cal C}]$ [see Engel (2019), equation (17) ff.]. In this tiling, for each class representative $[\Sigma _{s}^{i}]$ are determined all the neighbouring Σ's adjacent to $[\Sigma _{s}^{i}]$ by a common wall, in order to find new $[\Sigma _{r}^{j}]$ . Proceeding in this way, for each class $[{\cal E}_{\Sigma _{s}^{k}}]$ can be found at least one representative $[\Sigma _{s}^{k}]$ along a finite path of adjacent Σ's.

As a main result we obtain by this adjacency procedure:

For s = 0, 1, 2, 3 there exist 1, 1, 6, 58 Σ_s classes, and for s = 18, 19, 20, 21 there exist 15, 3, 1, 1 Σ_s classes in $[{\cal C}]$ .

2. Determination of the Σ_s classes

Most concepts used in what follows were described by Engel (2019).

Beginning with Σ₀ as a representative of its class $[{\cal E}_{\Sigma _{0}}]$ , and its subdivision into combinatorial Φ types, the Σ_s classes for s = 0, 1, 2, 3 are successively determined. Recall that Σ₀ has 216 walls $[{\sf W}_{i}^{\Sigma _{0}}]$ , $[i = 1,\cdots,216]$ , which all are equivalent under the group $[{\cal G}_{E_{6}^{*}}]$ (see Engel, 2019). The neighbouring Σ_s adjacent to Σ₀ are equivalent too, and were determined along the following steps:

Step S1: Within the class of equivalent walls, one wall $[{\sf W}_{l}^{\Sigma _{0}}]$ , 1 ≤ l ≤ 216, is selected, and for any $[\Phi _{j} \subset \Sigma _{0}]$ leaning on $[{\sf W}_{l}^{\Sigma _{0}}]$ , the neighbouring Φ_k opposite to that wall is taken. Let $[{\sf Q} \in \Phi _{k}^{+}]$ . The determination of $[\Sigma _{s} ({\sf Q})]$ requires first the computation of the primitive parallelohedron,

$[{\sf P} \left ( {\sf Q} \right ) \, : = \bigcap _{\forall \, {\bf t} \in \Lambda^{d} \setminus \{ O\}} {\sf H}_{\bf t} . \eqno(2)]$

Note that in E⁶ every primitive parallelohedron has 126 facet vectors. The set of facet vectors of $[{\sf P}]$ is denoted by

$[{\cal F}_{\sf P} \, : = \{ {\bf f}_{1}, \cdots, {\bf f}_{126} \} . \eqno(3)]$

This shows that $[{\sf P} ({\sf Q})]$ has one closed zone with zone vector z* = (0, 0, 0, 0, 0, 1), and thus it belongs to $[{\Sigma}_{1} ({\sf Q})]$ . Because of symmetry, for each equivalent wall an equivalent result will be obtained.

Step S2: Next all triplets $[{\bf f}_{i},{\bf f}_{j},{\bf f}_{k} \in {\cal F}_{\sf P}]$ that fulfil the belt condition

$[{\bf f}_{i} + {\bf f}_{j} + {\bf f}_{k} = 0 , \eqno(4)]$

are determined. Their number is N_b = 371, and thus,

$[\Sigma _{1} \left ( {\sf Q} \right ) \, : = \bigcap _{h = 1}^{3N_{b}} {\sf H}_{h} , \eqno(5)]$

is obtained. Because of the large number of half-spaces $[{\sf H}_{h}]$ , the direct calculation of Σ₁ is not practicable. Instead, the calculation of the Φ subcones inside Σ₁ will reveal the walls $[{\sf W}_{i}^{\Sigma _{1}}]$ . Recall that $[{\sf Q}]$ is interior to Σ₁ if

$[{\sf Q} \in {\sf H}_{h}^{+}, \quad h = 1, \cdots, 3N_{b} . \eqno(6)]$

This allows the calculation of all $[\Phi _{k} \subset \Sigma _{1}]$ without explicitly knowing Σ₁, and for the walls of Σ₁ it holds that:

A wall $[{\sf W}_{j}^{\Phi _{k}}]$ of $[\Phi _{k} \subset \Sigma _{1}]$ is a wall of Σ₁ if there exists a wall $[{\sf H}_{h}^{0}]$ , 1 ≤ h ≤ 3N_b, such that

$[{\sf W}_{j}^{\Phi _{k}} = {\sf H}_{h}^{0} . \eqno(7)]$

By calculating a sufficiently large number of $[\Phi _{k} \subset \Sigma _{1}]$ , most of the walls of Σ₁ can be determined. The process converts relatively quickly.

Step S3: In order to verify the result, the induced symmetry of Σ₁ is applied:

For any $[{\sf Q} \in \Sigma _{1}]$ the induced symmetry of Σ₁ is defined by

$[{\cal G}_{\Sigma _{1}} \, : = \left \{ {\sf S}_{i} \mid {\sf Q}_{i} = {\sf S}_{i} {\sf Q} {\sf S}_{i}^{t} \in \Sigma _{1}, \quad \forall \, {\sf S}_{i} \in {\cal G}_{E_{6}^{*}} \right \} . \eqno(8)]$

The centre

$[{\sf C} \, : = \sum _{\forall \, {\sf Q}_{i} \in \Sigma _{1}} {\sf Q}_{i} , \eqno(9)]$

is invariant under the group $[{\cal G}_{\Sigma _{1}}]$ and lies in Σ₁. Applying the symmetry $[{\cal G}_{\Sigma _{1}}]$ to the walls of Σ₁ proves that there are 166 walls which belong to seven classes under $[{\cal G}_{\Sigma _{1}}]$ , and these are shown in Table 1. The neighbouring Σ_s, s = 0, 1, 2, are given in Table 2.

Table 1
Wall normals n_i for the wall classes of Σ₁ in E⁶

Class	Order	n₁₁, …, n₁₆ / n₂₂, …, n₂₆ / n₃₃, … / n₆₆	Neighbour
1	1	0 −1 1 0 0 1 / 0 0 −1 1 −1 / 0 1 −1 1 / 0 0 1 / 0 −1 / −1	Σ₀
2	5	0 −1 1 −1 1 / 0 0 0 0 0 / 0 0 0 0 / 0 0 0 / 0 0 / 0	Σ₁
3	30	0 0 0 0 0 0 / 0 0 0 0 0 / 0 1 0 0 / 0 0 0 / 0 0 / 0	$[\Sigma _{2}^{1}]$
4	40	0 0 0 0 0 1 / 0 0 0 0 0 / 0 0 0 0 / 0 0 0 / 0 0 / 0	$[\Sigma _{2}^{2}]$
5	10	0 1 0 0 0 0 / 0 0 0 0 0 / 0 0 0 0 / 0 0 0 / 0 0 / 0	$[\Sigma _{2}^{3}]$
6	20	0 0 0 0 0 0 / 0 −1 0 0 0 / 1 0 0 1 / 0 0 0 / 0 0 / 0	$[\Sigma _{2}^{4}]$
7	60	0 0 0 0 0 0 / 0 0 0 0 0 / 0 0 0 0 / 0 0 0 / 0 1 / 0	$[\Sigma _{2}^{5}]$

Table 2
Σ classes for levels 0–3 in E⁶

s	Class	Subcone	Order	Neighbours
0	Σ₀	216	103860	[216]Σ₁
1	Σ₁	166	480	Σ₀, [5]Σ₁, $[[30]\Sigma _{2}^{1}]$ , $[[40]\Sigma _{2}^{2}]$ , $[[10]\Sigma _{2}^{3}]$ , $[[20]\Sigma _{2}^{4}]$ , $[[60]\Sigma _{2}^{5}]$
2	$[\Sigma _{2}^{1}]$	129	32	[2]Σ₁, $[[4]\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{3}]$ , $[[8]\Sigma _{3}^{1}]$ , $[[8]\Sigma _{3}^{2}]$ , $[[8]\Sigma _{3}^{3}]$ , $[[8]\Sigma _{3}^{4}]$ , $[[8]\Sigma _{3}^{5}]$ , $[[4]\Sigma _{3}^{6}]$ , $[[16]\Sigma _{3}^{7}]$ , $[[16]\Sigma _{3}^{8}]$ , $[[8]\Sigma _{3}^{9}]$ , $[[8+8]\Sigma _{3}^{10}]$ , $[[4]\Sigma _{3}^{11}]$ , $[[4]\Sigma _{3}^{12}]$ , $[[8]\Sigma _{3}^{13}]$ , $[[4]\Sigma _{3}^{14}]$ , $[\Sigma _{3}^{44}]$
	$[\Sigma _{2}^{2}]$	129	24	[2]Σ₁, $[[2]\Sigma _{2}^{2}]$ , $[[2]\Sigma _{2}^{4}]$ , $[[6]\Sigma _{2}^{5}]$ , $[[12]\Sigma _{3}^{2}]$ , $[[3]\Sigma _{3}^{3}]$ , $[[6]\Sigma _{3}^{4}]$ , $[[6]\Sigma _{3}^{5}]$ , $[[12]\Sigma _{3}^{8}]$ , $[[6]\Sigma _{3}^{14}]$ , $[[6]\Sigma _{3}^{15}]$ , $[[6]\Sigma _{3}^{16}]$ , $[[6]\Sigma _{3}^{17}]$ , $[[6+6]\Sigma _{3}^{18}]$ , $[[6]\Sigma _{3}^{19}]$ , $[[6]\Sigma _{3}^{20}]$ , $[[6]\Sigma _{3}^{21}]$ , $[[6]\Sigma _{3}^{22}]$ , $[[12]\Sigma _{3}^{23}]$ , $[[3]\Sigma _{3}^{45}]$ , $[[2]\Sigma _{3}^{46}]$ , $[\Sigma _{3}^{47}]$
	$[\Sigma _{2}^{3}]$	138	96	[2]Σ₁, $[[6]\Sigma _{2}^{1}]$ , $[[4]\Sigma _{2}^{6}]$ , $[[24]\Sigma _{3}^{5}]$ , $[[12]\Sigma _{3}^{6}]$ , $[[24]\Sigma _{3}^{19}]$ , $[[24]\Sigma _{3}^{22}]$ , $[[12]\Sigma _{3}^{24}]$ , $[[24]\Sigma _{3}^{25}]$ , $[[6]\Sigma _{3}^{44}]$
	$[\Sigma _{2}^{4}]$	134	48	[2]Σ₁, $[[4]\Sigma _{2}^{2}]$ , $[[6]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{2}^{6}]$ , $[[24]\Sigma _{3}^{7}]$ , $[[6]\Sigma _{3}^{9}]$ , $[[12]\Sigma _{3}^{11}]$ , $[[3]\Sigma _{3}^{12}]$ , $[[6]\Sigma _{3}^{15}]$ , $[[12]\Sigma _{3}^{19}]$ , $[[12]\Sigma _{3}^{20}]$ , $[[6]\Sigma _{3}^{21}]$ , $[[3]\Sigma _{3}^{24}]$ , $[[6]\Sigma _{3}^{26}]$ , $[[6]\Sigma _{3}^{27}]$ , $[[12]\Sigma _{3}^{28}]$ $[[6]\Sigma _{3}^{29}]$ , $[[2]\Sigma _{3}^{46}]$ , $[\Sigma _{3}^{48}]$ , $[[3]\Sigma _{3}^{49}]$
	$[\Sigma _{2}^{5}]$	125	16	[2]Σ₁, $[[4]\Sigma _{2}^{2}]$ , $[[2]\Sigma _{2}^{4}]$ , $[[4]\Sigma _{2}^{5}]$ , $[[8]\Sigma _{3}^{1}]$ , $[[4]\Sigma _{3}^{4}]$ , $[[2]\Sigma _{3}^{6}]$ , $[[8]\Sigma _{3}^{7}]$ , $[[8]\Sigma _{3}^{8}]$ , $[[4]\Sigma _{3}^{10}]$ , $[[8]\Sigma _{3}^{13}]$ , $[[4+4]\Sigma _{3}^{16}]$ , $[[8]\Sigma _{3}^{17}]$ , $[[4]\Sigma _{3}^{20}]$ , $[[4]\Sigma _{3}^{22}]$ , $[[4]\Sigma _{3}^{23}]$ , $[[8]\Sigma _{3}^{25}]$ , $[[4]\Sigma _{3}^{26}]$ , $[\Sigma _{3}^{27}]$ , $[[8]\Sigma _{3}^{28}]$ , $[[8+4]\Sigma _{3}^{30}]$ , $[[4]\Sigma _{3}^{45}]$ , $[[2]\Sigma _{3}^{49}]$
	$[\Sigma _{2}^{6}]$	105.256280	12	$[\Sigma _{2}^{3}]$ , $[\Sigma _{2}^{4}]$ , $[\Sigma _{2}^{6}]$ , $[[3]\Sigma _{3}^{12}]$ , $[[6]\Sigma _{3}^{31}]$ , $[[6]\Sigma _{3}^{32}]$ , $[[12]\Sigma _{3}^{33}]$ , $[[6]\Sigma _{3}^{34}]$ , $[[3]\Sigma _{3}^{35}]$ , $[[12]\Sigma _{3}^{36}]$ , $[[6]\Sigma _{3}^{37}]$ , $[[6]\Sigma _{3}^{38}]$ , $[[6]\Sigma _{3}^{39}]$ , $[[12]\Sigma _{3}^{40}]$ , $[[6]\Sigma _{3}^{41}]$ , $[[3]\Sigma _{3}^{42}]$ , $[[6]\Sigma _{3}^{43}]$ , $[[3]\Sigma _{3}^{50}]$ , $[[6]\Sigma _{3}^{51}]$
3	$[\Sigma _{3}^{1}]$	129	4	$[\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{1}]$ , $[\Sigma _{3}^{2}]$ , $[[2]\Sigma _{3}^{4}]$ , $[[2]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{8}]$ , $[[2]\Sigma _{3}^{22}]$ , [115]Σ₄
	$[\Sigma _{3}^{2}]$	103.497315	4	$[[2]\Sigma _{2}^{2}]$ , $[\Sigma _{3}^{1}]$ , $[[2]\Sigma _{3}^{2}]$ , $[[2]\Sigma _{3}^{4}]$ , $[[2]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{8}]$ , $[[2]\Sigma _{3}^{22}]$ , [90]Σ₄
	$[\Sigma _{3}^{3}]$	99.344740	8	$[[2]\Sigma _{2}^{1}]$ , $[[4]\Sigma _{3}^{5}]$ , $[[2]\Sigma _{3}^{9}]$ , $[[2]\Sigma _{3}^{12}]$ , [89]Σ₄
	$[\Sigma _{3}^{4}]$	100.361510	4	$[\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{1}]$ , $[[2]\Sigma _{3}^{2}]$ , $[\Sigma _{3}^{4}]$ , $[\Sigma _{3}^{11}]$ , $[[2]\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{14}]$ , $[[1+1]\Sigma _{3}^{22}]$ , [87]Σ₄
	$[\Sigma _{3}^{5}]$	105.390562	4	$[\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{3}]$ , $[[1+1]\Sigma _{3}^{3}]$ , $[[1+1]\Sigma _{3}^{5}]$ , $[\Sigma _{3}^{6}]$ , $[\Sigma _{3}^{9}]$ , $[\Sigma _{3}^{10}]$ , $[\Sigma _{3}^{24}]$ , $[[2]\Sigma _{3}^{36}]$ , $[\Sigma _{3}^{39}]$ , [89]Σ₄
	$[\Sigma _{3}^{6}]$	107.485226	8	$[\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{3}]$ , $[[2]\Sigma _{3}^{5}]$ , $[[2+2]\Sigma _{3}^{10}]$ , $[\Sigma _{3}^{12}]$ , $[\Sigma _{3}^{24}]$ , $[[2]\Sigma _{3}^{38}]$ , $[\Sigma _{3}^{42}]$ , [94]Σ₄
	$[\Sigma _{3}^{7}]$	103.401973	2	$[\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{4}]$ , $[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{1}]$ , $[\Sigma _{3}^{2}]$ , $[[1+1]\Sigma _{3}^{7}]$ , $[[1+1+1]\Sigma _{3}^{8}]$ , $[\Sigma _{3}^{11}]$ , $[\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{25}]$ , $[\Sigma _{3}^{33}]$ , $[\Sigma _{3}^{36}]$ , [87]Σ₄
	$[\Sigma _{3}^{8}]$	98.307538	2	$[\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{1}]$ , $[\Sigma _{3}^{2}]$ , $[[1+1+1]\Sigma _{3}^{7}]$ , $[[1+1]\Sigma _{3}^{8}]$ , $[\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{14}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{25}]$ , [84]Σ₄
	$[\Sigma _{3}^{9}]$	99.282678	8	$[[2]\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{3}^{3}]$ , $[[2]\Sigma _{3}^{5}]$ , $[\Sigma _{3}^{9}]$ , $[[2]\Sigma _{3}^{10}]$ , $[\Sigma _{3}^{24}]$ , [88]Σ₄
	$[\Sigma _{3}^{10}]$	99.311756	4	$[[1+1]\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{3}]$ , $[\Sigma _{3}^{5}]$ , $[[1+1]\Sigma _{3}^{6}]$ , $[\Sigma _{3}^{9}]$ , $[[1+1]\Sigma _{3}^{10}]$ , $[\Sigma _{3}^{12}]$ , [88]Σ₄
	$[\Sigma _{3}^{11}]$	109.471017	8	$[\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{4}]$ , $[[2]\Sigma _{3}^{4}]$ , $[[4]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{14}]$ , $[[2]\Sigma _{3}^{19}]$ , $[[2]\Sigma _{3}^{32}]$ , $[[2]\Sigma _{3}^{38}]$ , [91]Σ₄
	$[\Sigma _{3}^{12}]$	112.627736	16	$[[2]\Sigma _{2}^{1}]$ , $[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{2}^{6}]$ , $[[2]\Sigma _{3}^{6}]$ , $[[4]\Sigma _{3}^{10}]$ , $[[2]\Sigma _{3}^{12}]$ , $[[2]\Sigma _{3}^{52}]$ , [97]Σ₄
	$[\Sigma _{3}^{13}]$	95.226124	4	$[\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{4}]$ , $[[2]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{8}]$ , $[\Sigma _{3}^{11}]$ , $[\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{14}]$ , $[[2]\Sigma _{3}^{25}]$ , [82]Σ₄
	$[\Sigma _{3}^{14}]$	98.258158	8	$[\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{2}]$ , $[[4]\Sigma _{3}^{8}]$ , $[[2]\Sigma _{3}^{4}]$ , $[[2]\Sigma _{3}^{19}]$ , $[[2]\Sigma _{3}^{13}]$ , $[\Sigma _{3}^{11}]$ , [84]Σ₄
	$[\Sigma _{3}^{15}]$	108.550026	8	$[[2]\Sigma _{2}^{2}]$ , $[\Sigma _{3}^{15}]$ , $[[2]\Sigma _{3}^{16}]$ , $[[2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{20}]$ , $[[4]\Sigma _{3}^{23}]$ , $[[2]\Sigma _{3}^{26}]$ , $[[2]\Sigma _{3}^{27}]$ , $[[2]\Sigma _{3}^{39}]$ , [89]Σ₄
	$[\Sigma _{3}^{16}]$	94.214427	4	$[\Sigma _{2}^{2}]$ , $[[1+1]\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{15}]$ , $[[1+2]\Sigma _{3}^{16}]$ , $[[1+1]\Sigma _{3}^{18}]$ , $[[1+1]\Sigma _{3}^{20}]$ , $[[2]\Sigma _{3}^{23}]$ , $[\Sigma _{3}^{26}]$ , $[\Sigma _{3}^{27}]$ , $[[2]\Sigma _{3}^{30}]$ , [77]Σ₄
	$[\Sigma _{3}^{17}]$	96.285460	4	$[\Sigma _{2}^{2}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{17}]$ , $[[2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{21}]$ , $[[2]\Sigma _{3}^{23}]$ , $[[2+2]\Sigma _{3}^{28}]$ , $[[2]\Sigma _{3}^{30}]$ , [79]Σ₄
	$[\Sigma _{3}^{18}]$	98.321970	4	$[[1+1]\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{15}]$ , $[[1+1]\Sigma _{3}^{16}]$ , $[[2]\Sigma _{3}^{17}]$ , $[\Sigma _{3}^{18}]$ , $[\Sigma _{3}^{20}]$ , $[[1+1[\Sigma _{3}^{21}]$ , $[[2]\Sigma _{3}^{23}]$ , $[\Sigma _{3}^{26}]$ , $[[2]\Sigma _{3}^{30}]$ , [81]Σ₄
	$[\Sigma _{3}^{19}]$	112.567544	4	$[\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{3}]$ , $[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{8}]$ , $[\Sigma _{3}^{11}]$ , $[\Sigma _{3}^{14}]$ , $[\Sigma _{3}^{19}]$ , $[[1+1]\Sigma _{3}^{22}]$ , $[[2]\Sigma _{3}^{25}]$ , $[\Sigma _{3}^{31}]$ , $[\Sigma _{3}^{32}]$ , $[\Sigma _{3}^{34}]$ , $[\Sigma _{3}^{37}]$ , $[\Sigma _{3}^{41}]$ , [93]Σ₄
	$[\Sigma _{3}^{20}]$	103.395491	4	$[\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{4}]$ , $[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{15}]$ , $[\Sigma _{3}^{16}]$ , $[[1+1]\Sigma _{3}^{18}]$ , $[\Sigma _{3}^{20}]$ , $[\Sigma _{3}^{21}]$ , $[[2]\Sigma _{3}^{23}]$ , $[\Sigma _{3}^{26}]$ , $[[2]\Sigma _{3}^{28}]$ , $[\Sigma _{3}^{29}]$ , $[[2]\Sigma _{3}^{30}]$ , $[\Sigma _{3}^{34}]$ , $[\Sigma _{3}^{43}]$ , [84]Σ₄
	$[\Sigma _{3}^{21}]$	103.387689	8	$[[2]\Sigma _{2}^{2}]$ , $[]\Sigma _{2}^{4}]$ , $[[4]\Sigma _{3}^{17}]$ , $[[2+2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{20}]$ , $[[2]\Sigma _{3}^{21}]$ , $[[2]\Sigma _{3}^{28}]$ , $[\Sigma _{3}^{29}]$ , $[[2]\Sigma _{3}^{51}]$ , [83]Σ₄
	$[\Sigma _{3}^{22}]$	108.531296	4	$[\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{3}]$ , $[\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{1}]$ , $[[2]\Sigma _{3}^{2}]$ , $[[1+1]\Sigma _{3}^{4}]$ , $[[1+1]\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{22}]$ , $[[2]\Sigma _{3}^{25}]$ , $[\Sigma _{3}^{37}]$ , $[\Sigma _{3}^{43}]$ , $[\Sigma _{3}^{51}]$ , [91]Σ₄
	$[\Sigma _{3}^{23}]$	99.343145	4	$[[2]\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{15}]$ , $[[2]\Sigma _{3}^{16}]$ , $[[2]\Sigma _{3}^{17}]$ , $[[2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{20}]$ , $[[2]\Sigma _{3}^{23}]$ , $[[2]\Sigma _{3}^{28}]$ , $[\Sigma _{3}^{30}]$ , [81]Σ₄
	$[\Sigma _{3}^{24}]$	114.572071	16	$[[2]\Sigma _{2}^{3}]$ , $[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{3}^{6}]$ , $[[4]\Sigma _{3}^{5}]$ , $[[2]\Sigma _{3}^{9}]$ , $[[4]\Sigma _{3}^{31}]$ , $[[2]\Sigma _{3}^{35}]$ , $[[2]\Sigma _{3}^{50}]$ , [95]Σ₄
	$[\Sigma _{3}^{25}]$	104.367659	4	$[\Sigma _{2}^{3}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{7}]$ , $[[2]\Sigma _{3}^{8}]$ , $[[2]\Sigma _{3}^{13}]$ , $[[2]\Sigma _{3}^{19}]$ , $[[2]\Sigma _{3}^{22}]$ , $[\Sigma _{3}^{25}]$ , $[[2]\Sigma _{3}^{33}]$ , $[[2]\Sigma _{3}^{40}]$ , [86]Σ₄
	$[\Sigma _{3}^{26}]$	98.295360	8	$[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{15}]$ , $[[2]\Sigma _{3}^{16}]$ , $[[2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{20}]$ , $[[2]\Sigma _{3}^{27}]$ , $[\Sigma _{3}^{26}]$ , $[[4]\Sigma _{3}^{30}]$ , [80]Σ₄
	$[\Sigma _{3}^{27}]$	111.490894	16	$[[2]\Sigma _{2}^{4}]$ , $[\Sigma _{2}^{4}]$ , $[[4]\Sigma _{3}^{15}]$ , $[[4]\Sigma _{3}^{16}]$ , $[[4]\Sigma _{3}^{26}]$ , $[[2]\Sigma _{3}^{27}]$ , $[[2]\Sigma _{3}^{35}]$ , $[[2]\Sigma _{3}^{42}]$ , $[[2]\Sigma _{3}^{52}]$ , [88]Σ₄
	$[\Sigma _{3}^{28}]$	99.298816	4	$[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2+2]\Sigma _{3}^{17}]$ , $[[2]\Sigma _{3}^{20}]$ , $[\Sigma _{3}^{21}]$ , $[[2]\Sigma _{3}^{23}]$ , $[[2+2]\Sigma _{3}^{28}]$ , $[\Sigma _{3}^{29}]$ , $[[2]\Sigma _{3}^{38}]$ , $[[2]\Sigma _{3}^{40}]$ , [78]Σ₄
	$[\Sigma _{3}^{29}]$	113.641446	24	$[[3]\Sigma _{2}^{4}]$ , $[[6]\Sigma _{3}^{20}]$ , $[[3]\Sigma _{3}^{21}]$ , $[[6]\Sigma _{3}^{28}]$ , $[[6]\Sigma _{3}^{41}]$ , [89]Σ₄
	$[\Sigma _{3}^{30}]$	90.184843	4	$[[2+1]\Sigma _{2}^{5}]$ , $[[2]\Sigma _{3}^{16}]$ , $[[2]\Sigma _{3}^{17}]$ , $[[2]\Sigma _{3}^{18}]$ , $[[2]\Sigma _{3}^{20}]$ , $[\Sigma _{3}^{23}]$ , $[[2]\Sigma _{3}^{26}]$ , $[[2]\Sigma _{3}^{28}]$ , $[[2]\Sigma _{3}^{30}]$ , [72]Σ₄
	$[\Sigma _{3}^{31}]$	86.61862	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{24}]$ , $[[2]\Sigma _{3}^{36}]$ , $[\Sigma _{3}^{37}]$ , $[\Sigma _{3}^{38}]$ , $[\Sigma _{3}^{53}]$ , $[\Sigma _{3}^{56}]$ , [77]Σ₄
	$[\Sigma _{3}^{32}]$	86.59029	4	$[\Sigma _{2}^{6}]$ , $[[2]\Sigma _{3}^{33}]$ , $[\Sigma _{3}^{37}]$ , $[\Sigma _{3}^{11}]$ , $[\Sigma _{3}^{56}]$ , $[\Sigma _{3}^{57}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{38}]$ , [77]Σ₄
	$[\Sigma _{3}^{33}]$	78.36807	2	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{7}]$ , $[\Sigma _{3}^{25}]$ , $[\Sigma _{3}^{32}]$ , $[\Sigma _{3}^{33}]$ , $[\Sigma _{3}^{36}]$ , $[\Sigma _{3}^{37}]$ , [71]Σ₄
	$[\Sigma _{3}^{34}]$	84.57367	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{20}]$ , $[[2]\Sigma _{3}^{40}]$ , $[\Sigma _{3}^{41}]$ , $[[1+1]\Sigma _{3}^{43}]$ , $[\Sigma _{3}^{51}]$ , [75]Σ₄
	$[\Sigma _{3}^{35}]$	90.79939	8	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{24}]$ , $[\Sigma _{3}^{27}]$ , $[\Sigma _{3}^{35}]$ , $[[2]\Sigma _{3}^{39}]$ , $[[1+1]\Sigma _{3}^{42}]$ , $[\Sigma _{3}^{52}]$ , [81]Σ₄
	$[\Sigma _{3}^{36}]$	78.36807	2	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{5}]$ , $[\Sigma _{3}^{7}]$ , $[\Sigma _{3}^{31}]$ , $[\Sigma _{3}^{33}]$ , $[\Sigma _{3}^{36}]$ , $[\Sigma _{3}^{38}]$ , [71]Σ₄
	$[\Sigma _{3}^{37}]$	86.61862	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{22}]$ , $[\Sigma _{3}^{31}]$ , $[\Sigma _{3}^{32}]$ , $[[2]\Sigma _{3}^{33}]$ , $[\Sigma _{3}^{54}]$ , $[\Sigma _{3}^{57}]$ , [77]Σ₄
	$[\Sigma _{3}^{38}]$	86.57513	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{57}]$ , $[\Sigma _{3}^{31}]$ , $[[2]\Sigma _{3}^{36}]$ , $[\Sigma _{3}^{11}]$ , $[\Sigma _{3}^{6}]$ , $[\Sigma _{3}^{32}]$ , [78]Σ₄
	$[\Sigma _{3}^{39}]$	83.39029	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{5}]$ , $[\Sigma _{3}^{15}]$ , $[\Sigma _{3}^{35}]$ , $[[1+1]\Sigma _{3}^{39}]$ , $[\Sigma _{3}^{42}]$ , [76]Σ₄
	$[\Sigma _{3}^{40}]$	73.25013	2	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{25}]$ , $[\Sigma _{3}^{28}]$ , $[\Sigma _{3}^{34}]$ , $[[1+1]\Sigma _{3}^{40}]$ , $[\Sigma _{3}^{41}]$ , $[\Sigma _{3}^{43}]$ , $[\Sigma _{3}^{51}]$ , [64]Σ₄
	$[\Sigma _{3}^{41}]$	89.74566	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{19}]$ , $[\Sigma _{3}^{29}]$ , $[\Sigma _{3}^{34}]$ , $[[2]\Sigma _{3}^{40}]$ , $[\Sigma _{3}^{41}]$ , $[\Sigma _{3}^{43}]$ , $[\Sigma _{3}^{51}]$ , $[\Sigma _{3}^{53}]$ , $[[1+1]\Sigma _{3}^{54}]$ , [77]Σ₄
	$[\Sigma _{3}^{42}]$	89.73349	8	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{6}]$ , $[\Sigma _{3}^{27}]$ , $[[2]\Sigma _{3}^{34}]$ , $[[1+1]\Sigma _{3}^{35}]$ , $[[1+1]\Sigma _{3}^{52}]$ , [80]Σ₄
	$[\Sigma _{3}^{43}]$	84.57367	4	$[\Sigma _{2}^{5}]$ , $[\Sigma _{3}^{20}]$ , $[\Sigma _{3}^{22}]$ , $[[1+1]\Sigma _{3}^{34}]$ , $[[2]\Sigma _{3}^{40}]$ , $[\Sigma _{3}^{41}]$ , $[\Sigma _{3}^{51}]$ , [75]Σ₄
	$[\Sigma _{3}^{44}]$	113.559629	32	$[\Sigma _{2}^{1}]$ , $[[2]\Sigma _{2}^{3}]$ , $[[2]\Sigma _{3}^{44}]$ , $[[4]\Sigma _{3}^{50}]$ , [104]Σ₄
	$[\Sigma _{3}^{45}]$	93.276584	8	$[\Sigma _{2}^{2}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[2+2]\Sigma _{3}^{45}]$ , $[[2]\Sigma _{3}^{46}]$ , $[[2]\Sigma _{3}^{47}]$ , $[[2+2+4]\Sigma _{3}^{49}]$ , [74]Σ₄
	$[\Sigma _{3}^{46}]$	99.399178	24	$[[2]\Sigma _{2}^{2}]$ , $[\Sigma _{2}^{4}]$ , $[[6]\Sigma _{3}^{45}]$ , $[[1+2]\Sigma _{3}^{46}]$ , $[[2]\Sigma _{3}^{47}]$ , $[\Sigma _{3}^{48}]$ , $[[3]\Sigma _{3}^{49}]$ , [81]Σ₄
	$[\Sigma _{3}^{47}]$	99.417350	72	$[[3]\Sigma _{2}^{2}]$ , $[[9]\Sigma _{3}^{45}]$ , $[[6]\Sigma _{3}^{46}]$ , [81]Σ₄
	$[\Sigma _{3}^{48}]$	102.398634	144	$[[3]\Sigma _{2}^{4}]$ , $[[6]\Sigma _{3}^{46}]$ , $[[9]\Sigma _{3}^{49}]$ , [84]Σ₄
	$[\Sigma _{3}^{49}]$	96.338198	16	$[\Sigma _{2}^{4}]$ , $[[2]\Sigma _{2}^{5}]$ , $[[4+4]\Sigma _{3}^{45}]$ , $[[2]\Sigma _{3}^{46}]$ , $[\Sigma _{3}^{48}]$ , $[[2+2]\Sigma _{3}^{49}]$ , [78]Σ₄
	$[\Sigma _{3}^{50}]$	89.80719	8	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{24}]$ , $[\Sigma _{3}^{44}]$ , $[\Sigma _{3}^{50}]$ , $[\Sigma _{3}^{55}]$ , $[\Sigma _{3}^{58}]$ , [83]Σ₄
	$[\Sigma _{3}^{51}]$	81.41501	4	$[\Sigma _{2}^{6}]$ , $[\Sigma _{3}^{21}]$ , $[\Sigma _{3}^{22}]$ , $[\Sigma _{3}^{34}]$ , $[[2]\Sigma _{3}^{40}]$ , $[\Sigma _{3}^{41}]$ , $[\Sigma _{3}^{43}]$ , $[\Sigma _{3}^{51}]$ , [72]Σ₄
	$[\Sigma _{3}^{52}]$	83.35433	8	$[\Sigma _{3}^{12}]$ , $[\Sigma _{3}^{27}]$ , $[\Sigma _{3}^{35}]$ , $[[1+1]\Sigma _{3}^{42}]$ , $[\Sigma _{3}^{52}]$ , [77]Σ₄
	$[\Sigma _{3}^{53}]$	73.14453	8	$[[2]\Sigma _{3}^{41}]$ , $[[2]\Sigma _{3}^{31}]$ , $[[2]\Sigma _{3}^{54}]$ , [67]Σ₄
	$[\Sigma _{3}^{54}]$	71.12163	4	$[\Sigma _{3}^{37}]$ , $[[1+1]\Sigma _{3}^{41}]$ , $[\Sigma _{3}^{53}]$ , [67]Σ₄
	$[\Sigma _{3}^{55}]$	78.28445	16	$[[2]\Sigma _{3}^{35}]$ , $[[2]\Sigma _{3}^{50}]$ , $[[2+2]\Sigma _{3}^{58}]$ , [70]Σ₄
	$[\Sigma _{3}^{56}]$	69.13979	8	$[[2]\Sigma _{3}^{31}]$ , $[[2]\Sigma _{3}^{32}]$ , $[[2]\Sigma _{3}^{57}]$ , [63]Σ₄
	$[\Sigma _{3}^{57}]$	68.12463	4	$[\Sigma _{3}^{56}]$ , $[\Sigma _{3}^{38}]$ , $[\Sigma _{3}^{32}]$ , $[\Sigma _{3}^{37}]$ , [64]Σ₄
	$[\Sigma _{3}^{58}]$	75.21855	8	$[\Sigma _{3}^{35}]$ , $[\Sigma _{3}^{42}]$ , $[\Sigma _{3}^{50}]$ , $[[1+1]\Sigma _{3}^{55}]$ , [70]Σ₄

Step S4: For each Σ_s obtained, we proceed analogously to steps S1 to S3 in order to get further Σ_s. For each new Σ_s we have to check their equivalence:

$[\Sigma _{s}^{k}]$ and $[\Sigma _{s}^{l}]$ are arithmetically equivalent and belong to the same equivalence class $[{\cal E}_{\Sigma _{s}^{k}}]$ if there exists $[{\sf A} \in GL_{d} ({\bb Z})]$ such that for any $[{\sf Q}_{j} \in \Sigma _{s}^{l}]$ it holds that

$[{\sf Q}_{i} = {\sf A} {\sf Q}_{j} {\sf A}^{t} \in \Sigma _{s}^{k} . \eqno(10)]$

Because $[GL_{d} ({\bb Z})]$ is of infinite order, the above equation is not practicable. However, if optimal bases are admitted to the forms $[{\sf Q}]$ only, then the number of transformations $[{\sf A}]$ that have to be taken into account becomes finite. It was discovered for every Σ_s at maximal path length 5 from Σ₀ (see Fig. 1) that it is sufficient to consider $[{\sf A} \in {\cal G}_{E_{6}^{*}}]$ only, in order to verify equivalence. If equivalence is proved for any $[{\sf Q}_{j}]$ then it holds for all $[{\sf Q} \in \Sigma _{s}^{l}]$ .

Figure 1
Shortest paths among the Σ_s classes for levels s = 0–3 in E⁶.

Alternatively, the combinatorial equivalence of parallelohedra may be compared:

$[\Sigma _{s}^{k}]$ and $[\Sigma _{s}^{l}]$ are equivalent and belong to the same equivalence class if there exist $[{\sf Q}_{i} \in \Sigma _{s}^{k}]$ and $[{\sf Q}_{j} \in \Sigma _{s}^{l}]$ such that

$[{\sf P} \left ( {\sf Q}_{i} \right )_{\simeq}^{\rm comb} \, {\sf P} \left ( {\sf Q}_{j} \right ) . \eqno(11)]$

The latter procedure requires a sufficiently large number of Φ cones to be determined in order to find at least one equivalent pair.

Analogously, using the procedures described in steps S1 to S4 the Σ_s cones, s = 21, 20, 19, 18, were successively determined starting with Σ₂₁ as a representative of its class $[{\cal E}_{\Sigma _{21}}]$ . Recall that Σ₂₁ has 21 walls $[{\sf W}_{i}^{\Sigma _{21}}]$ , $[i = 1, \cdots, 21]$ , which all are equivalent under the group $[{\cal G}_{A_{6}^{*}}]$ (see Engel, 2019).

3. Results

In Table 2 are given the Σ_s classes, s = 0, 1, 2, 3, under the general linear group $[GL_{d} ({\bb Z})]$ . Each equivalence class $[{\cal E}_{\Sigma _{s}^{i}}]$ is given by its representative $[\Sigma _{s}^{i}]$ . $[\Sigma _{s}^{i} ({\sf Q})]$ is chosen such that $[{\sf Q}]$ becomes optimal. Under the heading `s' is given the number of closed zones. Under the heading `Subcone' is stated the number of walls of $[\Sigma _{s}^{i}]$ . In cases where the complete Σ_s cone was calculated, the numbers of walls and edges are indicated as N_w and N_e, respectively. Under the heading `Order' is given the order of the induced symmetry under $[{\cal G}_{E_{6}^{*}}]$ . Under the heading `Neighbours' are stated the neighbouring $[\Sigma _{s}^{j}]$ , each of them preceded, in brackets, by the number of equivalent subcones under the group $[{\cal G}_{\Sigma _{s}^{i}}]$ . If more than one number is given, it means that they are equivalent under $[{\cal G}_{E_{6}^{*}}]$ . Remarkably, $[\Sigma _{r}^{i}]$ has neighbours with s = r − 1, r, r + 1 only. Note that Σ₄ cones were not determined and the preceding number gives an upper bound for the number of equivalent subcones only. In Fig. 1 are drawn the shortest paths from Σ₀ to each other $[\Sigma _{s}^{i}]$ .

In Table 3 are given the Σ_s classes, s = 21, 20, 19, 18. Under the heading `Subcone' are given the numbers of walls N_w and edges N_e. Most of the cones are simple with Φ and Σ cones identical. In cases where the cone is not simple, two numbers are shown as `a/b' under the heading `Types', where `a' indicates the number of Φ types and `b' gives the total number of Φ cones in $[\Sigma _{s}^{i}]$ . The numbers of Φ types for s = 21, 20, 19, 18 correspond to the numbers given by Baburin & Engel (2013). All these Φ types correspond to principal primitive parallelohedra. Under the heading `Order' is given the order of the induced symmetry under $[{\cal G}_{A_{6}^{*}}]$ . Under the heading `Neighbours' are listed the neighbouring $[\Sigma _{s}^{j}]$ which are preceded, in brackets, by the number of equivalent types under the group $[{\cal G}_{\Sigma _{s}^{i}}]$ . Note that the Σ₁₇ cones were not determined and the preceding number gives an upper bound for the number of equivalent types only. In Fig. 2 are drawn the shortest paths from Σ₂₁ to each other $[\Sigma _{s}^{i}]$ .

Table 3
Σ classes for levels 21–18 in E⁶

s	Class	Subcone	Types	Order	Neighbours
21	Σ₂₁	21.21	1	10080	[21]Σ₂₀
20	Σ₂₀	21.21	1	480	Σ₂₁, $[[10]\Sigma _{19}^{1}]$ , $[[10]\Sigma _{19}^{2}]$
19	$[\Sigma _{19}^{1}]$	25.22	1/2	48	[2]Σ₂₀, $[[8]\Sigma _{18}^{1}]$ , $[[12]\Sigma _{18}^{2}]$ , $[[3]\Sigma _{18}^{3}]$
	$[\Sigma _{19}^{2}]$	21.21	1	48	Σ₂₀, $[\Sigma _{19}^{2}]$ , $[\Sigma _{19}^{3}]$ , $[[4]\Sigma _{18}^{1}]$ , $[[6]\Sigma _{18}^{2}]$ , $[[4]\Sigma _{18}^{4}]$ , $[[4]\Sigma _{18}^{5}]$
	$[\Sigma _{19}^{3}]$	21.21	1	96	$[[2]\Sigma _{19}^{2}]$ , $[\Sigma _{18}^{6}]$ , $[[6]\Sigma _{18}^{7}]$ , $[[8]\Sigma _{18}^{8}]$ , $[[4]\Sigma _{18}^{9}]$
18	$[\Sigma _{18}^{1}]$	21.21	1	12	$[\Sigma _{19}^{1}]$ , $[\Sigma _{19}^{2}]$ , $[\Sigma _{18}^{1}]$ , $[\Sigma _{18}^{10}]$ , [17]Σ₁₇
	$[\Sigma _{18}^{2}]$	25.23	3/3	8	$[\Sigma _{19}^{1}]$ , $[\Sigma _{19}^{2}]$ , $[\Sigma _{18}^{2}]$ , $[\Sigma _{18}^{7}]$ , [21]Σ₁₇
	$[\Sigma _{18}^{3}]$	33.25	2/12	16	$[[3]\Sigma _{19}^{1}]$ , [30]Σ₁₇
	$[\Sigma _{18}^{4}]$	25.22	2/2	24	$[[2]\Sigma _{19}^{2}]$ , $[[2]\Sigma _{18}^{5}]$ , $[[2]\Sigma _{18}^{8}]$ , [19]Σ₁₇
	$[\Sigma _{18}^{5}]$	21.21	1	24	$[[2]\Sigma _{19}^{2}]$ , $[[2]\Sigma _{18}^{4}]$ , $[\Sigma _{18}^{11}]$ , [16]Σ₁₇
	$[\Sigma _{18}^{6}]$	21.21	1	96	$[\Sigma _{19}^{3}]$ , $[[2]\Sigma _{18}^{6}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{7}]$	27.24	4/5	16	$[\Sigma _{19}^{3}]$ , $[[2]\Sigma _{18}^{2}]$ , [24]Σ₁₇
	$[\Sigma _{18}^{8}]$	21.21	1	12	$[\Sigma _{19}^{3}]$ , $[\Sigma _{18}^{4}]$ , $[\Sigma _{18}^{12}]$ , [19]Σ₁₇
	$[\Sigma _{18}^{9}]$	21.21	1	24	$[\Sigma _{19}^{3}]$ , $[\Sigma _{18}^{13}]$ , [19]Σ₁₇
	$[\Sigma _{18}^{10}]$	21.21	1	24	$[[2]\Sigma _{18}^{1}]$ , $[\Sigma _{18}^{12}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{11}]$	21.21	1	24	$[\Sigma _{18}^{5}]$ , $[[2]\Sigma _{18}^{13}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{12}]$	21.21	1	24	$[[2]\Sigma _{18}^{8}]$ , $[\Sigma _{18}^{10}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{13}]$	21.21	1	12	$[\Sigma _{18}^{9}]$ , $[\Sigma _{18}^{11}]$ , $[\Sigma _{18}^{14}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{14}]$	21.21	1	24	$[[2]\Sigma _{18}^{13}]$ , $[\Sigma _{18}^{15}]$ , [18]Σ₁₇
	$[\Sigma _{18}^{15}]$	21.21	1	72	$[[3]\Sigma _{18}^{14}]$ , [18]Σ₁₇

Figure 2
Shortest paths among the Σ_s classes for levels s = 21–18 in E⁶.

References

Baburin, I. A. & Engel, P. (2013). Acta Cryst. A69, 510–516. Web of Science CrossRef IUCr Journals Google Scholar
Engel, P. (2015). Cryst. Res. Technol. 50, 929–943. Web of Science CrossRef Google Scholar
Engel, P. (2019). Acta Cryst. A75, 574–583. CrossRef IUCr Journals Google Scholar
Ryshkov, S. S. (1973). Sov. Math. Dokl. 14, 1314–1318. Google Scholar

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ISSN: 2053-2733

Volume 76| Part 5| September 2020| Pages 622-626

https://doi.org/10.1107/S2053273320009936

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