On Σ-Classes in E8. I. The Neighborhood of E8
Abstract
1. Introduction
2. Basic Notations
3. The Parallelohedron of the Lattice
4. The Neighborhood of
5. Results
Supplementary Materials
Funding
Conflicts of Interest
References
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No. | Group | Order | Comb. Type | Zones | Belts |
---|---|---|---|---|---|
16.256 | 8(8) | ||||
112.272 | 136(0) | ||||
48.576 | 24(0) | ||||
272.1120 | 56(0) | ||||
240.19449 | 8760(0) | ||||
264.5304 | 36(0) | ||||
24.1296 | 12(12) | ||||
348.3588 | 93(0) | ||||
60.10404 | 18(18) | ||||
72.510 | 9(9) | ||||
186.5940 | 174(0) | ||||
402.94254 | 1497(0) | ||||
510.362880 | 36(36) | ||||
40.900 | 10(10) | ||||
360.3840 | 180(0) | ||||
60.14400 | 20(20) | ||||
290.43310 | 1195(0) | ||||
390.106200 | 450(0) | ||||
180.6510 | 15(15) | ||||
1152 | 296.13696 | 580(0) | |||
1152 | 200.2992 | 304(0) | |||
456.28632 | 264(0) | ||||
672 | 398.90384 | 297(0) | |||
673 | 426.37638 | 21(21) | |||
672 | 308.8028 | 1011(0) | |||
24 | 398.127848 | 1036(0) |
1 0 0 0 0 0 0 0 | 0 1 0 0 0 0 0 0 | 1 1 0 0 0 0 0 0 | |||
0 0 1 0 0 0 0 0 | 1 0 1 0 0 0 0 0 | 0 1 −1 0 0 0 0 0 | |||
0 0 0 1 0 0 0 0 | 0 0 1 1 0 0 0 0 | 1 0 1 1 0 0 0 0 | |||
0 1 −1 −1 0 0 0 0 | 0 0 0 0 1 0 0 0 | 1 0 0 0 1 0 0 0 | |||
0 1 0 0 −1 0 0 0 | 0 0 1 0 −1 0 0 0 | 0 0 1 1 −1 0 0 0 | |||
0 0 0 0 0 1 0 0 | 0 1 0 0 0 1 0 0 | 0 1 −1 0 0 1 0 0 | |||
0 0 0 1 0 1 0 0 | 0 1 0 1 0 1 0 0 | 0 0 1 1 0 1 0 0 | |||
0 0 0 0 1 1 0 0 | 0 1 −1 0 1 1 0 0 | 0 0 0 1 1 1 0 0 | |||
1 0 0 0 0 −1 0 0 | 1 0 1 0 0 −1 0 0 | 1 −1 1 0 0 −1 0 0 | |||
1 0 0 −1 0 −1 0 0 | 0 0 1 0 −1 −1 0 0 | 1 0 1 0 −1 −1 0 0 | |||
0 0 0 0 0 0 1 0 | 1 0 0 0 0 0 1 0 | 0 0 0 1 0 0 1 0 | |||
1 0 0 1 0 0 1 0 | 1 0 1 1 0 0 1 0 | 1 −1 1 1 0 0 1 0 | |||
1 0 0 0 1 0 1 0 | 1 −1 0 0 1 0 1 0 | 1 0 −1 0 1 0 1 0 | |||
1 0 0 1 1 0 1 0 | 1 −1 0 1 1 0 1 0 | 1 −1 1 1 1 0 1 0 | |||
0 0 0 1 0 1 1 0 | 0 0 0 1 1 1 1 0 | 1 0 0 1 1 1 1 0 | |||
1 0 0 0 0 −1 1 0 | 1 −1 0 0 0 −1 1 0 | 1 −1 1 0 0 −1 1 0 | |||
1 −1 1 1 0 −1 1 0 | 1 −1 0 0 1 −1 1 0 | 0 1 0 0 0 0 −1 0 | |||
0 0 1 0 0 0 −1 0 | 0 1 0 −1 0 0 −1 0 | 0 1 −1 −1 0 0 −1 0 | |||
0 1 0 0 −1 0 −1 0 | 0 0 1 0 −1 0 −1 0 | 0 1 1 0 −1 0 −1 0 | |||
0 1 0 −1 −1 0 −1 0 | 0 1 0 0 0 1 −1 0 | 0 0 1 0 −1 −1 −1 0 | |||
0 1 0 −1 −1 −1 −1 0 | 0 0 1 −1 −1 −1 −1 0 | 1 −1 0 1 1 0 2 0 | |||
0 0 0 0 0 0 0 1 | 1 0 0 0 0 0 0 1 | 1 1 0 0 0 0 0 1 | |||
0 1 −1 0 0 0 0 1 | 1 1 −1 0 0 0 0 1 | 0 0 0 1 0 0 0 1 | |||
1 0 0 1 0 0 0 1 | 1 1 0 1 0 0 0 1 | 1 0 1 1 0 0 0 1 | |||
1 0 0 0 1 0 0 1 | 1 0 −1 0 1 0 0 1 | 1 1 −1 0 1 0 0 1 | |||
1 0 0 1 1 0 0 1 | 0 1 −1 0 0 1 0 1 | 0 0 0 1 0 1 0 1 | |||
0 1 0 1 0 1 0 1 | 1 1 0 1 0 1 0 1 | 0 1 −1 1 0 1 0 1 | |||
0 1 −1 0 1 1 0 1 | 1 1 −1 0 1 1 0 1 | 0 1 −2 0 1 1 0 1 | |||
0 0 0 1 1 1 0 1 | 1 0 0 1 1 1 0 1 | 1 1 0 1 1 1 0 1 | |||
0 1 −1 1 1 1 0 1 | 1 1 −1 1 1 1 0 1 | 1 0 0 0 0 −1 0 1 | |||
0 1 −1 1 1 2 0 1 | 1 0 0 1 0 0 1 1 | 1 0 −1 0 1 0 1 1 | |||
1 0 −1 1 1 0 1 1 | 1 0 0 1 1 1 1 1 | 1 0 −1 1 1 1 1 1 | |||
1 1 −1 1 1 1 1 1 | 1 0 0 2 1 1 1 1 | 1 0 −1 1 2 1 1 1 | |||
0 1 0 0 0 0 −1 1 | 1 1 0 0 0 0 −1 1 | 0 1 −1 0 0 0 −1 1 | |||
0 1 −1 −1 0 0 −1 1 | 0 1 0 0 −1 0 −1 1 | 0 1 0 0 0 1 −1 1 | |||
0 1 −1 0 0 1 −1 1 | 0 2 −1 0 0 1 −1 1 | 0 1 0 1 0 1 −1 1 | |||
0 1 −1 0 1 1 −1 1 | 0 0 1 0 0 0 0 −1 | 0 0 1 0 −1 0 0 −1 | |||
0 0 1 0 −1 −1 0 −1 | 0 0 1 −1 −1 −1 0 −1 | 0 0 0 0 0 0 1 −1 | |||
1 −1 1 0 0 −1 1 −1 | 0 0 1 −1 −1 −1 −1 −1 | 1 1 −1 1 1 1 0 2 |
1 0 0 0 0 0 0 0 | 0 1 0 0 0 0 0 0 | 1 1 0 0 0 0 0 0 | |||
0 0 1 0 0 0 0 0 | 1 0 1 0 0 0 0 0 | 0 1 −1 0 0 0 0 0 | |||
0 0 0 1 0 0 0 0 | 0 0 1 1 0 0 0 0 | 1 0 1 1 0 0 0 0 | |||
0 1 −1 −1 0 0 0 0 | 0 0 0 0 1 0 0 0 | 1 0 0 0 1 0 0 0 | |||
0 1 0 0 −1 0 0 0 | 0 0 1 0 −1 0 0 0 | 0 0 1 1 −1 0 0 0 | |||
0 0 0 0 0 1 0 0 | 0 1 0 0 0 1 0 0 | 0 1 −1 0 0 1 0 0 | |||
0 0 0 1 0 1 0 0 | 0 1 0 1 0 1 0 0 | 0 0 1 1 0 1 0 0 | |||
0 0 0 0 1 1 0 0 | 0 1 −1 0 1 1 0 0 | 0 0 0 1 1 1 0 0 | |||
1 0 0 0 0 −1 0 0 | 1 0 1 0 0 −1 0 0 | 1 −1 1 0 0 −1 0 0 | |||
1 0 0 −1 0 −1 0 0 | 0 0 1 0 −1 −1 0 0 | 1 0 1 0 −1 −1 0 0 | |||
0 0 0 0 0 0 1 0 | 1 0 0 0 0 0 1 0 | 0 0 0 1 0 0 1 0 | |||
1 0 0 1 0 0 1 0 | 1 0 1 1 0 0 1 0 | 1 −1 1 1 0 0 1 0 | |||
1 0 0 0 1 0 1 0 | 1 −1 0 0 1 0 1 0 | 1 0 −1 0 1 0 1 0 | |||
1 0 0 1 1 0 1 0 | 1 −1 0 1 1 0 1 0 | 1 −1 1 1 1 0 1 0 | |||
2 −1 1 1 1 0 1 0 | 1 −2 1 1 1 0 1 0 | 0 0 0 1 0 1 1 0 | |||
0 0 0 1 1 1 1 0 | 1 0 0 1 1 1 1 0 | 1 −1 1 2 1 1 1 0 | |||
1 0 0 0 0 −1 1 0 | 1 −1 0 0 0 −1 1 0 | 1 −1 1 0 0 −1 1 0 | |||
1 −1 1 1 0 −1 1 0 | 1 −1 2 1 0 −1 1 0 | 1 −1 0 0 1 −1 1 0 | |||
2 −1 1 0 1 −1 1 0 | 2 −1 1 1 1 −1 1 0 | 1 −2 1 1 1 −1 1 0 | |||
1 −1 1 1 −1 −1 1 0 | 1 −1 2 1 −1 −1 1 0 | 1 −1 1 0 −1 −2 1 0 | |||
0 1 0 0 0 0 −1 0 | 0 0 1 0 0 0 −1 0 | 0 1 0 −1 0 0 −1 0 | |||
0 1 −1 −1 0 0 −1 0 | 0 1 0 0 −1 0 −1 0 | 0 0 1 0 −1 0 −1 0 | |||
0 1 1 0 −1 0 −1 0 | 0 1 0 −1 −1 0 −1 0 | 0 1 0 0 0 1 −1 0 | |||
0 0 1 0 −1 −1 −1 0 | 0 1 0 −1 −1 −1 −1 0 | 0 0 1 −1 −1 −1 −1 0 | |||
1 −1 1 2 0 0 2 0 | 1 −1 0 1 1 0 2 0 | 1 −2 0 1 1 0 2 0 | |||
1 −1 1 1 1 0 2 0 | 2 −1 1 1 1 0 2 0 | 1 −2 1 1 1 0 2 0 | |||
1 −2 1 2 1 0 2 0 | 2 −1 0 1 2 0 2 0 | 1 −2 0 1 2 0 2 0 | |||
1 0 0 2 1 1 2 0 | 1 −1 0 2 1 1 2 0 | 1 −1 1 2 1 1 2 0 | |||
1 −1 0 2 2 1 2 0 | 1 −1 1 1 0 −1 2 0 | 2 −1 1 1 0 −1 2 0 | |||
1 −2 1 1 0 −1 2 0 | 1 −1 0 1 1 −1 2 0 | 2 −1 0 1 1 −1 2 0 | |||
1 −2 0 1 1 −1 2 0 | 2 −1 1 1 1 −1 2 0 | 1 −2 1 1 1 −1 2 0 | |||
2 −2 1 1 1 −1 2 0 | 0 2 0 −1 −1 0 −2 0 | 0 1 0 −2 −1 −1 −2 0 | |||
0 0 0 0 0 0 0 1 | 1 0 0 0 0 0 0 1 | 1 1 0 0 0 0 0 1 | |||
0 1 −1 0 0 0 0 1 | 1 1 −1 0 0 0 0 1 | 0 0 0 1 0 0 0 1 | |||
1 0 0 1 0 0 0 1 | 1 1 0 1 0 0 0 1 | 1 0 1 1 0 0 0 1 | |||
1 0 0 0 1 0 0 1 | 1 0 −1 0 1 0 0 1 | 1 1 −1 0 1 0 0 1 | |||
1 0 0 1 1 0 0 1 | 0 1 −1 0 0 1 0 1 | 0 0 0 1 0 1 0 1 | |||
0 1 0 1 0 1 0 1 | 1 1 0 1 0 1 0 1 | 0 1 −1 1 0 1 0 1 | |||
0 1 −1 0 1 1 0 1 | 1 1 −1 0 1 1 0 1 | 0 1 −2 0 1 1 0 1 | |||
0 0 0 1 1 1 0 1 | 1 0 0 1 1 1 0 1 | 1 1 0 1 1 1 0 1 | |||
0 1 −1 1 1 1 0 1 | 1 1 −1 1 1 1 0 1 | 1 0 0 0 0 −1 0 1 | |||
0 1 −1 1 1 2 0 1 | 1 0 0 1 0 0 1 1 | 1 −1 1 2 0 0 1 1 | |||
1 0 −1 0 1 0 1 1 | 1 0 0 1 1 0 1 1 | 2 0 0 1 1 0 1 1 | |||
1 −1 0 1 1 0 1 1 | 2 −1 0 1 1 0 1 1 | 2 −1 1 1 1 0 1 1 | |||
1 0 −1 1 1 0 1 1 | 1 −1 −1 1 1 0 1 1 | 1 −1 1 2 1 0 1 1 | |||
2 −1 1 2 1 0 1 1 | 2 −1 0 1 2 0 1 1 | 2 0 −1 1 2 0 1 1 | |||
1 −1 −1 1 2 0 1 1 | 1 0 0 1 1 1 1 1 | 1 0 −1 1 1 1 1 1 | |||
1 1 −1 1 1 1 1 1 | 1 0 0 2 1 1 1 1 | 2 0 0 2 1 1 1 1 | |||
1 −1 0 2 1 1 1 1 | 1 −1 1 2 1 1 1 1 | 1 0 −1 2 1 1 1 1 | |||
1 0 −2 0 2 1 1 1 | 2 0 0 1 2 1 1 1 | 1 −1 0 1 2 1 1 1 | |||
1 0 −1 1 2 1 1 1 | 2 0 −1 1 2 1 1 1 | 1 −1 −1 1 2 1 1 1 | |||
1 0 −2 1 2 1 1 1 | 2 0 0 2 2 1 1 1 | 1 −1 0 2 2 1 1 1 | |||
1 0 −1 2 2 1 1 1 | 2 −1 1 1 0 −1 1 1 | 2 −1 0 0 1 −1 1 1 | |||
2 0 −1 0 1 −1 1 1 | 2 −1 0 1 1 −1 1 1 | 2 −1 1 1 1 −1 1 1 | |||
1 0 −1 1 2 2 1 1 | 1 1 −2 1 2 2 1 1 | 1 0 −1 2 2 2 1 1 | |||
0 1 0 0 0 0 −1 1 | 1 1 0 0 0 0 −1 1 | 0 1 −1 0 0 0 −1 1 | |||
0 1 −1 −1 0 0 −1 1 | 0 1 0 0 −1 0 −1 1 | 0 1 0 0 0 1 −1 1 | |||
0 1 −1 0 0 1 −1 1 | 0 2 −1 0 0 1 −1 1 | 0 1 0 1 0 1 −1 1 | |||
0 1 −1 0 1 1 −1 1 | 0 2 −2 −1 1 1 −1 1 | 0 2 −2 0 1 2 −1 1 | |||
2 0 0 1 1 0 2 1 | 2 −1 0 1 1 0 2 1 | 2 0 −1 1 1 0 2 1 | |||
1 −1 −1 1 1 0 2 1 | 2 0 0 2 1 0 2 1 | 1 −1 0 2 1 0 2 1 | |||
2 −1 0 2 1 0 2 1 | 2 −1 1 2 1 0 2 1 | 2 −1 0 1 2 0 2 1 | |||
2 0 −1 1 2 0 2 1 | 1 −1 −1 1 2 0 2 1 | 2 −1 −1 1 2 0 2 1 | |||
2 −1 0 2 2 0 2 1 | 1 0 0 2 1 1 2 1 | 2 0 0 2 1 1 2 1 | |||
1 −1 0 2 1 1 2 1 | 1 0 −1 2 1 1 2 1 | 2 0 −1 1 2 1 2 1 | |||
1 −1 −1 1 2 1 2 1 | 2 0 0 2 2 1 2 1 | 1 −1 0 2 2 1 2 1 | |||
2 −1 0 2 2 1 2 1 | 1 0 −1 2 2 1 2 1 | 2 0 −1 2 2 1 2 1 | |||
1 −1 −1 2 2 1 2 1 | 2 −1 0 1 1 −1 2 1 | 1 0 −1 2 2 2 2 1 | |||
0 0 1 0 0 0 0 −1 | 0 0 1 0 −1 0 0 −1 | 0 0 1 0 −1 −1 0 −1 | |||
0 0 1 −1 −1 −1 0 −1 | 1 −1 2 −1 −1 −2 0 −1 | 0 0 0 0 0 0 1 −1 | |||
1 −1 1 0 0 −1 1 −1 | 1 −1 2 1 −1 −1 1 −1 | 1 −2 2 0 0 −2 1 −1 | |||
1 −1 2 0 −1 −2 1 −1 | 1 −2 2 0 −1 −2 1 −1 | 0 0 1 −1 −1 −1 −1 −1 | |||
0 0 1 −1 −1 −2 −1 −1 | 0 0 1 −2 −1 −2 −1 −1 | 0 0 2 −1 −2 −2 −1 −1 | |||
0 0 1 −2 −2 −2 −1 −1 | 1 −2 1 1 1 0 2 −1 | 1 −2 1 1 0 −1 2 −1 | |||
1 −2 2 1 0 −1 2 −1 | 1 −2 1 1 1 −1 2 −1 | 1 1 −1 1 1 1 0 2 | |||
1 1 −2 1 2 1 0 2 | 0 1 −2 1 1 2 0 2 | 1 1 −1 1 2 2 0 2 | |||
1 1 −2 1 2 2 0 2 | 2 0 0 1 1 0 1 2 | 2 0 −1 1 1 0 1 2 | |||
2 0 −1 1 2 0 1 2 | 2 0 0 2 1 1 1 2 | 1 0 −1 2 1 1 1 2 | |||
1 0 −1 1 2 1 1 2 | 2 0 −1 1 2 1 1 2 | 2 1 −1 1 2 1 1 2 | |||
1 0 −2 1 2 1 1 2 | 2 1 −2 1 2 1 1 2 | 2 0 0 2 2 1 1 2 | |||
1 0 −1 2 2 1 1 2 | 2 0 −1 2 2 1 1 2 | 2 1 −1 2 2 1 1 2 | |||
1 1 −2 1 2 2 1 2 | 1 0 −1 2 2 2 1 2 | 1 1 −1 2 2 2 1 2 | |||
2 1 −1 2 2 2 1 2 | 1 1 −2 2 2 2 1 2 | 0 1 −2 0 1 1 −1 2 | |||
1 2 −2 0 1 1 −1 2 | 0 2 −2 0 1 2 −1 2 | 2 0 −1 2 2 1 2 2 |
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Engel, P. On Σ-Classes in E8. I. The Neighborhood of E8. Crystals 2023, 13, 246. https://doi.org/10.3390/cryst13020246
Engel P. On Σ-Classes in E8. I. The Neighborhood of E8. Crystals. 2023; 13(2):246. https://doi.org/10.3390/cryst13020246
Chicago/Turabian StyleEngel, Peter. 2023. "On Σ-Classes in E8. I. The Neighborhood of E8" Crystals 13, no. 2: 246. https://doi.org/10.3390/cryst13020246
APA StyleEngel, P. (2023). On Σ-Classes in E8. I. The Neighborhood of E8. Crystals, 13(2), 246. https://doi.org/10.3390/cryst13020246