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| Euclidean Patterns in Non-Euclidean Tilings |
We list here all 14 kaleidoscopic subgroups of *246 that are compatible with the genus-3 translational unit (ooo) for the P, G, and D surfaces. A kaleidoscopic subgroup is one with an orbifold that has a single boundary component and no other symmetry features. There are a total of 131 subgroups of *246 that retain the ooo translational symmetry. A complete list is available here.
Click on the small image of each subgroup to see a larger image of that subgroup or click on the subgroup name to see a page of details for that subgroup.
| Domain | Subgroup | Normal | Index | Curvature | P Spacegroup | G Spacegroup | D Spacegroup |
|---|---|---|---|---|---|---|---|
|
*246 | Y | 1 | -1/24 | Im-3m | Ia-3d | Pn-3m |
|
*266 | Y | 2 | -1/12 | I-43m | Ia-3 | P-43m |
|
*344 | Y | 2 | -1/12 | Im-3 | I-43d | P-43m |
|
*2223 | Y | 2 | -1/12 | Pm-3m | I4(1)32 | P4(2)32 |
|
*2224 | N | 3 | -1/8 | I4/mmm | I4(1)/acd | P4(2)/nnm |
|
*2323 | Y | 4 | -1/6 | I23 | I2(1)3 | F-43m |
|
*2244a | N | 6 | -1/4 | Immm | I-42d | P-42m |
|
*2244b | N | 6 | -1/4 | I4mm | I4(1)/a | Cmma |
|
*22222a | N | 6 | -1/4 | P4/mmm | I4(1)22 | P4(2)22 |
|
*22222b | N | 6 | -1/4 | Fmmm | Fddd | P4(2)nm |
|
*2626 | N | 8 | -1/3 | R3m | R-3 | R3m |
|
*222222a | N | 12 | -1/2 | Fmm2 | C2/c | Cmm2 |
|
*4444 | N | 12 | -1/2 | Imm2 | I-4 | Cmm2 |
|
*222222b | Y | 12 | -1/2 | Pmmm | I2(1)2(1)2(1) | P222 |
