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38 2. GROUPS AND SYMMETRY oo ooo o o oU o ooo ooo o o o o o o W ooo (1/2)v ooo ooo A (1/2)u Such a fundamental region F is useful because the position vector p every point in the plane can be uniquely expressed in the form p = p + mu + nv for some point p ∈ F and m, n ∈ Z. This allows us to tile the plane with copies of a figure located within a fundamental region. 7 is obtained ♣ ♦ contained in the by translating (using vectors mu + nv for (m, n ∈ Z)) the pattern ♥ ♠ fundamental region {su + tv : −1/2 s < 1/2, −1/2 t < 1/2} to tile the whole plane.
5. 6. Hexagonal lattice with W at one of the vertices where W has position vector w + u + v; this is the subset {w + su + tv : 0 s < 1, 0 t < 1} ⊆ R2 . An alternative is the fundamental region centred at W , {w + su + tv : −1/2 s < 1/2, −1/2 t < 1/2} ⊆ R2 . 38 2. GROUPS AND SYMMETRY oo ooo o o oU o ooo ooo o o o o o o W ooo (1/2)v ooo ooo A (1/2)u Such a fundamental region F is useful because the position vector p every point in the plane can be uniquely expressed in the form p = p + mu + nv for some point p ∈ F and m, n ∈ Z.
The fundamental region has a half rotation symmetry, so the holohedry group is {(I | 0), (−I | 0)}. 40 2. 11. 12. 13. pgm The full symmetry group is {(I | mu + nv) : m, n ∈ Z} ∪ {(−I | mu + nv) : m, n ∈ Z} 8. 14. 15. cm which consists of the translations together with half rotations about the points mu + nv, (m + 1/2)u + nv, mu + (n + 1/2)v, (m + 1/2)u + (n + 1/2)v (m, n ∈ Z). Pattern pm. The fundamental region has a reflection in the x-axis, so the holohedry group is {(I | 0), (Sx | 0)}, 1 0 where Sx = .