F: …°°°°°°°°°°°°°°°°°°…This has no symmetry, other than translational symmetry in one direction. Its symmetry group is the infinite cyclic group, C∞= °T | ±generated by a translation to the right by one unit. E: …±±±±±±±±±±±±±±±±±…This pattern has mirror symmetry in an axis parallel to the direction of the translation. Its symmetry group is the infinite dihedral group: D∞= °T, M | M2= I, MT = T-1M±where T is a generator for the translations and M is the reflection in the axis. A:…²²²²²²²²²²²²²²²²…Like the pattern E, above, this has mirror symmetry, as well as the translation symmetry. But it has infinitely many axes of mirror symmetry, not just one. Its symmetry group is: °T, M | M2= I, MT = TM±where T is a generator for the translations and M is any reflection. It is isomorphic to C∞×C2.
104 pb: …³´³´³´³´³´³´³´³´…This pattern has no mirror symmetry but it does have glide symmetry. The axis of the glide is the centre line. A reflection in this line, followed by a translation through half a unit to the right, is a glide which fixes the whole pattern. Every other glide is the product of this one and a suitable translation. The symmetry group for this pattern is also the infinite cyclic group C∞= °G | ±generated by the glide described above. Every translation in the symmetry group is an even power of this glide while the odd powers are other glides. N: …µµµµµµµµµµµµµµ…This pattern has no mirror or glide symmetry. But it has rotational symmetry about the centre of each letter N. If we pick one of these rotations, the others can be obtained by multiplying it by a suitable translation. So the symmetry group is again the infinite dihedral group: D∞= °T, R | R2= I, R-1TR = T-1±where T is the generating translation and R is one of the rotations. MW: This pattern has 2-fold rotational symmetry about the points on the horizontal axes half-way between successive scrolls. In addition there’s glide symmetry in the horizontal axis. The symmetry group is generated by this glide, G, and the 2-fold rotation, R. The translations are even powers of G, while the odd powers of G are other glides. The reflections are simply the product of a glide and the rotation. The symmetry group is again the infinite dihedral group: D∞= °G, R | R2= 1, M2= 1, M-1GM = G-1±where G is the glide described above and R is one of the rotations. H: …¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶…This pattern has both rotational and mirror symmetry in two directions. The symmetry group also contains glides, but these are just products of the reflection in the horizontal axis and a translation. The symmetry group is generated by a translation, T, through 1 unit, the reflection M in the horizontal axis, and the 180°reflection about one of the centres of 2-
105 fold rotation. (The reflections in the vertical axes can be expressed in terms of these generators. The symmetry group of this pattern is thus: °T, M, R | M2= R2= 1, MT = TM, MR = RM, R-1TR = T-1±where T is a generating translation, R is a rotation and M is a reflection. This group is
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- Fall '19
- Symmetry group, Rotational symmetry, symmetry groups, The Symmetry Group