Course MT3818 Topics in Geometry

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Shift vectors

To complete the data we need for classifying the groups, we need one more concept to add to the lattice L, the point group P, and the action of P on L. This is to allow for the fact that some reflections in P act as glides rather than as reflections in G.

Definition

Let A belongs P be a reflection with Tv comp A belongs G. Then the vector a = v + A(v) is called a shift vector of A.

Properties

  1. A shift vector lies in the lattice L.

    Proof
    (Tv comp A)2(x) = Tv comp A(v + A(x)) = Tv(A(v) + x) since A2= I and this is Tv+A(v)(x) and so v + A(v) is in the lattice.


  2. A shift vector of A is left fixed by A.

    Proof
    A(a) = A(v + A(v)) = A(v) + A2(v) = A(v) + v since A2 = I and this is a again.


Remark

One can in fact define a shift vector for any element of P.
For example, if R is rotation by 2p/n and Tv comp R belongs G then take a = v + R(v) + R2(v) + ... + Rn-1(v). However, since a is fixed by R it must be 0 and so is not very interesting!

Examples

  1. Consider the group of symmetries of this pattern (pm).
    The lattice is rectangular.
    The point group is D1 which acts on the lattice as vertical reflection R.
    In this case since R is a symmetry, 0 is a shift vector.
    To get another shift vector, take v as shown so that Tv is in the group (and v is in the lattice) and take a = v + R(v) as shown.
    Note that a is on the mirror and so is left fixed by R.
    In this case, if w is a vertical basis element of the lattice, any shift vector of R is of the form 2nw where n belongs Z.


  2. The group of symmetries of this pattern (pg).
    The lattice is again rectangular.
    The point group is D1 which acts on the lattice as vertical reflection R.
    In this case R is not a symmetry of the pattern.
    To get another shift vector, take v as shown so that Tv is in the group (and v is in the lattice) and take a = v + R(v) as shown.
    As before a is on the mirror and so is left fixed by R.
    Then, if w is a vertical basis element of the lattice, any shift vector of R is of the form (2n+1)w where n belongs Z.


  3. The group of symmetries of this pattern (cm).
    This time lattice is rhomboid.
    The point group is still D1 and still acts on the lattice as vertical reflection R.
    The vector 0 is a shift vector of R.
    As before, take v as shown so that Tv is in the group (and v is in the lattice) and take a = v + R(v) as shown.
    As before a is on the mirror and so is left fixed by R.
    Then, if w is a vertical basis element of the lattice, then shift vector of R is of the form nw where n belongs Z. In this case all the lattice points on the mirror are shift vectors.




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(The point groups)
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JOC March 2003