M336 GE4 Exercise 2.7
Shirleen Stibbe s.stibbe@open.ac.uk
g
They've given you the standard form, t[d]q[ϕ], where d is the point the origin is mapped to by the symmetry (P&S 3.2, p14), and q[ϕ] is a linear reflection.
h
Your job in this question is to get it into the form t[g]q[c, ϕ], then check whether the reflection component, q[c, ϕ] !
is a symmetry of W (the axis of the indirect symmetry is a reflection axis)
!
is not a symmetry of W (the axis is a glide axis).
f
The trick is that the axis of an indirect symmetry (glide or reflection) which maps the origin to a point d passes through the point d/2. So to dismantle standard form, you can always take c = d/2, then work out the translation component later. a) f = t[(4, 2)]q[0] d = (4, 2), so c = (2, 1), ϕ = 0, so the reflection component is q[(2, 1), 0]. Reflection is in the horizontal line passing through (2, 1) i.e. the line y = 1, which is a symmetry of W, so the axis of f is a reflection axis. The reflection component maps the origin to (0, 2). So the translation component must map (0, 2) to (4, 2), i.e translation component is t[(4, 0)], also a symmetry of W. Note: As for lattices, it's always all or nothing; i.e. either both components are symmetries, or neither is a symmetry of W. b) g = t[(2, 0)]q[π/2] d = (2, 0), so c = (1, 0), ϕ = π/2, so reflection component is q[(1, 0), π/2]. Reflection is in the vertical line passing through (1, 0) i.e. the line x = 1, which is a symmetry of W, so the axis of g is a reflection axis. The reflection component maps the origin to (2, 0), so the translation component is (0, 0), which shows that the indirect symmetry is a reflection, not a glide.
h = t[(0, 4)] q[π/4] d = (0, 4), so c = (0, 2), ϕ = π/4, so reflection component is q[(0, 2), π/4] Reflection is in the line at angle π/4 to the x-axis, passing through the point (0, 2). This is not a symmetry of W so the axis of h is a glide axis. The reflection component maps the origin to (-2, 2), so the translation component maps (-2, 2) to (0, 4), i.e. t[(2, 2)], which is also not a symmetry of W (as expected!).
And by the way: In general, the axis of an indirect symmetry (glide or reflection) which maps the point (a, b) to the point (c, d) passes through the midpoint of the line joining (a, b) to (c,d), i.e. through the point [1/2(a + c), 1/2(b + d)]. www.shirleenstibbe.co.uk/m336