Course MT3818 Topics in Geometry

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Frieze groups

We now look at some infinite subgroups of I(R2).

We will look at subgroups which map a strip, say the subset R cross [-1, 1], and are discrete. That is, there are no elements which move points an arbitrarily small distance.
[For example, the additive group of reals R acting on the line by addition is not a discrete group.]

The finite symmetry group of the strip is D2. The elements of this group are: the identity I, a reflection R in the horizontal, a reflection V in a vertical line (through 0 say) and a half turn H (about 0 say).
We have R2 = V2 = H2 = I , RV = VR = H etc.

What else could be in an infinite group?
It could contain a translation T (which we will take to be by a shortest distance in the group) and/or a glide reflection G (which we can take to be G = TR = RT).
We then get the following possible groups generated by some of these elements. In each case we give a pattern which is mapped to itself by the group.
(See Exercises 4 Question 3)

Group generated by:Pattern left invariantGroup
(i)T.... ....Cinfinity
(ii)G.... ....Cinfinity
(iii)T, R.... ....Cinfinity cross D1
(iv)T, V.... ....Dinfinity
(v)T, H.... ....Dinfinity
(vi)G, V.... ....Dinfinity
(vii)everything.... ....Dinfinity cross D1

These seven groups are the only possibilities -- up to conjugacy in the group I(R2).


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(Symmetry groups of Platonic solids)

JOC March 2003