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 × [-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 invariant Group (i) T .... .... C∞ (ii) G .... .... C∞ (iii) T, R .... .... C∞ × D1 (iv) T, V .... .... D∞ (v) T, H .... .... D∞ (vi) G, V .... .... D∞ (vii) everything .... .... D∞ × D1

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

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JOC March 2003