Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation
isometry is denoted by T. As we have noticed in lectures the symmetry group of the
fundamental pattern
consists of all powers of T and is thus isomorphic to which group?

Q 2. Now use Lemma F from your lecture notes to prove the following result:
Proposition
a) If there is a rotation R present in the symmetry group of any frieze pattern then
R is a turn of magnitude 0 or ¼.

b) If there is a re°ection m present in the symmetry group of any frieze pattern
then m is either parallel or perpendicular to the direction of T.

Q 3. A turn of magnitude 0 is equal to what element of the group?

Q 4. There can be only one re°ection axis parallel to T present in the symmetry
group. Why?

Q 5. Show that if T is a translation through d units, then the presence of one
perpendicular mirror generates a further perpendicular re°ection axis in the frieze
pattern, 1/2d
units away from the axis of the first mirror.

Q 6. Given the motif
Draw pictures of
a) the frieze pattern that corresponds to a symmetry group generated by T and a
parallel re°ection m, having the basic motif given in the above diagram (i.e. by
the bold letter `F'). Apart from T and m there are no other symmetries in this
frieze pattern.

b) the frieze pattern that corresponds to a symmetry group generated by T and a
perpendicular re°ection m, having the basic motif given in the above diagram.
Apart from T and m there are no other symmetries in this frieze pattern.

Q 7.
a) Show that the product of two half-turn isometries in the plane is a translation
through twice the distance between the centres of the turns.

b) Suppose that the symmetry group of a frieze pattern contains a half-turn, ¿, (as
well as the translation T, of course). What implication does the result in part
a) of this question have for the symmetry of a frieze pattern having a half-turn
¿ and a translation T.

c) Draw, using the motif from Q 6, the frieze pattern corresponding to a symmetry
group generated by T and a half turn ¿ and no other symmetry.

Q 8.
a) Suppose that a parallel mirror and a perpendicular mirror are both present in
the symmetry group of a frieze pattern. What type of symmetry is the product
of the two mirrors?

b) Draw the frieze pattern that corresponds to a symmetry group generated by T,
a perpendicular re°ection m, and a parallel mirror m0, having the basic motif
given in Q 6. Indicate clearly on your diagram the three symmetries.

Q 9.
a) Consider the symmetry group consisting of T and half-turn ¿. If there is a
parallel re°ection axis present it is unique and furthermore, the axis of the
parallel mirror symmetry must pass through the centres of the half-turns. Why
is this?

By considering the product of a re°ection and a half-turn, show that the symmetry
group consisting of T, a half-turn ¿ and a mirror symmetry parallel to the
direction of translation generates a pattern considered. in a previous question.
Which question?

b) Again consider the symmetry group consisting of T and half-turn ¿ but this
time introduce a mirror symmetry perpendicular to the direction of translation.
There are only two cases: CASE I: the axes of the mirror symmetries must
either pass through the centres of the half-turns or CASE II: the axes of the
mirror symmetries bisect the array of half-turn centres. Why?

c) Case I above leads to a symmetry group which is isomorphic to one already
considered. Which one?

d) Draw the frieze pattern corresponding to the symmetry group of CASE II above,
indicating clearly on your diagram the mirror axes, the centres of half-turn and
the translation distance.

Q 10 Finally, suppose that a single glide re°ection g is present in the symmetry group
of a frieze pattern only.
a) Draw the resulting pattern, again using the motif given in Q 6.

b) Which group is this symmetry group of this pattern isomorphic to?