Lesson 3 Math part

Math5 Lesson 3

Goals:
To understand groups acting on sets, fundamental region and subgroups.

Text: We know that there are certain properties which are true in general of sets of symmetries of mandalas. An important one is that when symmetry is broken the size of the resulting set always divides the size of the original set (order of a subgroup divides the order of the group).

Consider the following multiplication tables.

*  a  b  c  d
a  a  a  a  a
b  a  b  c  d
c  a  c  a  c
d  a  d  c  b

*  a  b  c  d
a  a  b  c  d
b  b  a  d  c
c  c  d  b  a
d  d  c  a  b

We want to know what kinds of tables can come from mandalas. Try to imagine what a mandala with the first multiplication table would look like: any motion composed with the motion a would have to result in a again. But there is only one motion which does this -- the "do-nothing" -- and the multiplication table claims there are three distinct such motions. Clearly there is no mandala with symmetries like that, so the first table doesn't come from a mandala.

The second table, on the other hand, probably looks familiar. Can you draw a picture of a mandala with this table?

Recall the definition of group and notice that the set with the first table is not a group (why?) while the second is.

So far we've only looked at patterns with finite sets of symmetries. However, we can take any one of those patterns and repeat it throughout the infinite plane. Depending on how we do this, we may maintain some or all of the original symmetries and we may create some new ones.

Wallpaper Groups (Plane Symmetry Groups)
David E. Joyce (djoyce@black.clarku.edu)

Notice that whenever a pattern is repeated, a new kind of symmetry is created -- translational symmetry. When we repeat a pattern infinitely, we get infinite translational symmetry. Hence we now have an infinite set of symmetries. In fact, this new infinite set is a group (how can we see this?). Recall that, in the mandala case, we could break or ignore some of the symmetries and be left with a smaller set, a subset of the complete set. Such a smaller set is called a subgroup of the original group. In the case of this new infinite group, we have infinite subgroups (find some).

Another new kind of symmetry, one that is a little less obvious, is displayed in this pattern:

From J. Bourgoin: "ARABIC ALLOVER PATTERNS"
Dover Publications, Inc., 1985

In this case, the symmetry is created by a combination of translation and mirror flip. Such a symmetry is called a glide reflection. Neither a mirror flip nor a translation alone will allow the red design to lie down on the blue design -- we need to do both.

We define repeat patterns in the plane to be patterns with infinite translational symmetry in two directions. Such patterns can be used to make strong artistic statements; for example, they were used in Islamic art to suggest the infinite.

Find the symmatries of each of the following patterns.

From Istvan Hargittai and Magdolna Hargittai: "SYMMETRY A Unifying Concept"
Shelter publications, Inc.
p.187

Attached to this sheet are three descriptions of patterns with different symmetry groups taken from three different sources. Feel free to work together with one or two friends on this assignment. If two people work together it is not necessary to hand in more than one copy. (With all your names on it.) Please also hand in your Group Mtork from the July 11th class as part of this assignment also.


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