Recall our rules of the game Here we examine a few helpful theorems using Geometer's Sketchpad.

**Theorem 6** If a pattern contains a
center of rotation, the angle of rotation at this center must be *2 pi
/ n* for some integer *n*.

A pattern is generated by a single rotation through angle *2 pi/ n*
is called cyclic symmetry of order *n* and is denoted **n**. (The
general notation will be explained later!)

Q: A pattern generated by a single translation is called an cyclic symmetry with infinite order. Why does this make sense?

**Theorem 7** If a pattern contains two
mirror lines, they must be parallel or meet at an angle of *pi / n*
for some integer n.

A pattern is generated by just two mirror lines meeting at angle p/n
is called dihedral symmetry of order n and is denoted ***n**. If the
mirror lines are parallel, the pattern is called dihedral symmetry of infinite
order.

Q6: Here are pictures of dihedral and cyclic symmetries.
Can you tell which is which instantly?

Examine the sketches **5** and ***5**
for examples of five-fold dihedral and cyclic symmetries. In particular,
how many copies of the motif are there?

Q7: In both of the above theorems, what if n is rational rather than an integer? Specifically, think about Rule 0 -- in the sketch the motif is really a sector with angle

**Theorem 8** If a pattern contains two or more centers of rotation,
then the angles of rotation at these centers can only be *2 pi / n*
where n is 2,3,4 or 6.

This requires two lemmas:

**Lemma 8.a** If a pattern contains two
or more centers of rotation, then any angles of rotation must be *2 pi
/ n* where n<=6.

*Suppose for contradiction that there is at least one center of rotation
with angle 2 pi / n with n > 6; let C be the set of all these points.
Then C has at least two elements (Why?) Choose a pair x,y of points in C
that are closer to each other than to any other points in C . Now x is a
center of rotation with angle 2 pi / n, with n > 6, and so around x are
arranged at least 6 copies of y. Since y is an element of C, so are all
these copies of y. So: were x and y really closer to each other than to
any other points in C? *

So centers with angle *2 pi / n* where n>6 are ruled out. Next
we turn to angles of *2 pi / 5*. Interestingly, there are many beatiful
ways of filling the plane with local five-fold symme- try. However each
five-fold symmetry extends only a finite distance before being broken, and
ultimately violates our First Rule.

**Lemma 8.b** If a pattern contains two
or more centers of rotation, then any angles of rotation must be *2 pi
/ n* where n = 2, 3, 4, 6 (i.e. n is smaller than 6 and n isn't equal
to 5 )

This is similar to the proof of Lemma 8a: *Suppose for contradiction
that the set of centers of rotaton with angle 2 pi / 5 is not empty; call
this set C. Then C contains at least two points. (Why?) Choose a pair x,y
of points in C that are closer to each other than to any other points in
C . Around x are arranged 5 images of y, each of which is an element of
C. Let y' be any one of these images; around y are arranged five images
(called say, y'') of y' So: could one of these y'' coincide with x? If not,
are any of the y'' closer to x than y? *

Thus Theorem 8 is proved.

BUT This still begs the question: What exactly are the planar patterns? We will delve into this and related mysteries in the next three weeks, culminating with a surprising connection to

Motifs of Planar Patterns

to outline

Chaim Goodman-StraussDept. Mathematics Univ. Arkansas Fayetteville, AR 72701strauss@comp.uark.edu501-575-6332