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?
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.
Chaim Goodman-Strauss Dept. Mathematics Univ. Arkansas Fayetteville, AR 72701 firstname.lastname@example.org 501-575-6332