ACCIDENTAL SYMMETRY ------------------- It may happen that you wish to draw a plane pattern with a given symmetry type. Even if you select the correct template, decorate it, and use the correct rules to replicate the decoration about the plane, you still might wind up with a pattern having the wrong type! Certainly your pattern will have the symmetries you intended, but if you choose your decoration in an unfortunate way, you might create have additional and unintended symmetries when you finish. Example: You plan to generate a 2222. Your template looks like ._____._____. | | | | | | | | ._____._____. where the dots represent order 2 gyration points. Everything works fine when you decoration leads to a pattern like ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | | | | | | | | | | | | | | _| | _| | _| | _| | _| | _| | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | _ | _ | _ | _ | _ | _ | | | | | | | | | | | | | | | | | | | | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | | | | | | | | | | | | | | _| | _| | _| | _| | _| | _| | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | _ | _ | _ | _ | _ | _ | | | | | | | | | | | | | | | | | | | | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. but nothing prevents you from picking a decoration that leads to a pattern like ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | | | | | | | | | | | | | | _|_ | _|_ | _|_ | _|_ | _|_ | _|_ | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | | | | | | | | | | | | | | _|_ | _|_ | _|_ | _|_ | _|_ | _|_ | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. | | | | | | | | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | ._____._____._____._____._____._____._____._____._____._____._____._____. which has type *2222 instead. This example raise the question of all the ways one might fill in the blanks in the following sentence: If you try to make a pattern with symmetry type _______, you might end up with a pattern with symmetry type _______ depending on how you decorate the template. The following chart summarizes the answers *632 Y N N N N N N N N N N N N N N N N *333 Y Y Y N N N N N N N N N N N N N N 3*3 Y Y Y N N N N N N N N N N N N N N *442 N N N Y N N N N N N N N N N N N N *2222 Y N N Y Y Y Y N N N N N N N N N N When trying 4*2 N N N Y N Y N N N N N N N N N N N to make a pattern 2*22 Y N N Y Y Y Y N N N N N N N N N N with (Y=yes; N=no) symmetry 22* Y N N Y Y Y Y Y N N N N N N N N N type... ** Y Y Y Y Y Y Y Y Y Y N N N N N N N *x Y Y Y Y Y Y Y Y Y Y N N N N N N N 22x Y N N Y Y Y Y Y N N Y N N N N N N xx Y Y Y Y Y Y Y Y Y Y Y Y N N N N N 632 Y N N N N N N N N N N N Y N N N N 333 Y Y Y N N N N N N N N N Y Y N N N 442 N N N Y N N N N N N N N N N Y N N 2222 Y N N Y Y Y Y Y N N Y N Y N Y Y N O Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y * * 3 * * 4 2 2 * * 2 x 6 3 4 2 O ...could you wind up 6 3 * 4 2 * * 2 * x 2 x 3 3 4 2 with symmetry type... 3 3 3 4 2 2 2 * x 2 3 2 2 ???? 2 3 2 2 2 2 2 This chart contains 17 x 17 = 289 entries and each of them requires justification! Obvious we could have a messy story on our hands if we don't take care to organize the details. Why might the question have the answer ``no''? We list 4 possible reasons: 1) Suppose you intend a symmetry type which contains reflections (so that the name has a *). Then you'll certainly get the reflection lines you intend and possibly others you didn't and other symmetries as well. But you can't wind up with a pattern having no reflections at all. 2) Similarly, if you plan to make a point a corner point of order n, you must get a corner there, though its order might equal a multiple of n (since unintended reflection lines might pass through the point). 3) If you plan to make a point a gyration point of order n, you will get a gyration point or possibly a corner point at that place, with order n or a multiple of n. 4) If you plan for a line of glide reflection, you'll either get it or a full fledged line of reflection. The following chart summarizes the justifications for all the ``no'' answers: * * 3 * * 4 2 2 * * 2 x 6 3 4 2 O | 6 3 * 4 2 * * 2 * x 2 x 3 3 4 2 | 3 3 3 4 2 2 2 * x 2 3 2 2 | 2 3 2 2 2 2 | 2 | -----------------------------------------------------------| | *632 *6 *6 *6 *6 *6 *6 *6 *6 *6 * * * * * * * | Key: | | *333 *3 *3 *3 *3 *3 *3 *3 * * * * * * * | *: | | The first pattern has a reflection 3*3 *3 *3 *3 *3 *3 *3 *3 * * * * * * * | but the second pattern lacks | reflections | *442 *4 *4 *4 *4 *4 *4 *4 *4 *4 * * * * * * * | | x: | *2222 *2 *2 *2 *2 *2 * * * * * * * | The first pattern has a glide | reflection but the second pattern | lacks both glide reflections and 4*2 4 4 4 4 4 4 4 4 * * * * * * * | reflections | | 2*22 *2 *2 *2 *2 *2 * * * * * * * | *2, *3, *4, *6: | | The first pattern has a corner 22* 2 2 2 2 * * * * * * * | point, but the second lacks a | corner point of that order or any | multiple of it. ** * * * * * * * | | | 2, 3, 4, 6: *x * * * * * * * | | The first pattern has a gyration | point, but the second lacks a 22x 2 2 2 2 2 x x x x x | gyration point or corner point of | that order or any multiple of it. | xx x x x x x | | | 632 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 | | | 333 3 3 3 3 3 3 3 3 3 3 3 3 | | | 442 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 | | | 2222 2 2 2 2 2 2 2 | | | O | | -----------------------------------------------------------| Justifying the ``yes'' answers naturally calls for examples, indeed 101 of them! Observe however that if attempting symmetry type A can accidently give you symmetry type B and attempting symmetry type B can accidently give you symmetry type C then attempting symmetry type A can accidently give you symmetry type C. If you don't immediately find this obvious, it helps to think about it backwards. Saying that ``attempting symmetry type A can accidently give you symmetry type B'' amounts to saying that we can group together several occurrances of the type B template to form a type A template which generates the same pattern. Clearly if a grouping a type C templates produces a type B template and a grouping a type B templates produces a type A template, then a grouping of type C templates and produce a type A template. With this insight, the following graphics summarize all the ``yes'' answers: ** <----> *x Pairs such that attempting either symmetry type might produce the other. *2222 <----> 2*22 3*3 <----> *333 O Attempting a symmetry type might produce a symmetry type located further below in the diagram (along / | \ some path). / | \ / | \ xx \ 333 | \ 2222 | \/ | \ ** /\ | \ | \/ \ 22x \ /\ / \ | *333 632 \ | 442 22* \ | | | \ | *2222 | \ | / | | | | *632 4*2 | \ | \ *442 Verifying the information contained in these graphics requires a mere 26 examples. This we leave to the reader.