FRIEZE SYMMETRY (PART II) ------------------------- EXTENDING FRIEZE PATTERNS TO PLANE PATTERNS ------------------------------------------- We've learned two schemes for naming frieze symmetry types and seen interesting parallels with plane symmetry types, but we still lack a convincing demonstration that our 7 types exhaust all the possibilities. One way to do the job involves converting frieze patterns to plane patterns and then using the classification that we already know. Before we start, let's look at some compact examples of frieze patterns: _ _ _ _ _ _ _ WALK: | |_ | |_ | |_ | |_ | |_ | |_ | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ JUMP: |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ HOP: | | | | | | | | | | | | | | _ _ _ _ _ _ _ _ _ _ _ SIDLE: | | | | | | | | | | | | | | | | | | | | | SPIN JUMP: |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_ | | | | | | | | | | | | | | | | | | | | | _ _ _ _ _ _ _ _ _ _ _ _ _ _ SPIN HOP: _| _| _| _| _| _| _| _| _| _| _| _| _| _| _ _ _ _ _ _ _ _ _ _ SPIN SIDLE: _| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| You may find that these patterns look sufficiently different from the footprint patterns that you want to take a few minutes to locate their reflections and gyrations. Now we'll convert a frieze pattern to a plane pattern by declaring as reflective the boundaries of the strip which the frieze decorates. We present the results: WALK -----> *x |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ _ _ _ _ _ _ _ _ _ _ _ _ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | _ _ _ _ _ _ _ _ _ _ _ |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ _ _ _ _ _ _ _ _ _ _ _ _ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | |_ | JUMP -----> ** _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ HOP ------> ** |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | SIDLE ----->*2222 |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | SPIN-JUMP -----> *2222 |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | SPIN-HOP ------> 22* _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| _| SPIN-SIDLE ------>2*22 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_| |_ For convenience, we repeat the symmetry conversions without the patterns: WALK ------------> *x JUMP ------------> ** HOP -------------> ** SIDLE -----------> *2222 SPIN-JUMP -------> *2222 SPIN-HOP --------> 22* SPIN-SIDLE ------> 2*22 Notice that 7 different frieze symmetries only give rise to 5 plane symmetries. A bit of thought shows that only these 5 could occur. To see this, first, from now on, IMAGINE MENTALLY THAT WE PRINT FRIEZE PATTERNS USING BLUE INK ALONG THE CENTER LINE, WITH THE COLOR BECOMING PURPLER UNTIL WE GET THE THE BOUNDARY WHERE WE PRINT BRIGHT RED. This color scheme could never interfere with the symmetries, because NO MOTION OF THE FRIEZE PATTERN CHANGES THE DISTANCE TO THE CENTER LINE OF ANY POINT. (The significance of this becomes clear further on.) Now of the seventeen plane symmetry types 632 333 442 2222 *632 *333 *442 *2222 3*3 4*2 22* 2*22 O ** *x 22x xx we can surely eliminate as a possibility any type that lacks a * since our conversion explicitly INTRODUCES reflections. This leaves only *632 *333 *442 *2222 3*3 4*2 22* 2*22 ** *x Next, getting any cone points or corner points of order higher than 2 would contradict our color scheme. This brings the list down to *2222 22* 2*22 ** *x and these constitute just exactly the 5 types that actually occur! RULING OUT OTHER POSSIBILITIES ------------------------------ But why do ** and *2222 occur twice? How can we rule out some as yet undiscovered frieze type also giving rise to one of these 5 types? Let's work backwards. Given a plane pattern with one of the types *2222 22* 2*22 ** *x how could we isolate within the pattern a frieze that generates it. Obviously we'd have to cut out a strip between two parallel reflection lines. Because of our assumption about frieze pattern coloring, we may assume that ANY SYMMETRY OF THE PLANE PATTERN EITHER CONSTITUTES A SYMMETRY OF THIS STRIP OR ELSE MOVES THE STRIP ENTIRELY ONTO A DISJOINT STRIP. This means that any symmetry feature located within the strip (gyration points, reflection lines, glide reflections line) must occur EXACTLY IN THE MIDDLE. Now let's go down the list of five (these pictures show the reflection lines and gyration points of the patterns, not the patterns themselves!!): *2222 has just reflection lines and corner points: | | | | | | | | | | so we can extract 1 or 2 | | | | | | | | | | adjacent narrow strips, ------------------------------------------- but any more and we've have | | | | | | | | | | an off-center reflection line. | | | | | | | | | | ------------------------------------------- | | | | | | | | | | | | | | | | | | | | ------------------------------------------- | | | | | | | | | | | | | | | | | | | | 22* has just reflection lines and gyration points: . . . . . . . . . so we can only extract 1 narrow strip, since any ------------------------------------------- more and we'll have off-center . . . . . . . . . gyration points. ------------------------------------------- . . . . . . . . . ------------------------------------------- . . . . . . . . . 2*22 has reflection lines, corner points and gyration points: | . | . | . | . | . | . | . | . | . | so we can only extract | | | | | | | | | | 1 narrow strip, since any ------------------------------------------- more and we'll have off-center | . | . | . | . | . | . | . | . | . | gyration points. | | | | | | | | | | ------------------------------------------- | . | . | . | . | . | . | . | . | . | | | | | | | | | | | ------------------------------------------- | . | . | . | . | . | . | . | . | . | | | | | | | | | | | ** has just reflection lines: so we can extract 1 or 2 adjacent narrow strips, ------------------------------------------- but any more and we've have an off-center reflection line. ------------------------------------------- ------------------------------------------- *x has reflection lines and glide reflection lines: ------------------------------------------- so we can only extract - - - - - - - - - - - - - - - - - - - - - - 1 narrow strip, since any more and we'll have off-center ------------------------------------------- glide-reflection line. - - - - - - - - - - - - - - - - - - - - - - ------------------------------------------- - - - - - - - - - - - - - - - - - - - - - - ------------------------------------------- RELATING FRIEZE PATTERNS TO SPHERE PATTERNS ------------------------------------------- Given a frieze pattern, in infinitely many ways we could cut out from the infinite strip a finite rectangle (taking care that the ends match up) and tape the ends together to obtain a decorated cylinder. Shrinking the top and bottom edges of the cylinder down to single points, we'd get a decorated sphere. In this way, each frieze symmetry type gives rise to an infinite family of sphere symmetry types, finite because the number of repetitions of the frieze motif we choose to cut out will determine the order of the gyration point or corner point we create at the north and south poles of the sphere. In practical terms, one might find it very hard to distinguish between a frieze pattern sitting in a plane and a VERY LARGE sphere pattern with the design concentrated near the equator. This suggests another approach to deriving the 7 frieze types: classify spherical patterns, and look for symmetry types that occur in infinite families varying only according the the behavior near the poles. In the next lecture we shall carry this out.