HOW TO RECOGNIZE THE SYMMETRY TYPE OF A PLANE PATTERN ----------------------------------------------------- DOES THE PATTERN HAVE REFLECTIONS LINES? ---------------------------------------- If YES: (in which case the symmetry must have one of the 10 types: *632, *442, *333, *2222, 4*2, 3*3, 22*, 2*22, **, *x) DO REFLECTION LINES EVER CROSS? ------------------------------- If YES: (in which case the symmetry must have one of the 7 types: *632, *442, *333, *2222, 4*2, 3*3, 2*22) WHAT IS THE LARGEST NUMBER OF REFLECTION LINES THAT MEET AT A POINT? -------------------------------------------------------------------- If 6: Pattern has type *632 ! If 4: Pattern has type *442 ! If 3: (in which case pattern has type *333 or 3*3) DOES THE PATTERN HAVE AN ORDER 3 GYRATION AT THE CENTER OF THE EQUILATERAL TRIANGLES FORMED BY THE REFLECTION LINES? -------------------------------------------------------------- If YES: Pattern has type 3*3 ! If NO: Pattern has type *333 ! If 2: (in which case symmetry has either type *2222, 4*2 or 2*22) WHAT IS THE ORDER OF GYRATION AT THE CENTER OF THE RECTANGLES FORMED BY THE REFLECTION LINES? ------------------------------------------------------------- If 4: Pattern has type 4*2 ! If 2: Pattern has type 2*22 ! If 1 (i.e., no gyration point): Pattern has type *2222 ! If NO, (in which case the symmetry has either type 22*, ** or *x) DOES THE PATTERN HAVE GLIDE REFLECTIONS BETWEEN NEARBY REFLECTION LINES? ------------------------------------------------------------------------ If YES: Pattern has type *x ! If NO: (in which case symmetry has either type ** or 22*) DOES THE PATTERN HAVE ORDER 2 GYRATION POINTS BETWEEN NEARBY REFLECTION LINES? ------------------------------------------------------------ If YES: Pattern has type 22* ! If NO: Pattern has type ** ! If NO: (in which case the symmetry must have one of the 7 types: 632 442 333 2222 22x xx o) DOES THE PATTERN HAVE GYRATION POINTS? -------------------------------------- IF YES: (in which case the symmetry must have one of the 5 types: 632 442 333 2222 22x) WHAT IS THE GYRATION POINT OF HIGHEST ORDER? -------------------------------------------- If 6: Pattern has type 632 ! If 4: Pattern has type 442 ! If 3: Pattern has type 333 ! If 2: (in which case symmetry has either type 2222 or 22x) DO GLIDE REFLECTION LINES BISECT THE PARALELLOGRAMS WHICH HAVE THEIR CORNERS AT THE GYRATION POINTS? --------------------------------------------------------- If YES: Pattern has type 22x ! If NO: Pattern has type 2222 ! If NO: (in which case the symmetry has either type xx or o) DOES THE PATTERN HAVE GLIDE RELECTIONS? --------------------------------------- If YES: Pattern has type xx ! If NO: Pattern has type o ! NOTES: 1) One calls the sort of procedure describe above a ``decision tree.'' One could devise many different decision trees to accomplish the same purpose. In this case, we can identify any pattern by asking no more than 4 questions. Thus this decision tree has height (or sometimes ``depth'') 4. 2) The parenthetical remarks merely summarize the partial information obtained at each stage of the decision process. We could omit them. 3) Reflections lines divide the plane of a *632 into 30-60-90 triangles (each triangle forms a template); the plane of a *442 into 45-45-90 triangles; (each triangle forms a template); the plane of a *333 or 3*3 into 60-60-60 triangles (equilateral); (in the *333, each triangle forms a template in the 3*3, each triangle consists of 3 templates); the plane of a *2222 or 2*22 into rectangles (possibly squares); (in the *2222, each rectangle forms a template in the 2*22, each rectangle consists of 2 templates); the plane of a 4*2 into squares (each square consists of 4 templates). 4) The set of gyration points in a 632 form the corners of 30-60-90 triangles (and templates consist of 2 adjacent triangles); in a 442 form the corners of 45-45-90 triangles; (and templates consist of 2 adjacent triangles); in a 333 form the corners of 30-30-30 triangles; (and templates consist of 2 adjacent triangles); in a 2222 form the corners of parallelograms; (and templates consist of 2 adjacent parallelograms); in a 22x form the corners of rectanges (and templates consist of 1 such rectangle). 5) To detect feature such as gyrations, reflections and glide reflections, one must inspect the ENTIRE pattern. 6) If one can spot a feature with a definite sense (clockwise-counterclockwise, left-right) that NEVER occurs mirror reversed, then the pattern does NOT possess any reflections or glide reflections (and so has type 632, 442, 333, 2222 or o). But caution, an artist might choose to decorate a template with a glyph and its mirror image (e.g. something like a []), so merely finding a feature mirror reversed does not guarantee any type of reflection in the pattern as a whole. 7) Most people find higher order gyrations easier to see than lower order gyrations. If you've found the order 6 gyration points in a 632, they form the corners of equilateral triangles. Then the order 3 gyration points sit in the middle of those triangles (and form hexagons). The order 2 gyration points occur halfway between nearby gyration points of order 3 (and also between nearby gyration points of order 6). If you've found the order 4 gyration points (of both sorts) in a 442, they form the corners of squares. Then the order 2 gyration points sit in the middle of those squares. Ditto for the corner points in a *442. If you've found order 4 gyration points of just one sort in a 442, they form the corners of squares. Then the order 4 gyration points of the other sort lie in the middle of the squares with the order 2 gyration points at the midpoints of the sides. A 333 has gyration points of order 3 of three sorts. Sometimes you'll spot only 1 of the three sorts. The points of this sort will form the corners of equilateral triangles. In the middle of any such triangle you'll find an order 3 gyration point of another sort, and in the middle of an adjacent triangle an order 3 gyration point of the remaining sort.