diff --git a/src/Advanced/Wallpaper/main.typ b/src/Advanced/Wallpaper/main.typ index 7614037..0d8b264 100644 --- a/src/Advanced/Wallpaper/main.typ +++ b/src/Advanced/Wallpaper/main.typ @@ -1,5 +1,11 @@ #import "@local/handout:0.1.0": * +// Resources: +// +// https://eschermath.org/wiki/Wallpaper_Patterns.html +// https://mathworld.wolfram.com/WallpaperGroups.html +// https://en.wikipedia.org/wiki/Wallpaper_group + #show: handout.with( title: [Wallpaper Symmetry], by: "Mark", @@ -10,3 +16,6 @@ #include "parts/01 reflect.typ" #pagebreak() + +#include "parts/02 rotate.typ" +#pagebreak() diff --git a/src/Advanced/Wallpaper/parts/02 rotate.typ b/src/Advanced/Wallpaper/parts/02 rotate.typ new file mode 100644 index 0000000..837c17d --- /dev/null +++ b/src/Advanced/Wallpaper/parts/02 rotate.typ @@ -0,0 +1,115 @@ +#import "@local/handout:0.1.0": * +#import "@preview/cetz:0.3.1" + += Rotational Symmetry + + +Another symmetry is n-fold rotational symmetry about a point, whose signature is written n. Multiple bold numbers means multiple points of rotational symmetry. + +Two points of rotational symmetry are considered the same if we can perform a translation + rotation sending one to the other, while leaving the pattern the same. + +There are also patterns with both kinds of symmetries. To classify such patterns, first find all the mirror symmetries, then all the rotational symmetries that are not accounted +for by the mirror symmetries. + +By convention we write the rotational symmetries before +the `*`. + + +#table( + stroke: none, + align: center, + columns: (1fr, 1fr), + rows: 50mm, + image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%), +) + + +#problem() +Mark the three rotation points in Figure 1. + +#problem() +Find the signature of the pattern in Figure 2. + +#solution([`3 *3`]) + +#pagebreak() + + +Some exceptional cases: It is possible to have two different parallel mirror lines. In +this situation the signature is ∗ ∗ + + +#table( + stroke: none, + align: center, + columns: 1fr, + rows: 60mm, + image("../res/**.png", height: 100%), +) + +#problem() +Draw another wallpaper pattern with signature `**` + + +#pagebreak() + +There are two other types of symmetries. The first called a miracle whose signature is +written ×. It is the result of a glide reflection, which is translation along a line followed +by reflection about that line. +This occurs when there is orientation-reversing symmetry not accounted for by a mirror. +For example, if we modify Figure 3 slightly we get a signature of ∗× + +#table( + stroke: none, + align: center, + columns: (1fr, 1fr), + rows: 60mm, + image("../res/*x-b.png", height: 100%), + image("../res/*x-a.png", height: 100%), +) + +Signature ∗×. There is a glide reflection (shown by the by the dotted line) +taking the clockwise spiral to the counter-clockwise spiral, reversing orientation + + +#pagebreak() + + +#problem() +Find the signatures of the following patterns: + +#table( + stroke: none, + align: center, + columns: (1fr, 1fr), + rows: 60mm, + image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%), + image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%), +) + + +#pagebreak() + +There is another exceptional case with two miracles, where there are two glide reflection +symmetries along distinct lines. There are other glide reflections, but they can be obtained +by composing the two marked in the diagram. + + +#table( + stroke: none, + align: center, + columns: (1fr, 1fr), + rows: 60mm, + image("../res/xx-b.png", height: 100%), + image("../res/xx-a.png", height: 100%), +) + +Figure 7: There are two distinct mirrorless crossings, so the signature is `xx`. +Lastly, if none of the above symmetries appear in the pattern, then there is only regular +translational symmetry, which we denote by O. + +In summary, to find the signature of a pattern: +- Find the mirror lines (∗) and the distinct intersections +- Find the rotational points of symmetry not account for by reflections. +- Look for any miracles (×) i.e. glide reflections that do not cross a mirror line. +- If you found none of the above, it is just O diff --git a/src/Advanced/Wallpaper/res/**.png b/src/Advanced/Wallpaper/res/**.png new file mode 100644 index 0000000..d6e3c80 Binary files /dev/null and b/src/Advanced/Wallpaper/res/**.png differ diff --git a/src/Advanced/Wallpaper/res/*x-a.png b/src/Advanced/Wallpaper/res/*x-a.png new file mode 100644 index 0000000..c238c28 Binary files /dev/null and b/src/Advanced/Wallpaper/res/*x-a.png differ diff --git a/src/Advanced/Wallpaper/res/*x-b.png b/src/Advanced/Wallpaper/res/*x-b.png new file mode 100644 index 0000000..22f0cf9 Binary files /dev/null and b/src/Advanced/Wallpaper/res/*x-b.png differ diff --git a/src/Advanced/Wallpaper/res/3*3.png b/src/Advanced/Wallpaper/res/3*3.png new file mode 100644 index 0000000..85bc171 Binary files /dev/null and b/src/Advanced/Wallpaper/res/3*3.png differ diff --git a/src/Advanced/Wallpaper/res/333.png b/src/Advanced/Wallpaper/res/333.png new file mode 100644 index 0000000..7a78cb0 Binary files /dev/null and b/src/Advanced/Wallpaper/res/333.png differ diff --git a/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-cm-4.jpg b/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-cm-4.jpg new file mode 100644 index 0000000..180b930 Binary files /dev/null and b/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-cm-4.jpg differ diff --git a/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-p4g-2.jpg b/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-p4g-2.jpg new file mode 100644 index 0000000..619892c Binary files /dev/null and b/src/Advanced/Wallpaper/res/wiki/Wallpaper_group-p4g-2.jpg differ diff --git a/src/Advanced/Wallpaper/res/xx-a.png b/src/Advanced/Wallpaper/res/xx-a.png new file mode 100644 index 0000000..cfcfbe5 Binary files /dev/null and b/src/Advanced/Wallpaper/res/xx-a.png differ diff --git a/src/Advanced/Wallpaper/res/xx-b.png b/src/Advanced/Wallpaper/res/xx-b.png new file mode 100644 index 0000000..4ab8e4f Binary files /dev/null and b/src/Advanced/Wallpaper/res/xx-b.png differ