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"parts/01 reflect.typ" +#pagebreak() diff --git a/src/Advanced/Wallpaper/meta.toml b/src/Advanced/Wallpaper/meta.toml new file mode 100644 index 0000000..f0630ae --- /dev/null +++ b/src/Advanced/Wallpaper/meta.toml @@ -0,0 +1,7 @@ +[metadata] +title = "Wallpaper Symmetries" + + +[publish] +handout = true +solutions = true diff --git a/src/Advanced/Wallpaper/parts/00 intro.typ b/src/Advanced/Wallpaper/parts/00 intro.typ new file mode 100644 index 0000000..b3bfc7c --- /dev/null +++ b/src/Advanced/Wallpaper/parts/00 intro.typ @@ -0,0 +1,131 @@ +#import "@local/handout:0.1.0": * +#import "@preview/cetz:0.3.1" + += Wallpaper Symmetries + +#definition() +A _Euclidean isometry_ is a transformation of the plane that preserves distances. \ +Intuituvely, an isometry moves objects on the plane without deforming them. + +There are four classes of _Euclidean isometries_: +- Translation +- Reflection +- Rotation +- Glide reflection +#note([We can prove that there are no others, but this is beyond the scope of this handout.]) \ +A simple example of each isometry is shown below: + +#let demo(c) = { + let s = 0.5 + cetz.draw.line( + (0, 0), + (3 * s, 0), + (3 * s, 1 * s), + (1 * s, 1 * s), + (1 * s, 2 * s), + (0, 2 * s), + close: true, + fill: c, + stroke: black + 0mm * s, + ) +} + +#table( + stroke: none, + align: center, + columns: (1fr, 1fr), + rows: (3.5cm, 3.5cm), + row-gutter: 2mm, + [ + #cetz.canvas({ + import cetz.draw: * + + demo(ored) + translate(x: 0, y: -1.5) + demo(oblue) + }) + #v(1fr) + Translation + ], + [ + #cetz.canvas({ + import cetz.draw: * + + circle((-2, 0), radius: 0.1, stroke: none, fill: black) + arc( + (-2, 0), + radius: 1, + anchor: "origin", + start: 0deg, + stop: -30deg, + mode: "PIE", + ) + + demo(ored) + rotate(z: -30deg, origin: (-2, 0)) + demo(oblue) + }) + #v(1fr) + Rotation + ], + + [ + #cetz.canvas({ + import cetz.draw: * + + line((-2, 0), (4, 0)) + + translate(x: 0, y: 0.25) + demo(ored) + set-transform(none) + + set-transform(( + (1, 0, 0, 0), + (0, 1, 0, 0), + (0, 0, 1, 0), + (0, 0, 0, 1), + )) + + translate(x: 0, y: 0.25) + demo(oblue) + }) + #v(1fr) + Reflection + ], + [ + #cetz.canvas({ + import cetz.draw: * + + + demo(ored) + + set-transform(( + (1, 0, 0, 0), + (0, 1, 0, 0), + (0, 0, 0, 0), + (0, 0, 0, 0), + )) + translate(x: 1.5, y: 0) + + demo(oblue) + + set-transform(none) + line((-1, 0), (5, 0)) + }) + #v(1fr) + Glide reflection + ], +) + +#definition() +A _wallpaper_ is a two-dimensional pattern that... +- has translational symmetry in at least two directions + #note([(and therefore fills the plane)]) +- has a countable number of reflection, rotation, or glide symmetries. \ + +#problem() +Is the empty plane a valid wallpaper? + +#solution([ + No, since it has uncountably many symmetries. +]) diff --git a/src/Advanced/Wallpaper/parts/01 reflect.typ b/src/Advanced/Wallpaper/parts/01 reflect.typ new file mode 100644 index 0000000..c35f8bf --- /dev/null +++ b/src/Advanced/Wallpaper/parts/01 reflect.typ @@ -0,0 +1,99 @@ +#import "@local/handout:0.1.0": * +#import "@preview/cetz:0.3.1" + += Mirror Symmetry + +#definition() +A _mirror symmetry_ is a reflection about a line. \ +If $n$ mirror symmetries intersect at a point, we say that point is an _$n$-fold mirror node_. + +#v(3mm) + +Two mirror nodes are identical if we can map one to the other with a translation and a rotation \ +while preserving the pattern on the wallpaper. + +#problem(label: "pat333") +Find all three three distinct mirror nodes in the following pattern. \ +What is the order of each intersection? \ +#hint([ + You may notice rotational symmetry in this pattern. \ + Don't worry about that for now. +]) + +#table( + stroke: none, + align: center, + columns: 1fr, + rows: 50mm, + image("../res/wolfram/p3m1.svg", height: 100%), +) + +#solution([This is `*333`]) + +#v(1fr) + + +#definition() +_Orbifold notation_ gives us a way to write down the symmetries of a wallpaper. \ +We will introduce orbifold notation one symmetry at a time. + +#definition() +In orbifold notation, mirror nodes are denoted by a `*` followed by at least one integer. \ +Every integer $n$ following a `*` denotes a mirror node of order $n$. + +#v(3mm) + +The order of these integers doesn't matter. `*234` and `*423` are the same signature. \ +However, we usually denote $n$-fold symmetries in descending order (that is, like `*432`). + +#problem() +What is the signature of the wallpaper in @pat333? +#solution([It is `*333`]) + + +// MARK: page +#v(1fr) +#pagebreak() + +#problem() +Find the signature of the following pattern. + +#table( + stroke: none, + align: center, + columns: 1fr, + rows: 50mm, + image("../res/*632-a.png", height: 100%), +) + + +#solution([ + It is `*632`: + #table( + stroke: none, + align: center, + columns: 1fr, + rows: 40mm, + image("../res/*632-b.png", height: 100%), + ) +]) + +#v(1fr) + +#problem() +Draw a wallpaper pattern with signature `*2222` + +#solution([ + Sample solutions are below. + + #table( + stroke: none, + 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zy&Pq5`}@kw+#!2(2ir=X@qh + + +pmm +Exported by Tess 1.70. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -- 2.47.2 From 2b8f25944c9608d92d2334921824a7353d584f0e Mon Sep 17 00:00:00 2001 From: rm-dr <96270320+rm-dr@users.noreply.github.com> Date: Tue, 22 Apr 2025 19:54:48 -0700 Subject: [PATCH 3/7] TMP --- src/Advanced/Wallpaper/main.typ | 9 ++ src/Advanced/Wallpaper/parts/02 rotate.typ | 115 ++++++++++++++++++ src/Advanced/Wallpaper/res/**.png | Bin 0 -> 547667 bytes src/Advanced/Wallpaper/res/*x-a.png | Bin 0 -> 681794 bytes src/Advanced/Wallpaper/res/*x-b.png | Bin 0 -> 163286 bytes src/Advanced/Wallpaper/res/3*3.png | Bin 0 -> 546077 bytes src/Advanced/Wallpaper/res/333.png | Bin 0 -> 581904 bytes .../res/wiki/Wallpaper_group-cm-4.jpg | Bin 0 -> 819696 bytes .../res/wiki/Wallpaper_group-p4g-2.jpg | Bin 0 -> 446444 bytes src/Advanced/Wallpaper/res/xx-a.png | Bin 0 -> 82614 bytes src/Advanced/Wallpaper/res/xx-b.png | Bin 0 -> 415426 bytes 11 files changed, 124 insertions(+) create mode 100644 src/Advanced/Wallpaper/parts/02 rotate.typ create mode 100644 src/Advanced/Wallpaper/res/**.png create mode 100644 src/Advanced/Wallpaper/res/*x-a.png create mode 100644 src/Advanced/Wallpaper/res/*x-b.png create mode 100644 src/Advanced/Wallpaper/res/3*3.png create mode 100644 src/Advanced/Wallpaper/res/333.png create mode 100644 src/Advanced/Wallpaper/res/wiki/Wallpaper_group-cm-4.jpg create mode 100644 src/Advanced/Wallpaper/res/wiki/Wallpaper_group-p4g-2.jpg create mode 100644 src/Advanced/Wallpaper/res/xx-a.png create mode 100644 src/Advanced/Wallpaper/res/xx-b.png 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 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