Math 2020: Wallpaper patternsAny pattern on an (infinite) plane surface, which is periodic in two directions is called a “wallpaperpattern”. To be periodic in a certain direction means that if you move it a certain nonzero amount inthat direction, the pattern exactly matches up with itself.Isometries:An isometry of the plane is a map from the plane to itself the preserves distances. E.g., if two pointsare 5 inches from each other before the transformation, they are still 5 inches from each other after thetransformation.There are only 4 different kinds of isometries:• Translation: means to move a certain amount in a certain direction.This can be described by vectors: move the point (x, y) to (x + a, y + b).• Rotation: means to rotate about a fixed point through a certain angle.• Reflection: means to reflect about a fixed line.• Glide Reflection: means to reflect about a line, and then moveparallel to this line by some amount.translate by this amountrotate 30 degreescenter of rotationaxis of reflectionaxis of glide reflectionWallpaper groups:The symmetry group of a wallpaper pattern is the set of all the transformations of the plane thattake the wallpaper pattern to itself. A symmetry group of a wallpaper pattern is called a wallpapergroup. It turns out that there are only 17 different types of wallpaper group. The first pattern only hastranslations, and no rotations, reflections or glide reflections. See second handout sheet for examples ofall the rest.Symmetry group:translations by (a,b)for all integers a,bSymmetry group:translations by (a,b)for all integers a,bSymmetry group:for all integers a,btranslations by (2a,b)exactly the same symmetry groupsSame ”type” of symmetry groupsdifferent from first three -has rotationsSymmetry group:rotations of 90 degrees,center (2a,2b), andfor all integers a,brotations of 90 degrees,center (2a+1,2b+1),translations by (2a,2b) andIn all these pictures the bottom left corner is(0,0) in the x-y plane. The patterns should beimagined to continue infinitelyGoal for students: By able to look at a wallpaper pattern, and identify which symmetries preservethe pattern. Be able to locate centers of rotations, and axis of reflections. Be able to say whether twodifferent patterns have the same type of symmetry groups.Fundamental unit:For any wallpaper pattern, there is a smallest “tile” or unit that can be used to make the pattern, byrepeating the unit over and over.Goal for students: Be able to look at a wallpaper pattern and be able to identify a fundamental unit.Useful references:Hollister (Hop) David’s 17 Wallpaper Groups page, with java applets:http://www.clowder.net/hop/17walppr/17walppr.htmlThe geometry center has a detailed description of all patterns with animations andlink to the kali java applet for drawing your own patterns:http://www.scienceu.com/geometry/articles/tiling/wallpaper.htmlDavid Joyce’s wallpaper page, with examples and descriptions, and a page for each wallpaper patternhttp://www.clarku.edu/ djoyce/wallpaper/Mathworld page, with examples of each of the 17 types of patterns:http://mathworld.wolfram.com/WallpaperGroups.htmlXah Lee’s page with pictures of all wallpaper patterns:http://www.xahlee.org/Wallpaperdir/c5 17WallpaperGroups.htmlThis page has a list of steps you can follow to determine which kind of symmetry group a wallpaper pattern has:http://plato.acadiau.ca/courses/educ/reid/Geometry/Symmetry/symmetry.htmlMy own wallpaper program, for drawing your own patterns:http://hverrill.net/Wallpaper/wallindex.html1Imagine you are looking through small windows at infinitely repeating wallpaper patterns.Exercise:Match up the patterns on the left with patterns on the right which have the same symmetry groups. (Thesymmetry group is the list of all possible symmetries, e.g., rotations, (glide) reflections, translations.)These images were created with the java applet at: