Tiling of the Plane With A Tree
Illia Stadnyk
Wallpapers have a unique patterns where they can be formed using the same pattern over and over again. The shapes are usually constructed using one shape but can be done with more than one shape as long as the follow that they have no overlap and no gaps, such as simple walkway using squares. In mathematics this is done usually using translational symmetry and rotations, but in the end the patterns share a symmetry. 
These tiling shares a lot with crystals who they share patterns with. That the rotations are only in groups of 2, 3, 4, or 6. That the symmetry in the crystal is that of the tiling groups and is referred to as plane crystallographic group. They come in total of unique 17 groups. Each group has a name that refers to the number of rotations, if it has a mirror or a glide reflection, and if it is primitive cell or centered cell. The proof for these 17 groups was written by Evgraf Fedorov in 1891.
The group we will look today is pg, or the full name being p1g1. It is a primitive cell that has the highest order of rotation of 1, and that it has a single glide rotation. It is relatively simple wallpaper, the pattern can be seen through a lot of patterns such two rectangles that make the end of odd looking arrow. It can be made with many shapes as long as they can be fitted in as there is no rotations needed.
To make create wallpaper group pg can be most easily done with a square or with a triangle as they can easily be used for tiling. A more interesting is combining these shapes to create is something that resembles a Christmas tree. To make sure it works properly the shape has to be made so that glided reflection has no gaps, if that is satisfied then it will be a pg group.
In my case the shape was similar that of a basic tree. The design originally did not have correct spacing so the shape can an opening where the bottom parts of the glided reflection was supposed to connect. Simple extension from a simple triangle shape to more of a rectangle shape allowed the extension to make sure there was no gap bellow.
Citations:
https://en.wikipedia.org/wiki/Wallpaper_group#Group_pg
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