“Wallpaper groups”? What does that mean anyway…

For the final project of my capstone class, my group and I decided to produce a tessellation coloring book. Since we are all future elementary math teachers, we thought this was a perfect, relaxing idea. Make random designs, generate them on a computer, bind them together, and BAM…a coloring book! However, our professor thought to challenge us and introduced us to the 17 Wallpaper Groups.

Each of the 17 symmetry groups is made up of translations, rotations, reflections, and glide-reflections, or any combination in between. I started researching about each symmetry group so I could decide how to incorporate it into our project. There was a lot of information on each group, such as: lattice type, IUC notation, rotation orders, and reflection axes. Here, is a spreadsheet to organize my findings.

After looking at all of these properties, I started to wonder about the history of these specific groups. Who was/were the mastermind behind our symmetrical wallpaper?

A brief summary of events, as told by Natalie:

It turns out the 17 plane symmetry groups were discovered during the 19th century. A collection of people throughout mathematical history added to the development: first, the Pythagoreans discovered that there are five regular solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron). These were used for a variety of purposes, but shined in architecture and decorative art. A man more famous in astronomy, Johannes Kepler has influenced work in mathematics as well. He is generally credited for the first systematic explanation of the set of tessellations known as *Semi-Regular* or *Archimedean*. There are 11 specific examples of these tessellations which are composed of regular polygons.

Next up, Robert Hooke looked at

different ways atoms could be arranged to form crystals. Then, in 1831, Hessel classified the 32 three-dimensional

point groups that correspond to the three-dimensional crystal classes. A crystal is an endless repetition of some point group operations in a 3D plane, or “wallpaper”. The wallpaper, or space lattice, is generated by tessellations of this symmetrical point group.

Now, in 1835, Frankenheim (possibly related to Frankenstein?) geometrically found all the symmetrical network of points that can have crystallographic symmetry.

A French mathematician by the name of Camille Jordan discussed a general method for defining all the possible ways of regular repeating identical groups of points (that’s a mouthful). Through this theory, he discovered 16 of the 17 wallpaper groups. It was not until 1891 when Fedorov, German mathematician Schonflies, and English geologists Barlow, proved that there were only 17 possible patterns in “The Symmetry of Regular Systems of Figures” (two-dimensional crystallography groups). And alas, the 17 Wallpaper Groups were born.

The discovery of these groups dates back a long time, with mathematicians and scientists from all over adding their insight. This was rather interesting to learn about and will be extremely helpful to my project!