# PenRace and Creating Tessellations (Pt 1)

Materials: Penrose Rhombus Tiles, Glue, Pencil, paper, eraser, electronic devices with an internet connection, favorably laptops, Wi-fi

Goals: Participants learn more about periodic and aperiodic tessellations, Penrose tilings

Topics: Tessellation, symmetries, translation, rotation, reflection, glide-reflection, (regular) polygon, (a)periodicity, the hat

- Refer to the slides attached to this article. Build the concept of tessellation through examples by asking participants for the similarities between the shapes. You can correct some ideas they may give by counterexamples: the tiles don’t have to be equal (the mosaic), they don’t have to be connected edge-to-edge (the pentagonal tiling), the tiles are not necessarily polygons (the giraffe spots)
- Introduce the concepts of isometry, and the four types of isometry after looking at some examples. Revisit the examples to reinforce the concepts
- Show how these ideas apply to tessellations with concepts: translational symmetry, n-fold rotational symmetry, (glide) reflection symmetry
- Give a (competitive) quiz to reinforce these concepts (even 17-year-olds are hyped by quizzes, trust me)
- Introduce the problem of plane tiling with one type of polygon or regular polygon by an interactive activity
- Introduce the concept of aperiodic tessellation and the history
- Reinforce by continuing the last quiz
- Present the Penrose tiles
.*(it’s crucial that you don’t reveal their names for now)* - Introduce the website https://sciencevsmagic.net/tiles/ for them to figure out the configuration, divide participants into teams, and see which team finishes tiling the designated number of Penrose tiles first.
- A tessellation-creating activity: drawing Escheresque tessellations from square grids. Introduce some basic symmetries for them to start.

Details:

- There is only a Vietnamese version for now. You can use the slides as a reference for what to include. A script will be down below
- They are translation, rotation, reflection, and glide-reflection. A verbal explanation is hard to grasp for newcomers, so visualize by an animation. Geogebra is an accessible and easy-to-use tool for this. https://www.geogebra.org/classic/hs4h2hhh
- There are examples on the slide. The flower tessellation is intended to exemplify for participants. Ask which tessellations are in it. There is rotational symmetry (6-fold, and in turn, 2-fold and 3-fold), translational symmetry (horizontal and roughly vertical translations, note how there are multiple translations), but no reflective symmetry since the flower petals are oriented in one direction. Sacrificing some symmetry of the original hexagonal tiling helps to give the artwork a more organic look.
- I use Quizziz for this. Since we have to retain this quiz for later use in the lesson, I use instructor-paced mode. For a whimsical flavor, do some sports commenting.
- For this part, use https://mathigon.org/polypad#polygons. The interface is for the most part interactive, but there are some things to notice. The zoom and pan functions are separate, note the downward right corner. Choose the custom polygons, grab the vertices, and move them around to edit. Double-click a vertice to remove it and click on the middle of a side to add a new vertice at that place. Check out below for a specific solution.
- For more details, check out the script. Just don’t mention Penrose tiles.
- This part checks the things we presented in 5 and 6. A two-round quiz gives more tension.
- For simplicity, print out 36-degree rhombi and 72-degree rhombi only (if possible, add patterns like those on the website at 9, don’t print out the entirety because that will make assembling the tiles a no-brainer task). For each team, about 90 72-degree and 60 36-degree should do.
- They’ll be sticking the pattern on a piece of paper as big as two pieces of A4 paper connected along one length. Try to prepare in advance ways to prevent sly participants from searching up the pattern (they can still find it without the name)
- This is one of the activities that can be a standalone. But one of the essential ideas is constructing the tessellation from a pattern of symmetry. Here, https://tiled.art/en/art/Foxes/, the tile is built by translating the left side to the right and translating the downside upwards. In our workshop, we let the participants work with pencil and paper., but the tiledart website has a tessellation maker as well. The designing functions for the tile’s interior on the website are a bit limited, however, so downloading Inkscape to support is an alternative. Check below for some more info.

To keep this article readable, further explanations are in part 2.