If we glide-reflect down from the rightmost square, would it match the translation right from the bottom square? Yes! Escher used this pattern a lot, as in this example from the Escher estate website. What would would happen with a translation in one direction (say horizontally) and a glide reflection in the other (vertically)? (Numbering or labeling the sides of the tile helps track of the motions.) We could just translate the square to each neighbor, as illustrated in this diagram. We could use midpoint rotations like I did for the triangle, but there are many other options. ![]() Let us consider some of the many possibilities with squares. I will show you how to make a square-based tiling, but you may preview my strategy by watching this video based on a triangle tiling. For example, any triangle tessellates with rotations around the midpoints of its sides. We’re especially interested in the motions that could take the original shape onto a replica that shares an edge with it. As we make a customizable tile, the same transformations will be involved in both making the tile and in replicating it to fill the plane. The first step is picking a starting tile that tessellates the plane (for example, a square or regular hexagon, any triangle or quadrilateral.) Once that decision is made, the next step is figuring out which motions you will use to take one tile to another. Starting out with a plain square in a checkerboard tiling, we will make changes to the edges, using our understanding of transformations to make sure the changed shape will still tile the plane. This post will guide the reader through using GeoGebra, a free dynamic algebra and geometry program, to make a customizable tiling. Escher’s work that caught the world’s imagination, and makes a great hook for a mathematics lesson. Dover has a great-looking collection of his designs. Moser was interested in all sorts of repeating motifs. In Robert Fathauer’s exhaustive new text Tessellations: Mathematics, Art and Recreation, he introduces Koloman Moser as possibly the first to make tessellations that looked like real-life objects. I was really honored to be asked to write for Henri’s site, which is really the most comprehensive site any one person has made to support math teachers, while also being of utmost quality. I asked him to share his approach to using software to “Escherize” tessellations. John Golden is a connoisseur of math-art and an expert user of GeoGebra. ![]() One reason this works is the connection with art, including the abstract patterns in Islamic art and the mind-bending creations of M.C. In my last two posts, I promoted the idea of using tiling (tessellation) as an interesting context in geometry class, especially for the introduction of some basic ideas of transformational geometry.
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