Alumni Gazette

Schattschneider offers several tessellation “recipes,” similar to those that Escher discovered after much investigation. In general, tessellations are created either by taking a simple shape and modifying it through geometric translation (moving corresponding endpoints of opposite sides by the same distance and in the same direction); or by modifying a shape through translation and reflection (the creation of a mirror image).
Follow these recipes to create your own tessellations, manually or with software programs such as The Geometer’s Sketchpad, GeoGebra, Cabri, or TesselManaic.
Using translation

Parallelogram
- Replace two adjacent sides (a and b above) with any series of lines or curves that connect the endpoints of the sides.
- Translate the new lines or curves to their opposite sides.

Par-hexagon
- Replace three adjacent sides (a, b, and c above) with curves that connect the endpoints of the sides.
- Translate the new lines or curves to their opposite sides.
- To tile the plane, translate the tile so that opposite edges match.

Using reflection and translation
- Begin with a rhombus to create a tile with reflection symmetry.
- Replace one side of the rhombus with a curve that connects the endpoints of the side (side a above).
- Reflect that curve in a diagonal of the rhombus that meets one of those endpoints (reflect to replace side b above).
- Translate the two curves to their opposite sides.
- To tile the plane, simply translate the tile so that opposite edges match.