Assessment activities: 5.3. Mosaics through transformations
A mosaic is a composition of flat figures that exhibit certain regularity, such as invariance under symmetries or rotations. Follow these steps to construct mosaics using GeoGebra.
- Choose any point A and draw two segments of equal length that meet at A, forming a 60° angle.
- Construct a non-convex trapezoid using the three points you already have. We will call it T.
- Perform two rotations of T with respect to point A (it doesn't matter the direction, but both should go in the same direction): one of 60° (which we will call T') and another of 120° (which we will call T*). We will call S the union of T, T', and T* (a new polygon).
- Use the longest side of S to perform the symmetry of the polygon with respect to that side, called S'. You will obtain a new polygon R, which is the union of S and S'.
- Check if R is invariant under a 60° rotation and find its axes of symmetry (if any).
- Select a diagonal of maximum length (in the direction that is most appropriate for your screen) as a vector and perform successive translations of the polygon according to that vector (successive means that if the result of translating R is R', you need to perform the translation again on R' to obtain R'', then translate R'' to obtain R''', and continue in this way until the translations go off the screen).
- Finally, select another diagonal that is different and appropriate, and repeat the procedure. This time, you must translate all the polygons you obtained in step 6).