It may not be obvious how an infinite frieze pattern can be created from a basic element. Follow these step-by-step instructions to create a frieze using Design 6 (reflection over a vertical line plus reflection over a horizontal line plus translation). The instructions use the letter p as a basic design element of the pattern. The printable design element page (PDF) contains several versions of a more complex design element. Print this page and cut out the design elements. Then create the frieze pattern using this design element. Alternatively, you can draw your own design element to create the frieze pattern.
Step 1: Start with a basic design element. It's best if it is a nonsymmetric design so that the symmetry created by the transformations is more apparent.
Step 2: All frieze patterns have translation symmetry, so we'll leave that for last. Once we create a basic unit that contains all of our required other symmetries, we can translate it infinitely in both directions. So in this case, we'll start with a vertical reflection. Take your basic design element and reflect it over a vertical line. It's best to choose a line that is close to, but not intersecting, your original element.
There are now two pieces to your basic design: the original element and its reflected image.
Step 3: The next symmetry is horizontal reflection, so take your basic design block (now two elements) and reflect them both over a horizontal line. Again, choose a line that is close to, but not intersecting, your original design.
Step 4: We now have a basic element with all of the required symmetries except for translation. Take your basic element and translate it by a fixed distance in both directions. You have created a frieze pattern!
The seven patterns given are certainly not all the possible combinations of transformations. How can they be the only possible frieze patterns? It turns out that other combinations fall into one of these categories as well. That is, they create equivalent patterns.
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