CONSTRUCTING THE SEVEN FRIEZE PATTERNS

A frieze pattern is a symmetrical pattern that extends indefinitely in one direction. In this project, you'll start by constructing the simplest frieze pattern, a pattern with only translation symmetry, then you'll apply isometries to it incrementally to create other symmetrical frieze patterns. You'll need to imagine these patterns are infinite, even though you'll only be constructing parts of them on your finite computer screen. You'll name the patterns you create using the two-character code xy. If x = m, there are vertical mirror reflections. If y = m, there is a horizontal mirror reflection. If y = g, there are glide reflections, but not a horizontal mirror reflection. If y = 2, there are rotations of 180° but not glide reflections. A 1 in either of these positions means the pattern doesn't have this type of symmetry. This description is based on a description provided by Doris Schattschneider, and matches the one used in the software program Kaleidomania!TM.

Sketch and Investigate

1. At the left side of your sketch, construct an asymmetrical polygon and its interior, like the flag shape shown at right.
2. Underneath this polygon, construct a horizontal segment EF, about an inch long.
3. Select, in order, point E and point F. Then choose Transform >> Mark Vector.
4. Translate the polygon interior by this marked vector.
5. Repeat until you have at least six polygons that go across your sketch.

You've constructed part of an example of the simplest frieze pattern: one with only translation symmetry. Experiment with changing your original polygon and your translation vector (EF) to see how the original polygon and its images are related. When you're done experimenting, make sure the translation vector is horizontal.

1. Choose File >> Save. Name the sketch 11.gsp.

Now construct a pattern with a horizontal line of symmetry.

1. Double-click on EF to mark it as a mirror for reflection.
2. Select the polygon interiors; then choose Transform >> Reflect.

You now have a second frieze pattern: one with reflection symmetry through a horizontal line in addition to translation symmetry. Experiment with changing your original polygon and your vector (which is now doubling as a mirror). Again, when you're done, make sure EF is horizontal.

1. Explain why this figure also has glide reflection symmetry.
2. Choose File >> Save As (not Save) and save the sketch as 1m.gsp.
3. Select and hide appropriate polygons so that the figure has glide reflection symmetry but no longer has reflection symmetry.

1. Save this sketch as 1g.gsp.

Next, you'll construct a pattern with vertical line of symmetry.

1. Reopen the sketch you saved earlier: 11.gsp.
2. Draw a vertical segment somewhere near the middle of this pattern.
3. Double click on the segment to mark it as a mirror for reflection.
4. Select the polygons, then choose Transform >> Reflect.

You should now have a pattern with reflection symmetry through a vertical line. Experiment with dragging different parts of the sketch.

1. Save this sketch as m1.gsp.
2. Drag a point so that EF is no longer perpendicular to sGH. What happens? Explain why the vertical reflection line must be perpendicular to the translation vector in this frieze pattern.

Let's apply a 180° rotation, also called a half turn, to your basic translation pattern.

1. Reopen the sketch 11.gsp.
2. Construct a point near the center of the pattern.
3. Double-click on the point to mark it as a center of rotation.
4. Select all the polygons; then choose Transform >> Rotate.
5. Enter 180°.

You should now have a figure with 180° symmetry. Experiment with dragging different parts of the sketch.

1. Save the sketch as 12.gsp.
2. Explain why the only symmetries a horizontal frieze pattern can have are translation symmetry, reflection symmetry through a vertical line, glide-reflection symmetry, reflection symmetry through a horizontal line, half-turn symmetry, and combinations of these.

We can create other possible frieze patterns using combinations of the isometries you've used so far.

1. Open the sketch 1g.gsp. This sketch has glide reflection symmetry.
2. Construct a vertical segment near the center of the pattern.
3. Double-click on the segment to mark it as a reflection mirror.
4. Reflect the polygons.

You now have a figure with glide reflection symmetry and reflection symmetry through a vertical line. Experiment by dragging different parts of the figure.

1. Save this sketch as mg.gsp.

1. Besides translation symmetry, glide-reflection symmetry, and reflection symmetry through vertical mirrors, what other kind of symmetry does this pattern have? Explain why.
2. Open the sketch m1.gsp. This sketch has vertical mirror reflections.
3. Reflect the polygons in this sketch over EF.
4. Save this sketch as mm.gsp.

1. Describe all the symmetries this figure has in addition to translations and vertical and horizontal mirror reflections, and explain why it has them.
2. Explain why the seven sketches you've made represent all the possible frieze pattern symmetries.
3. Experiment with changing colors of different polygons in this final sketch so that you can give it any of the seven frieze pattern symmetries.