Frieze Patterns
A frieze, or strip pattern, is a repeating pattern with translational symmetry in one direction. Additionally, the repeating patterns may have rotational, reflectional, or glide reflectional symmetry. Often these frieze patterns are seen as decorative borders in art and architecture around the world. And mathematically, it turns out, there are seven classes of frieze patterns that are generated by the isometries mentioned. They consist of the following structures:
- Class I – translations only
- Class II – translations and horizontal reflection
- Class III – translations and vertical reflections
- Class IV – translation and rotations by 180°
- Class V – contains all possible types of symmetries of a strip – translations, horizontal and vertical reflections and rotations by 180 o
- Class VI – translation and glide reflections and translations
- Class VII – glide reflections, translations and rotations by 180°
To help visualize these isometries, John Conway, a noted mathematician at Princeton University, developed a whimsical, yet accessible method of understanding these classes of frieze patterns. By using the idea of our footsteps, we can call each class by a corresponding move that we might make ourselves with our feet (or a single foot in two of the Classes). These movements identify the repeat or fundamental domain. 5
- Class I – Hop – one foot moving forward
- Class II – Jump – two feet together jumping forward
- Class III – Sidle – toes facing each other to heels facing each other
- Class IV – Spinning Hop – one foot to generate the rotation and the translation
- Class V - Spinning Jump – two feet together, spin around and forward
- Class VI – Step – basic walking forward
- Class VII – Spinning Sidle – heels facing each other, a spinning hop forward
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