Frieze-Patterns

Introduction

Frieze patterns are one-dimensional repeating patterns that are based on an equidistant point lattice. They have two-dimensional motifs that repeat symmetrically in one direction, making them periodic. Frieze patterns are part of a class of infinite discrete symmetric groups of patterns on a strip, which are called frieze groups. There are seven possible frieze groups that have infinite repeating symmetry within one dimension. These patterns can be mapped onto themselves by horizontal translation and other transformations.

Symmetries

• Translation (T): Translational symmetry is when the motif has undergone a movement, a shift or a slide, in a linear manner through a specified distance without any rotation or reflection.
• Glide (G): In 2-D geometry, glide reflection occurs when the motif is reflected over the line and then translated along the line as a single single operation.
• Vertical (V): When the motif can be divided into two equal halves, by a straight standing line, they are said to possess vertical symmetry.
• Horizontal (H): When the motif can be divided into two equal halves, by a straight sleeping line, they are said to possess vertical symmetry.
• Rotation (R): When the movement of the motif is fixed around an axis, the pattern contains rotational symmetry.

The Seven Frieze Groups

There are seven distinct frieze groups.

1. Hop - Translation Symmetry
2. Step - Translational and glide reflection symmetries
3. Sidle - Translational and vertical reflection symmetries
4. Jump - Tanslation and horizontal reflection symmetries
5. Spinning Hop - Translation and rotation (by a half-turn) symmetries
6. Spinning Sidle - translation, glide reflection and rotation (by a half-turn) symmetries
7. Spinning Hop - all symmetries- translation, horizontal & vertical reflection, and rotation

Methodology

The methodology was divided into two part:

1. Pattern Generation
2. Pattern Recognition