in Space

Hyperbolic Geometry

Frieze Patterns


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The most widespread systematic use of decorative patterns is the simplest: the creation of linear friezes. The range of alternatives is not as great as the wallpaper catalogues might lead one to believe. There are only seven linear patterns that can be repeated on a strip of paper to produce a frieze using two colours. When two colours (say, black and white) are used to produce a linear frieze, there are only four basic ingredients that can be employed to create a repeating pattern. The 1st is translation: just moving a pattern along the frieze, en bloc. The 2nd is reflection about a vertical or horizontal axis. The 3rd is rotation through 180° around a fixed point.

The 4th is glide reflection, which consists of a forward translation together with a reflection of the image about a line parallel to the direction in which it is translated, and results in the mirror images created by the reflection being slightly offset from one another, rather than being vertically aligned.

The seven different possible frieze patterns arise by acting upon some initial motif, which need have no symmetry, with one of the following operations:

(a) translation                      left top

(b) horizontal reflection             left bottom

(c) glide reflection                   centre top

(d) vertical reflection                centre

(e) rotation through 180°            centre bottom

(f) horizontal/vertical reflection       right top

(g) rotation/vertical reflection.        right bottom

Examples of the seven possible varieties of frieze patterns are found in decorations all over the ancient world: from the pottery of San Ildefonso, to the vases of the Incas, and traditional forms of Maori decoration.

Text taken from ‘The Artful Universe’ by John Barrow