Farn

Fractal Frieze Ornaments - Symmetry I

The Frieze Groups are represented with fractal patterns in my FriezeOrnaments Applet. It's easy to use, so have a try. Filling Frieze Ornaments along an axis in a regular manner, you only have seven types of moves to copy the basic pattern.

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Fractal Frieze Ornaments

Ornaments are very important in art and handicraft since a long time. It was already used, when the mathematical background wasn't explored yet. Therefor it is a surprise, that you could find all possible forms in historical art objects, p.e. The Alhambra. The symmetry of Frieze Ornaments is very attractive to nearly everyone.
In a Frieze Ornament there is an axis, to which moves of the basic pattern are made. These moves fill the pattern band. It is proved that there are exactly 7 symmetry groups, called the Frieze Groups, which are described below. The applet ornaments I call Fractal Frieze Ornaments, simply because the basic patterns are of fractal kind.

Complete list of possible transformations

 Type   DescriptionVisualization
  

1

The basic pattern is moved by a constant amount: It is called a translation or a move translation.

Type 1

2

The ornament is filled by continuous use of a combination of a mirror operation and a translation along the ornament's axis.

Type 1

3

The basic pattern is repeated only by 180°-rotations. The center of the rotations are located on the ornament's axis.

Type 1

4

Only reflections at lines orthogonal to the ornament's axis are needed for this result.

Type 1

5

Similiar to type 2, but the result of the mirror operation is part of the ornament.

Type 1

6

The start is like the beginning of type 3, followed by a reflection at a line orthogonal to the ornament's axis.

Type 1

7

The new Frieze Ornament's parts are produced by a combination of two mirror operations, one at the ornament's axis followed by a reflection at a line orthogonal to it.

Type 1

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© 2007 Ulrich Schwebinghaus