Fractional Beauty
Harrie Welles

Starting point for my work are the 17 types of plane symmetry: 17 ways to arrange a motif regularly in the plane (Wikipedia gives a beautiful description under 'wallpaper groups'). These repetitive patters has fascinated people from the beginning of mankind. Already the egyptian craftsmen of the days of the great Pharaohs were masters in playing with the symmetry constraints inherent in each of 'the seventeen'. The fascination how to 'make art' out of these constraints reached its peak in medieval Islamic decoration art, with the Alhambra in Granada as one of its most illustrious examples.
In the footsteps of these distant ancestors my whole life I am cursed with an obsessive interest in the regimes that reign in each of the 17 plane symmetries: How can beauty be elicited from them? The method I follow is to introduce ratio's within these plane symmetries, whereby the integers in the ratio are related to each other in a complex geometrical way. I call my approach 'fractional beauty', because the integers in their geometric relatedness can be unraveled in fragments by means of arithmetic operations, which makes possible to predict which types of elements occur in the build up of patterns and provides handles for the manipulation of its beauty. In particular 'continued subtraction' and 'continued fraction' are useful for the unravelling of this geometric relatedness

The complexity and beauty of shape and color that can be generated by this purely mathematical approach suprises me over and over again.

Over the years I developed three diffent approaches in the introduction of ratios in plane symmetries. The mathematical background is explained under the 'Eueler knobs' at the end of the site.

In the gallery below, pieces are categorized according to these three approaches.


  on the road with stripes

  playing with direction in crystals

  vary in Alhambra tiling

cultural historical roots

Part of the beauty I try to elicit from plane symmetries finds expression in stripy patterns. These can be positioned in a heritage of thousands of years of decorative art. In Antiquity the Egyptians embellished their temples and graves with stripy patterns, the Greek used them to decorate their famous pottery and the elite in the Roman Empiredid in decorating of their splendid villa's. But also in Islamic culture and in the Far Eeast (Japan,China) stripy patterns are deeply rooted in decorative art. Under the knob just right from this text you can find an elucidation of these historical roots.

papers and


exhibitions and

about the author contact

Mathematical and methodological principles

Under the first Euler knob the seventeen plane symmetries are elucidated. Under the next Eueler knobs three different ways are shown to evoke complexity within these plane symmetries by the introduction of ratios.

1 seventeen plane symmetries

2 the introduction of obliquity

3 the introduction of lines of deflection

4 the introduction of serially connected line segments