Ten-Pointed Star Patterns
Examples of Tessellation in Islamic Art

The geometric skill of the artists and designers of the Medieval Islamic world is universally recognized. The Arab scholars who rescued Greek and Roman mathematical learning from oblivion made possible a blending of aesthetic and mathematical sensibilities that led to artisitic achievements of marvelous skill and beauty. The average person who visits the an Islamic monument will be dazzled by the beauty and intricacy of the patterns executed in mosaic tiles and carved plaster. But a person with a mathematical bent will also be delighted by the geometric sophistication of the decoration. For example, repeating patterns can be analyzed by types of symmetry. Mathematicians have identified 17 distinct types of "plane symmetry groups", also called "wallpaper groups" Wikipedia article. Remarkably, examples of all seventeen symmetry groups can be found in the decoration of the Alhambra, the Moorish palace in Granada, Spain.

As an introdction to the study of geometry and symmetry in Islamic art, it is useful to study one of the simpler designs. I have chosen a ten point star pattern, one that is very common throughout the Islamic world and which is the basis of many variations.

This is a detail of a very beautiful mosaic panel from the Friday Mosque in Isfahan, Iran. Multicolored ten-pointed stars are in a pattern with blue pentagons and black, kite-shaped, trapezoids.

On first glance, this panel might seem "impossible" as it is a space-filling pattern based on a pentagon, and pentagons do not tessellate very well. On closer inspection, one notices that in a few key spots, the pentagons are replaced by a six-sided figure that is composed of two overlapping pentagons. This double-pentagon allows the whole pattern to resolve itself and fit into a rectangular framework.
 

Symmetry Operations

Mathematicians classify two-dimensional patterns by "symmetry operations". There are four of them: translation, reflection, glide reflection and rotation. To understand how they work, imagine that you have made a copy of the pattern on tracing paper and that you are moving, turning or flipping the paper to match it with the original in different ways.

Axes of Translation Symmetry

By "translation" we just mean "movement along a straight line". If you have your tracing, and move it in the direction of any of the axes, you will soon find it matches a "repeat" of the pattern. In this design, there are 3 distinct Axes of Translation Symmetry.

Axes of Reflection Symmetry

If you turn over your piece of tracing paper, and match any axis with its axis on the orginal, the rest of the pattern will match. This is more easily understood as "Mirror Symmetry". In this pattern there are 2 distinct Axes of Reflection Symmetry.

Axes of Glide Symmetry

Glide Reflection is a little harder to visualize. First flip over your tracing paper and line up an axis with its axis on the original, then slide the tracing along the axis until the patterns match.

A series of footprints, left and right, one after another, is an example of glide reflection. In this pattern there are 2 distinct Axes of Glide Symmetry.

Points of Rotational Symmetry

If you were to put a pin through your tracing paper at any of the points, you could then rotate your paper 180 degrees and it would match the original. In this pattern there are 2 distinct Points of two-fold Rotational Symmetry.
 

The Same Pattern "Moroccan Style"

The pattern we've been looking at, from Isfahan, Iran, is a typical Persian style pattern and it can be found in Mosques and Madrasas from Iraq to Afghanistan and even Pakistan. In the Western Islamic lands, North Africa and Southern Spain, a different aesthetic prevails. You will find the same basic pattern, but the shapes of the tiles are different:

  • The Pentagons become Five-Pointed Stars.
  • The Kites grow into six_sided shapes resembling Shields
  • The Ten-Pointed Stars become spinier

      • On the left is the Persian style pattern reduced to its most basic elements. On the right is the Moroccan style pattern. Notice that the corners of the pentagons occur in the same positions as the points of the five-pointed stars.

On the left is the Persian style pattern reduced to its most basic elements. On the right is the Moroccan style pattern. Notice that the corners of the pentagons occur in the same positions as the points of the five-pointed stars.

Here is a recent, exquisite example of the Moroccan style pattern. It is in the Mausoleum Mosque of King Mohammed V in Rabat, Morroco which was completed in 1971. The zla'iji (tilesetter) responsible for it was Maallem (Master) Mulay Hafid 'Alawi of Fes.
 

If you superimpose the Persian pattern onto the Moroccan, you find that almost all of the vertices match up, and a third pattern emerges, somewhat busy but still quite lovely. The new pattern fuses East and West. Strangely, I have only seen example of this pattern in use.
 
 

A Timurid Variation

On the left is a portion of a mosaic frieze from the Shrine of Abdul Ansari in Gazurgah, Afghanistan. The artist has used the same shapes as the Persian Ten Pointed Star pattern, but they have been stretched and squeezed to allow eight and twelve pointed stars to coexist in a pentagonal design.

With the help of Photoshop, I have transformed this from a frieze into an allover pattern, revealing further beauty.



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