Dynamic Symmetry, Geometric Progressions, and Islamic Geometric Patterns
Chris Heath 
Discover how geometric progressions, dynamic symmetry, and the traditional method of creating Islamic geometric patterns are inextricably related.
Dynamic Symmetry
Dynamic symmetry is synonymous with a small number of rectangles, in particular, the root-two rectangle, root-three rectangle, root-four rectangle, root-five rectangle, and the golden rectangle.
Geometric Progressions
While fractal patterns possess the property of 'self-similarity' and demonstrate a characteristic irregularity that can be seen at all scales, geometric progressions are 'self-same' and express a characteristic regularity at all scales.
What this means, is that at all scales, or any scale, a geometric progression looks exactly the same. In this sense, geometric progressions have no scale; they do introduce proportion.
If you study the image above, you will see the same polygons nested within the same polygons. From the outside in, the size of each polygon is related to the next smaller polygon by a continuous proportion.
In the square progression, the proportion is the square root of two (√2). In the triangle progression, the proportion is the square root of three (√3), and in the pentagon progression, the proportion is phi, or the golden section. The golden section is commonly referred to as the golden ratio.
Geometric Progressions & Dynamic Symmetry
If you construct a geometric progression, you will be embedding dynamic rectangles into the progression. For example, the triangle progression if drawn to infinity, will reveal infinitely more root-three rectangles.
The root-two and root-three proportions are well suited to drawing Islamic geometric patterns. If you look at the root-two and root-three rectangles above, the lines of the progressions and all the angles line up perfectly with the their respective progressions.
In the case of the pentagon progression, there is a rectangle that is better suited to drawing Islamic geometric patterns, and that is the 54˚rectangle. As you can see below, you will find this rectangle embedded in the pentagon progression many times over.
You might be wondering why I left out the root-four rectangle, and that was deliberate. The root-four rectangle is also know as the double square. If you place two squares side by side to create a rectangle, what you end up with is a root-four rectangle, and the square root of four is two.
Dynamic Rectangles
With the exception of the root-four rectangle, the ratio of sides, that is the length of a dynamic rectangle's long side divided by the length of the rectangle's short side equals an irrational number. An irrational number is a number that has no end. It runs on forever.
- For the root-two rectangle, the number is, 1.4142... which is the square root of two (√2). The root-two rectangle may be used as the unit of repetition in a four-fold pattern.
- For the root-three rectangle, the number is 1.732... which is the square root of three (√3). The root-three rectangle may be used as the unit of repetition in a six-fold pattern.
- For the root-five rectangle, this is 2.236..., which is the square root of five (√5), and for the golden rectangle, the number is 1.618..., where 1.618... : 1 : 0.618 is the golden section.
The 54˚rectangle is used for creating five-fold Islamic Geometric Patterns. It is often referred to by Mohamad Aljanabi as the 'standard rectangle' because it's arguably the easiest rectangle to use for Islamic five-fold patterns. There are other rectangles suitable for five-fold patterns, and although the golden rectangle isn't one of them, all of these rectangles that are used for five-fold patterns embed the golden ratio into the patterns that are created.
Here are the three rectangles mentioned above that are suitable for Islamic geometric patterns (or any repeating pattern).
Bringing it Altogether
The title of this post is 'Dynamic Symmetry, Geometric Progressions and Islamic Geometric Patterns', so now it's time to look at a pattern. The pattern is a six-fold pattern. As shown below, the pattern is made up of hexagons that are related in size by the square root of three. This is important, because this proportion is useful for verifying traditional six-fold Islamic geometric patterns.
After introducing you to the geometric progressions and dynamic rectangles, it should come as no surprise when I tell you that the pattern's unit of repetition can be either a triangle (equilateral triangle to be precise), or a root-three rectangle. When looking at the rectangular unit next to the triangular unit, it can at first glance be difficult to believe that when tessellated, both units will create the same pattern as shown on the right.
In both cases, despite the complexity of the underlying grids, each of these units of repetition, as well as the underlying grids, can be methodically drawn from scratch using nothing more than a pair of compasses and a straight edge.
Personally, I find the triangular version easier to draw once I have drawn the triangle progression. This is because not much needs to be added to the triangle progression to complete the pattern.
However, a triangular unit of repetition is not as practicable when it comes to tessellating the pattern over a surface. Tessellating rectangles are so much easier, probably because, for example, right-angled vertical and horizontal lines are easier to set out across a wall, compared to setting out a triangular net with 60˚angles.
Also, when drawing on paper, a rectangle is better at maximising the available space. This would have been an important consideration when resources like paper were scarce. Maximising the unit on the surface of a piece of paper also makes best use of the paper and reduces errors by improving precision. Drawing at a larger scale is easier than drawing at a smaller scale, e.g., within a circle or a triangle.
P.S. I don't know if there is a historic version of this pattern in existence. My focus these days is exploring the geometry behind patterns; not so much where the pattern may be found. If you have seen this pattern on a building somewhere, please let me know.
Categories: : Design Graphics, Dynamic Symmetry, Geometric Progression, Geometry, Golden Ratio, Islamic Geometry, Surface Pattern Design