Constructing grids with ruler and compass

In September 2018 I gave a presentation about patterns constructed using grids at the 4th International Workshop on Geometric Patterns in Islamic Art in Istanbul. For the presentation I prepared these timelapse videos demonstrating different methods for constructing grids using ruler and compass and I have posted them here to be of service to a wider audience.

Square and isometric (triangular) grids provide a rich and expansive foundation for the exploration of geometric patterns. They have arisen from the simplest regular divisions of the 2D plane. The triangle and the square are two of the three regular polygons which can tile a 2D plane by themselves. The hexagon is the third but can be seen as a combination of 6 triangles so equates directly with the isometric grid. These first two videos show the basic construction of the hexagon and square from the circle, fundamental constructions in the study of geometric art.

To explore the patterns we must first construct wider grids. It is insightul to draw these grids using the traditional geometric tools of ruler and compass to understand the role the circle (and therefore the compass) plays in their construction. Both grids have been widely explored as a foundation for pattern design in many cultures across the world and throughout history.

Isometric grid – Method 1

As we saw above, the circle effortlessly divides itself into six equal segments. This first method is based on a structure often referred to as the Flower of Life. Professor Keith Critchlow describes it as the Creation Diagram.

We start with a small circle at the centre and the pattern grows outwards from this starting point, theoretically indefinitely, with further circles centred on new intersections generated by previous circles. It is very satisfying to see this beautiful construction unfold as you draw it. However, from a practical point of view, inaccuracies can compound by starting small and working outwards. It is also tricky to generate a final grid of a specific size.

Isometric grid – Method 2

The second method starts with a large circle and the area within is subdivided (again, in theory indefinitely) into ever smaller triangles. Working from the outside inwards is more accurate and has the added advantage of allowing you to work within an original circle (and therefore hexagon) of fixed dimensions.

Square grid – Method 1

The square grid can be generated using two similar methods. Firstly starting with four circles around a central one and then expanding outwards by using intersections generated by each additional circle.

Square grid – Method 2

Again, starting with a large square and subdividing is more accurate. The interplay of static and dynamic (horizontal and diagonal) grids can lead to very interesting designs which explore the √2 proportional relationship between the length of the edge of a square and it’s diagonal.

Odd number divisions are also possible but that’s a story for another time.

Another method (not demonstrated here) would be to use a set square to construct either grid. The 30-60-90 set square is based on the angles of a triangle so can be used to construct an isometric grid and the 45-45-90 set square can be used for the square grid.

Once the grid has been constructed at the desired size, intersections of the grid can be used to design repeating patterns.

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