We have worked with several grids so far, all of which shared a certain logic as, whether Islamic or Christian, they share a common origin in the Ancient world. Today we're getting a taste of a radically different kind of grid, mastered and widely used further north and west: grids based on the division of the circle, and tangential circles. These are the basis of all Celtic art, with its characteristic spirals, trumpets, teardrops and more.
Such grids can get very complicated, though they all develop along the same lines. A simple one is the triskele motif we drew in Working With Circles. The grid we're working on today is of intermediate complexity.
Draw a circle, the size you want your motif, on a horizontal line. Be warned that we're going to draw quite small circles at one point, so you may want to start large.
Draw the arc below, through the centre of the circle, and connect the points it marks on the circle. This bisects the radius.
Draw the circle that has for its diameter the radius of the large circle.
Keep the compass opening and place the dry point on A to find point C.
Join B and C to find D on the horizon line.
Open the compass to AD and draw an arc to mark two points on the half-size circle.
Join these points to find E.
Draw the circle of diameter EA, and then keep moving the compass point to the next point marked on the horizon line till you have a row of tangent circles fitting in the original circle (I recommend only drawing the tangent circles; the intermediate circles, dotted here, only serve to find the next point).
Notice we have nine circles. The purpose of steps 2-7 was to make this possible: E is located at one-ninth of the diameter of the half-size circle, which is one-eighteenth of the original circle, and the measurement we needed before we could find the nine circles we need in this design. For another method of thus dividing the diameter, see Dividing a Segment in our very first lesson, The Basics; that is also a good method to fit a different number of circles on the diameter.
The half-size circle is no longer needed and can be rubbed out at this point.
From here on, we are completing the grid. Observe carefully, because the sheer number of circles now may get confusing. A rule of thumb: in the following steps, the nine circles are never cut by other circles. All of the larger circles we're about to draw are tangent to them.
We start with circles that contain two small circles. Use the points between the small circles as centres, and draw the eight circles below.
Next are circles that contain three small circles. Put the compass point on the centres of the small circles, adjust the opening by half a circle, and draw the seven circles below. Ignore the outermost small circles, because we only want to draw circles that fit entirely in the large one. As the inner circles grow, more and more of the sides will be ignored, so start from the central area and work outward.
Next are circles that contain four small circles. Place the compass point between the small circles, adjust the opening, and draw six circles.
Next are circles that contain five small circles. Place the compass point on the centres of the small circles, adjust the opening, and draw five circles.
Next are circles that contain six small circles. Place the compass point between the small circles, adjust the opening, and draw four circles.
Next are circles that contain seven small circles. Place the compass point on the centres of the small circles, adjust the opening, and draw three circles.
Finally, there are only two circles that contain eight small circles. Place the compass point on either side of the central small circle, adjusting the opening, to draw them.
Now for a final row of circles, the smallest of all. We could have
started with them, but it was easier to work on a more manageable scale,
and return to these at the very end. Bisect the diameter of one of the small circles.
Now draw the circle that has for its diameter the radius of the small circle.
Fill all the small circles with these tiny circles, two of which fit in each.
The finished grid.
Here are a few of the shapes generated by this grid, by simply highlighting arcs and semi-circles:
There are of course more, and endless variations on the above; you can also break away from the classic Celtic motifs and find your own, as every intersection is a possible place to break a line.
One finished motif (found in Celtic Pattern by Adam Tetlow):
A possible coloured version:
Another possible motif, freely inspired:
With this we are diversifying the way we think of geometric patterns, and of grids, as well as discovering that Celtic art is not only about knots! The next lesson will involve circles again, as we return to the Islamic world...