Moroccan Geometry: S6 Class: 8-fold with ball (Mtemen bil Kora)--single pattern and tessellation

 Adv S6 class Mtemen Bil Kora (8-fold with ball) (متمن بل كرة)

Notes on the basic pattern:

The pattern consists of a main circle that is then inscribed by a square. The square is then subdivided in quarters, and these quarters are divided into eighths. A circle is drawn in each quarter. These circles are already divided in eighths, and the divider is used to find the distance between sixteenths. One point of the divider is then placed at the point where the verticle of each quarter meets the top of the square (and the circle within the square), and the other point marks the radius of a smaller circle to be drawn from the center of each quarter. These four smaller circles should also just touch the large circle along its circumference, at 45°, 135°, 225°, and 315°, or π/4, 3π/4, 5π/4, and 7π/4. These smaller circles are then divided into sixteenths (using dividers), and the points that mark the sixteenths where they flank each of the previously drawn vertical, horizontal, and diagonal axes are then connected. Each of the medium circles then inscribes an octagon, created in steps E2-E4. To tessellate the design, the squares are simply arranged on a grid pattern.

Construction

A: Draw a large circle and inscribe it within a square

1. Set a compasss to make a circle nearly to the edge of the page.
2. Draw the horizontal axis through it to half the circle
3. Draw the vertical axis. With a compass point at 3:00 and 9:00, draw crossed arcs above and below the circle and draw a vertical through these intersections to quarter the circle.
4. Using the same radius compass as in step 1, from 12h, 3h, 6h, and 9h, draw crossed arcs outside the circle, and use the intersections of those crossed arcs to draw a square inscribing the quartered circle from step 1. Because the horizontal and vertical axes have already been defined for the circle, the inscribing square is already divided into 4 squares.

B. Subdivide the 4 squares 

1. In each of the 4 squares, draw diagonals in each direction for a total of 6 lines, a short, long, and short diagonal from SW to NE, and the same from NW to SE. Each square is now divided into 4 triangles

2. Now draw horizontal and vertical axes through the centers of the squares to divide each square into 8 equal right triangles.

C. Draw circles within the squares 

1. Draw circles in each of the 4 squares to the edge of each square. Because of the existing construction lines, these circles are already divided into 8ths.

2. Use the dividers to find the midpoint of each wedge so as to divide each circle into 16ths, although you needn't divide the circle at this point (alternatively, you can use the arch method with a compass).

D. Draw smaller circles within the four larger ones

1. Use the dividers as set in the previous step, placing one point at the top of the smaller circle to a point inside along the vertical, and then set another compass up with that radius to draw a circle within each circle. The new circle should touch the large circle from step one where the diagonal intercects.

2. Divide the new smaller circles into 16ths by setting your ruler at the center and the 16th mark on the larger circle that inscribes it. Double check with dividers, and just mark the 16th marks on each smaller circle.

E. Now draw the construction lines:

1. Draw pairs of parallel lines through each small circle or pair of small circles. These lines will cross through the 16th marks created in step 10. Draw the lines, 2 per circle, parallel to the vertical, horizontal, and each diagonal axis, so that you are connecting the 16th marks which are closest to each of the main axes (not the outer ones). This will create seals of solomon at the center of each of the 4 circles.

2. Draw 2 intersecting lines at the outside corner (where the diagonal runs through it) of each of the larger circles from step C1 (for a total of 8 lines), making sure that each line passes through the outside points of the horizontal and vertical lines from step B2, where these lines touch the C1 circle.

3. Draw another 2 lines intersecting connecting the adjacent points of the lines from the previous step, but continue the line segment only to the adjacent circle, where it should touch the parallel lines from step E1. There will be a total of 8 lines in this step.

4. Draw another 2 lines again from each circle for a total of 8 lines, now extending the lines to the opposite circles. This will create an octagon within each circle, connecting the main axis points from steps B1 and B2.

Here are the completed construction lines with the frame of the finished pattern highlighted:

Here is the highlighted pattern, which ends at the edge of the frame

The pattern with channels, created by drawing small circles at different point on each line

The completed class pattern, inked, and with seals of Solomon at the corners

The same pattern, but with reversed seals in corners. Note the reversed seals are a combination of the Solomon Seal with Looza:

The tesselated pattern created repeating the main square (or large circles) on a grid.

The tesselated pattern created repeating the main square (or large circles) on a grid.

The homework pattern is simply the class pattern shifted along the diagonal, such that the Solomon seal, at first in the corner, is now at the center of the design. The ball, formerly at the center of the pattern, has been changed into sfets at the corners of this pattern (inked in gold).