Moroccan Geometry: S3 HW: 12-fold bald pattern, dynamically tessellated

12-fold bald (Tanashri Qraa) -- تناشري قرع

For the basic pattern from class. Tanashri Kraa. 12-fold Bald. The Kraa is the corner shape in the class pattern.

Dynamically Tesselated pattern, turned 45 degrees:

Class pattern:

1. Find the center of the page, and draw a circle the circumference of which will be the height and width of the square pattern. Divide this circle into 4ths by first drawing the horizontal, then find the vertical using the arc methods with the compass point at 3:00 and 9:00.
2. Using the original radius with the compass point at the 3, 6, 9, and 12:00 points, draw intersecting arcs, and then connect these to form a square which inscribes the circle, touching the circle at those 4 points.
3. Using the original radius again with the compass point at the same locations, mark where the arcs cross the circle’s edge. (steps 2 and 3 can be combine into a single step. You may label the points 1 to 12 as if they were the numbers on a clock.

4. Connect points 2-4, 10-8, 11-1, 7-5, then 12-2, 12-10, 3-5, 3-1, 6-8, 6-4, 9-11, 9-7.

                    5. Now connect the opposite points as if cutting a pizza, 1-7, 2-8, 4-10, 5-11. No line needs to extend beyond the base of the triangle whose apex is at the numbered mark on the circle.

                    6. Mark the intersection of the lines running between 12-2 & 1-11 with a small circle. Do the same at the intersection between the lines 6-8 and 7-5. Lining a straight edge along those 2 points, mark where the ruler crosses the circle between 12 & 1. This can also be done with dividers, but this marks the division into 24ths.

       7. Set the compass on 1 and adjust the radius to the point you just marked on the circle, then swing the compass to make a mark along the line connecting the center to 1. This will mark the edge of the inner circle, which will circumscribe a 12-pointed star in the middle. Set another compass radius from the center to this point and draw the inner circle.

 

        8. Now draw the parallel lines that define the outside edges of 12 pairs of adjacent loozat. Do this by connecting points on the inner circle 11’ to 6’ and 12’ to 5’, then 12’ to 7’ and 1’ to 6’, then 8’ to 1’ and 2’ to 7’, then 9’ to 2’ and 3’ to 8’, then 10’ to 3’ and 4’ to 9’, then 11’ to 4’ and 5’ to 10’. Notice that you are skipping 5 points on the clock with each connection. For each of the lines marked in greeen, extend them beyond the arrows (kandeel shapes) and beyond the edge of the square that inscribes the outer circle, to the 2-4/5-7/8-10/11-1 line segments of the adjacent squares.

            9. From the previous step, follow the lines 8’ to 1’ and 2’ to 7’ to the upper right corner of the square, where they meet the edge of the square. Set the compass point at the corner, and the radius of the compass to the point where either line intersects, then draw a quarter circle (or a full circle if tessellating). Repeat this at the 3 other corners. 

            10. With a straight edge at the corners, mark where the straight-edge intersects each arc (or circle).

            11. Connect the endpoints of each of the small arcs where they intersect the edge of the square.

        12. Then connect these intersections by horizontal and vertical lines to their adjacent neighbors. The line segments need only connect to the lines drawn in the previous step.

The construction lines are now complete. Now highlight the pattern.

The names of the shapes are as follows:

1 - Khatem tnashri (12-fold seal) or Trenja (bitter orange)

2 - looza tnashri (12-fold almond)

3 - Sfet tnashri 

4 - Kandeel tnashri

5 - Sfet lherba (spear)

6 - Sfet sgheer 

7 - Lakraa (laqraa) (void, empty space) (The Bald) only occurs in 12-fold patterns

8 - Khatem slimaniya Ken (quarter seal of solomon)

When tessellating  the dynamic pattern, the pattern is turned 45 degrees, and we have a pattern of circles (which are circumscribed by the squares we draw around them), and the starting point is essentially the same as the tessellation in the S2 HW, as series of 5 circles (expand as required by the tessellation) arrange like the face of a die.

subdivide each of these circles into 12ths.