Once you’ve figured out a few geometric basics, you’re ready to learn all about pathways: those lines that go over and under each other to create a three-dimensional effect.

Drawing pathways is a labor of love, but – to me – worth the extra work when you’re playing with a design and want to emphasize the lines rather than the shapes.

# Islamic Geometry: A Rosette Tile from The Alhambra

A friend recently shared a picture with me from a visit they made to The Alhambra in Granada, Spain, a few years ago.

After playing with it for a few hours, I fell in love with this deceptively-simple rosette.

# Islamic Geometry: Five-Fold Model

You are no doubt an excellent constructor of four- and six-folded bases by now. It’s time to move on to something just a bit trickier: five-fold patterns. Constructing a five-fold base can be infuriating, but with a little practice, you’ll figure it out in no time. Don’t

You’ll notice that the instructions here are repetitive up until a point; this is so that anyone can start on this page without having to reference anything else.

1. Like any other geometric drawing, start by making a horizon line. Set your compass to any radius and keep it set at this width for now. Now place your compass anywhere along the line and draw a full circle. This circle is often called the “mother circle” since it gives birth to the rest of the forms in the drawing. You’ll notice that the horizon line divides the mother circle perfectly into two halves.

2. Place your compass on the easternmost point of the mother circle – where it intersects with the horizon line – and draw another full circle.

3. You won’t be drawing any more circles the size of the mother circle, so it’s time to change your compass radius. Adjust it to any size larger than ¾ the diameter of your mother circle. Place your compass on the easternmost point of the mother circle – where it intersects with the horizon line – and draw another full circle. Do the same on the westernmost point using the same compass radius.

4. As long as both the circles you made in step 3 are the same size – no matter how big or small they are – they will intersect at two points on the vertical bisector of the mother circle. Draw in the vertical bisector.

5. Draw in another vertical line that connects the points where the mother circle and east circle intersect.

6. Set your compass radius to the width that spans from the northernmost point of the mother circle, to the point where the blue vertical that you just drew intersects with the horizon line.

7. Place your compass on the point where the blue vertical intersects with the horizon line. Draw a full circle. Once you get good at this, you’ll only need an arc, but for now it’s helpful to see the full drawing.

8. Set your compass radius to the width that spans from the northernmost point of the mother circle, to the point where the purple circle that you just drew intersects with the horizon line.

9. I drew a circle at the point where the purple circle intersects with the horizon line just to illustrate the size of this circle compared to the mother circle. You don’t need to draw one. This compass radius will allow you to divide the mother circle into five equal parts.

10. Place your compass at the northernmost point of the mother circle. Draw a full circle. Move on to a point where the circle you just draw intersects with the mother circle. Draw a full circle. Continue on around the mother circle, drawing three more full circles. This is the part where you’ll find out if your measurements were precise enough and is often the most frustrating part. Don’t fret! Adjust your compass a bit if you need to until your five circles intersect perfectly around the edge of the mother circle.

11. Connect the five “petals” that the overlapping circles form inside the mother circle. This will make a pentagon. This kind of pentagon is called “static” since it since it has a base to sit on.

12. Alternatively, you can start by placing your compass at the southernmost point of the mother circle.

13. If you connect the petals in this arrangement, you’ll end up with another pentagon. This kind of pentagon is called “dynamic” since it looks like it could topple over at any moment.

14. Regardless of whether you started at the top or bottom of the mother circle, draw in the five mirror lines by going through the petals and the points of the star at the center of the mother circle.

15. If you overlap the static and dynamic pentagon, you make a ten-pointed star. Another way to think of this is that you connected each of the ten points by skipping every other point.

16. If you connect each of the ten points without skipping any points, you get a decagon.

17. If you connect each of the ten points by skipping two points, you get a star made out of one continuous line.

# Islamic Geometry: Four-Fold Model

Now that you’ve learned how to make a six-fold model, you’re just a few steps away from turning your mother circle into a four-fold foundation. Four-fold patterns often contain shapes like squares, diamonds, octagons, and wonky forms with curvy lines.

You’ll notice that the instructions here are repetitive up until a point; this is so that anyone can start on this page without having to reference anything else.

1. Like any other geometric drawing, start by making a horizon line. Set your compass to any radius and keep it set at that width for the rest of the drawing. Now place your compass anywhere along the line and draw a full circle. This circle is often called the “mother circle” since it gives birth to the rest of the forms in the drawing. You’ll notice that the horizon line divides the mother circle perfectly into two halves.

2. Place your compass on the westernmost point of the mother circle – where it intersects with the horizon line – and draw another full circle. Do the same on the easternmost point. The east and west circle kiss each other right at the center of the mother circle.

3. Now, place your compass at any of the four points where the east and west circles intersect with the mother circle. Draw a full circle. Do the same at the other three points. You now have a rosette pattern with arcs that divide the mother circle into six equal parts.

4. Connect the large north and south “petals” that grow out of the mother circle. You now have a vertical bisector.

5. Place your compass at one of the two points where the vertical line crosses mother circle and draw a circle. Do the same at the other point. Like the east and west circles that you draw back in step 2, these circle touch at the center of the mother circle.

6. Connect the points where the vertical and horizontal bisectors cross the mother circle. This will make a square. This kind of square is called “dynamic” since it looks like it could topple over at any moment.

7. Find one of the four points where the north/south and east/west circles intersect. Now find the opposite – the one diagonally across the circle. Connect these points to create a diagonal bisector. Do the same on the other side to create an X.

8. Connect the points where the diagonal bisectors cross the mother circle. This will make a square. This kind of square is called “static” since it has a base to sit on.

9. You can also create a larger static square outside the mother circle by connecting the points you used in step 7 to make the diagonal bisector.

10. If you overlap the smaller static and dynamic square, you make an eight-pointed star. Another way to think of this is that you connected each of the eight points by skipping every other point.

11. If you connect each of the eight points without skipping any points, you get an octagon.

12. If you connect each of the eight points by skipping two points, you get a star made out of one continuous line.

# Islamic Geometry: Six-Fold Model

If you’ve liked diving into Islamic geometric patterns, it’s helpful to take a step back and look at the underlying armature before going further. Today, we’re going to talk about how to construct six-fold divisions of a circle. It’s called six-fold because it has six mirror lines. To put it another way, if drawn on paper, this pattern could be folded six different ways to create symmetrical halves.

Six-fold patterns will often contain triangles, hexagons, and dodecagons, since 3, 6, and 12 are divisions and multiples of 6.

1. Like any other geometric drawing, start by making a horizon line. Set your compass to any radius and keep it set at that width for the rest of the drawing. Now place your compass anywhere along the line and draw a full circle. This circle is often called the “mother circle” since it gives birth to the rest of the forms in the drawing. You’ll notice that the horizon line divides the mother circle perfectly into two halves.

2. Place your compass on the westernmost point of the mother circle – where it intersects with the horizon line – and draw another full circle. Do the same on the easternmost point. The east and west circle kiss each other right at the center of the mother circle.

3. Now place your compass at any of the four points where the east and west circles intersect with the mother circle. Draw a full circle. Do the same at the other three points. You now have a rosette pattern with arcs that divide the mother circle into six equal parts.

4. Connect the six points of the six small “petals” formed by the intersecting circles inside the mother circle. You’ve made a hexagon. This kind of hexagon is called “static” since it has a base to sit on.

5. Draw two lines connecting the opposite points of the hexagon (the horizontal line is already there). You now have three mirror lines.

6. Alternatively, draw three lines through the six large petals outside the mother circle. You’ve made one vertical and two diagonal mirror lines.

7. Together, these mirror lines form the six folds of the mother circle. We can number them like a clock for easy reference.

8. Place your compass at one of the two points where the vertical line crosses mother circle and draw a circle. Do the same at the other point. You’ll notice that these circles cross the mother circle at points you’ve already found: 2, 4, 6, and 10. This kind of multiplicity of ways to get to the same information happens all the time in Islamic geometry.

9. Draw another hexagon connecting the even points on the clock. This kind of hexagon is called “dynamic” since it looks like it could topple over at any moment. Notice that when you formed both the static and dynamic hexagons, you went around the mother circle, skipping every other point.

10. From here, the mother circle is your oyster. You can draw an equilateral triangle… or two to create a “seal of Solomon” or “star of David”… or four to make a star. These triangles are made by going around the circle and skipping every three points.

11. You can also draw a square… or three to create a different type of star. The squares are made by going around the circle and skipping every two points.

12. If you connect each of the twelve points by skipping every four points, you get a star made out of one continuous line.

# Islamic Geometry: A Window from Ibn Tulun Masjid

Since we started with the first geometric pattern that I learned, I thought we’d jump right on over to the most recently one I’ve draw. This pattern is from a window grille in the Mosque of Ibn Tulun, the oldest mosque in Egypt and the oldest surviving mosque in Africa. We drew this with William Charles Riding and the Prince Foundation School of Traditional Arts.

It’s also another a six-fold pattern with a central star. This time, though, the angles in the star aren’t 60º, so it requires a few extra lines.