Constructing Bourgoin’s Figure 171 – Part 2

Now that we know our way around the pattern (go back to Part 1), it should be fairly straightforward to construct with a compass and straightedge. But be aware: any pattern that requires you to construct a pentagon is an advanced challenge. They are trickier to make than squares or hexagons.

Here’s what we want to draw:

There are different scenarios for beginning. You might know where you want to site two rosette centres, and that will then determine the size of the master triangles and the rest of the layout. This is the scenario I’ll go through here. But, alternatively, you might want to scale the pattern so a certain number of rosettes will appear in the space you have; or you might have an exact size that you want the diameter of a rosette (or a pentagram) to be. Each of these is, in a sense, a different problem.

The first nine steps will take us from drawing one leg of a master triangle to having a compass set to the radius of a rosette circle.

Figure 1


1. Pick two points that will be adjacent rosette centres, and draw a line through them. We know that one of these will be at the apex of a master triangle, and the other will be one of the remaining corners (Figure 1). For the moment, we’ll call the triangle side that connects them the main axis.

Figure 2

2. Bisect the main axis between the rosette centres, and establish a midpoint (Figure 2). The midpoint will be useful later when we want to draw diagonals.

Figure 3

3. Create circles centred on the two rosette centres, each with a radius that takes it to the main axis midpoint. (Figure 3).

Note that you could actually draw these two circles with any radius. Our purpose in drawing them is simply to give us the ability to draw 20 divisions of a circle (i.e., at a spacing of 18°), and once we have those we won’t be using these circles any more. Drawing them to meet at the main axis midpoint has the advantage that these are large circles, which should make the 20-fold division more accurate.

Now we need to construct twenty evenly-spaced rays from each circle centre.

Figure 4

4. Construct a pentagon in one of the circles so that one vertex touches the axis midpoint (Figure 4). (You can find methods of constructing a pentagon within a circle at many places on the internet, including Wikipedia’s page on “Pentagon.”)

In the process of doing this, your compass will become set to the length of a pentagon side.

Figure 5

5. Without changing the span of the compass, use it to draw a pentagon in the other circle (Figure 5).

Figure 6

6. Continue using the same span to draw a second pentagon in each circle, with one vertex touching the place where the axis exits the circle. You now have a 10 pointed star, or 10/2 star, in each circle (Figure 6).

Figure 7

7. Divide each circle in 20 sections by drawing lines from the rosette centre through every point of the 10/2 star, and through each of its 10 dimples. You now have a line meeting the circle every 18°. Be sure to extend, outside the circles, the first rays adjacent to the main axis until they intersect (Figure 7).

Figure 8

8. Create a circle centred on this intersection, using a radius that will take it exactly to the main axis midpoint (Figure 8). This is the pentagram radius.

Figure 9

9. Note the point where the first 18° ray from one of your circles enters the pentagram circle (Figure 9). The radius of the rosette circle is the distance from the rosette centre to here. Draw a circle with this radius around each rosette centre.

You can even erase the initial circles you drew.

We’re now ready to extend the grid of master triangles, to locate other rosette centres and to put rosette circles around them. This occurs in the next three steps.

Figure 10

10. Using the appropriate rays from the two rosettes you’ve already sited, extend the grid of master triangles (Figure 10). Draw a rosette circle around each vertex, and construct the twenty evenly-spaced rays. Remember, you already know the rosette circle radius, and ray spacing can be copied from one of the other rosette circles. (E.g., place one leg of your compass on the point where one ray leaves the circle, and the other leg where the fourth next ray leaves the circle. Use this distance on a new circle to set up rays.)

Figure 11

11. Each pair of rosette centres allow you to construct a third. In my case, I have room for four, and all other possible centres are off my page (Figure 11).

Figure 12

12. You already know the distance from a rosette centre to the midpoints of the master triangle legs, so set your compass to that and add in leg midpoints. You can then add the diagonals that connect the midpoints (Figure 12). For triangles with missing vertices, you can still place a midpoint on a leg from the nearest rosette centre. Notice that even though you’ve never figured out where the midpoint of a master triangle base is, the intersecting diagonals will lead you to it.

Rosettes in place, it’s time to construct 10/4 stars in them, and extend the lines from these stars. This takes place in the next four steps.

Figure 13

13. Although you have twenty rays from each rosette centre, only ten of them are important from now on. These are shown above, by circling their intersections with the rosette circle (Figure 13).

14. Connect each to the vertex four along (Figure 14). This is the “10/4 star”.

Figure 15

15. You want to extend some of the 10/4 star lines outside the rosette circle. How far to extend each line is a bit weird. It’s okay if you extend a line too far, because you will wind up erasing a lot of construction lines anyway. But ideally, it looks like this (Figure 15, above).

Lines at 12:00 and 6 o’clock (a) go until they hit the next master triangle base. Lines going out parallel to master triangle legs (b: at 1:00, 5:00, 7:00 and 11:00) go out only as far as a midpoint bisector of that leg—at which point they meet identical lines coming from the next rosette. Lines that cross at 3:00 and 9:00 (c) go out just a short way: as far as the vertical line coming up or down from an adjacent rosette. But their opposite ends (d) go a long way: all the way to the midpoint of the next master triangle leg they encounter.

Figure 16

16. Having extended the 10/4 stars in each rosette you’ll have something like this (Figure 16).

Figure 17

17. As always, the final pattern is made by selecting some of the construction lines for inking, and the rest for erasure. We can get partial success with the construction lines we have so far (Figure 17).

In a virtual space, you can just go on creating master triangles and rosette centres as far as you like. But in the real world, you come to the edge of the page, or wall, and there are still areas in the corners where the adjacent rosette centres are off-page, and you do not have the lines coming out of them to guide you. This is where it gets doubly interesting, as the physical limitations of the space in which you are working create additional geometric problems.

The remainder of the process now is just working out how we can extend the pattern into these spaces.

This process will be different for every space. I’ll show how I completed this for the rectangular space I’m working in.

Figure 18

18. Drawing additional pentagram circles is quite handy. Their centres are a known distance outside the rosette rays, and their radius is known from way back in step 8. We can even locate those in the far corners because their centres lie on a line passing through other pentagram circle centres, and the spacing between centres can be measured with the compass elsewhere in the pattern (Figure 18).

Figure 19

19. Each of these circles can have a pentagram inscribed in it—we know the spacing between the vertices from other, existing pentagrams—and then we can extend the sides of that pentagram to form the guidelines that we need (Figure 19).

Figure 20

20. Some more inking and erasing, and we’re almost done. There are only four small areas near the corners (marked with question marks in Figure 20) yet to be finished. We know what should be there, but we don’t yet have the construction lines. At this point I’m inclined to use the compass to measure spaces and lengths out of completed parts of the pattern and sketch/copy them into the areas that need to be filled.

Whew, done!

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Constructing Bourgoin’s Figure 171 – Part 1

Just veering off into geometry here….

In November I was watching Eric Broug, an Islamic geometric design guru, give a talk online at an Islamic art conference, and I noticed that behind him they were projecting an interesting pattern on the scrim. I froze the video and grabbed a screenshot…

What the heck is this? How do I look this pattern up? How do I find what pattern this is, and how to draw it?

I have a few books of geometric patterns, but this was not in Eric Broug’s book Islamic Geometric Patterns, nor in Daud Sutton’s Islamic Design. So I took the tack of searching Jules Bourgoin, the 19th century Frenchman who catalogued Islamic geometric patterns, and whose 1879 book, Les Eléménts de l’Art Arabe, is available for free on the internet. Bourgoin’s book provides that useful function, like Köchel numbers for the works of Mozart, of giving us a handy identifier, albeit a random number, for many patterns.

After some swimming back and forth in a sea of patterns, I finally recognized it as his Figure 171.

Bourgoin gave some enigmatic and tense (and French) instructions about how to construct this pattern, and I couldn’t make head or tails of them. So next, a web search on “Bourgoin Figure 171.”

This did not yield instructions about how to construct the pattern, but it opened a number of satisfying rabbit holes. One was a talk by Lars Erickson at the 2021 Bridges conference, who was constructing this pattern (among others) using an extended girih tile site. He gave several references about this pattern, including a page at http://tilingsearch.org dedicated to it. Here I could see that it is fairly common, turning up in the great mosque in Damascus, the Kalyan mosque in Bukhara, the tomb of the Mughal emperor Akbar in Agra, India, and even on the spine of one of my favourite cookbooks, Taste of Persia by Naomi Duguide.

A video posted by Samira Mian was also helpful, even though it was not about this exact pattern. It clarified that the pattern grows out of five-fold symmetry, and that subdividing the circle into 20 equal portions is key.

After long period of staring at the pattern, I think I can do an analysis. We’ll do construction in part 2.

Analysis

Figure A

Here is the pattern, both as a line drawing and as a tiled pattern (Figure A). Believe it or not, you can construct this with a compass and a straightedge. No measuring of lengths or angles required.

Figure B

The first thing to notice is that we have a regular, repeating pattern of rosette centres with 10-fold symmetry. Each rosette has a 10-pointed star in its centre (yellow, in Figure B) surrounded by ten points (orange).

Although it initially might appear that the rosette centres are placed at the vertices of equilateral triangles, they are in fact, as Bourgoin notes, at the vertices of isosceles triangles whose angles are 72°-54°-54°. (Bourgoin’s text about Figure 171 says, “Plan isocèle ou losange. Le triangle isocèle a son angle de base égal aux 3/5 d’un droit.” Isosceles or diamond plan. The isosceles triangle has a base angle equal to 3/5 of a right angle. I.e., 54°.)

These isosceles master triangles alternate, apex up, apex down. Be sure to recognize which sides are the legs of the isosceles triangles (equal in length, meeting at the apex), as opposed to the third side which is the base (joins the two 54° angles, and is slightly longer). Notice that the head-to-head kites (red) always occur on the base, whereas the legs run through two opposite-facing petals (green).

Figure C

In addition to the 54° and the 72° angles in the master triangles, the pattern is full of angles with measures like 18°, 36°, 108° and 144° (Figure C). These all reinforce the impression that the pattern will be constructed from pentagons and 5-fold symmetry. The full circle of 360°, divided by 5, 10 and 20, respectively, gives 72°, 36° and 18°. The angle 54° is, in turn, three 18’s, and 108° and 144° are doubles of 54° and 72°. So, all in a family. The “5” family.

Figure D

Each rosette centre features a 10/4 star, which gives us the rosette points (orange). I’m calling them “10/4 stars” after Magnus J. Weninnger, who used this kind of expression in his 1971 book Polyhedron Models. A “10/4” polygon is a star formed by connecting each decagon vertex to the fourth next vertex. A “5/2” polygon, by the same logic, is the star formed by connecting each vertex of a pentagon to the vertex two along: in other words, a pentagram.

As shown above in Figure D, the majority of construction lines for the pattern are extensions of the 10/4 stars. So once we draw these stars, we’re going to have most of the lines we will need to draw the pattern.

Figure E

These 10/4 stars themselves can be drawn if we can construct circles of the right radius around the rosette centres. It won’t do to use just any circle: the radius has to be just right so the extended lines from the 10/4 stars meet and form 5/2 stars, or pentagrams (purple, in Figure E above).

In fact, this is the key puzzle of drawing this pattern: the relationship between the length of a master triangle leg and the radius of the rosette circles.

Figure F

But, we don’t need to determine the radius of the rosette circles first! Instead we should first determine the radius of the circles enclosing the pentagrams, the pentagram circles.

If we construct 20 equally spaced rays coming from each rosette centre, the rays will be 18° apart (Figure F). These rays alternate in function: one coincides with a point of the 10/4 star (orange), and the next coincides with the centre of a “petal” (green) between two points. We can think of these as point rays and petal rays.

Notice that each pentagram occupies the space between two petal rays, and it does so simultaneously for two different rosettes. In other words, it occupies 36° of arc from the point of view of two different rosettes.

One petal ray lies along the master triangle leg connecting these two adjacent rosette centres (let’s call it the main axis). The next ray, a point ray, comes out at 18° from the main axis. Note where this ray intersects the corresponding ray from the other rosette. This location, halfway between two petal rays—and this is true looking from either rosette centre—is the centre of a pentagram circle.

The radius of the pentagram circle, and of all pentagram circles (r, in Figure F) must be the distance from that centre to the midpoint of the main axis.

Figure G

Once the pentagram circle is drawn, the radius for the inner circles (R, in Figure G) falls out. It is the distance from the rosette centre to where the first point ray meets the pentagram circle.

Now we see the the relationship between the length of a master triangle leg and the radius of the rosette circles. It’s governed by this 144°-18°-18° isosceles triangle, where the main axis is the base, and it involves subtracting the triangle’s height (r) from the length of one of its legs.

We can quantify this (it’s interesting, but not of practical value in constructing the pattern) and say that if the length of the main axis is 1, then

Figure H

If we look at the relationships between the pentagram circles and the inner circles, we can see they pack nicely. While most of our construction lines will be based off those 10/4 stars in the rosette circles, the pentagram circles may come in handy as we reach the edges of the space we’re working in, in places where we cannot draw a rosette circle because its centre is off the canvas, board, wall… or whatever medium we are working on.

Figure I

The kite pairs along the bases of the master triangles clue us in to a few construction lines which are not generated by the 10/4 stars at each rosette. These are diagonals which cut through the pattern. They form the edges of some pentagrams, and the “noses” of the kites.

These diagonals are actually a secondary grid, of the same spacing and direction as the master triangles. The symmetry of the features that lie along them tell us that they are based on connecting the midpoints of the legs of the master triangles.

Figure J

Finally, what wallpaper group does this pattern belong to? It has two axes of reflection (blue, in Figure J above), and three 180° rotational centres (red diamonds). So it’s cmm, also known as 2*22. The basic unit of repeat is shown above, a 36°-54° right triangle, or half of a master triangle.

It’s kind of unexpected to find that a pattern built on 5- and 10-fold symmetry has a repeat that is basically rectangular. Maybe I should have seen that coming, though, from the underlying pattern being isosceles triangles arranged in rows where they alternate apex-up, apex-down.

On to part 2, Construction.