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.

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