# 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.

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.

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.

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.

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.

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

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).

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).

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.

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.

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.)

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).

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.

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”.

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.

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

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.

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).

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).

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!

# 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

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.

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).

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.

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.

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.

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.

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

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.

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.

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.

# Octagons in Baku

Besides the window in the Divankhana, I saw a lot of geometric design in Baku based around octagons.

For example, consider this pattern in a window in the external courtyard wall at Baku’s Taza Pir mosque.

This beautiful pattern, with its eight-pointed stars set within octagons, turns up on plate 67 in Jules Bourgoin’s 1879 Les Éléments de L’Art Arabe (which you can download from archive.org).

It’s wallpaper group is the fairly common *442 (p4m) and it is generated by tessellating a square cell.

Construction of this pattern is straightforward. The eight-pointed star in the centre is inscribed in a circle whose radius is one quarter the side of the square. The vertices of the octagon are found by extending the sides of the star. The rest of the construction lines are extensions of the octagon sides, and lines connecting star dimples that are three apart.

But one enters the Taza Pir compound via a stairway from the street. The panels in the stairwell are related, but subtly different from the window design!

What did they do here?  There is the same eight-pointed star in the centre, and the same enclosing octagon, but in this case they’ve trimmed back,  to the borders of the octagon, the square that one repeats.

As a result, the square tile borders stand out strongly as lines, and around each point where four tiles meet we get a big diamond holding four small diamonds.

(This also belongs to the *442 wallpaper group.)

Now, in the Old City, I came across a piece of octagon-based decoration that illustrates what happens if one doesn’t follow best practices, as explained by Eric Broug.  This pattern involves starting with the same pattern as in the Taza Pir windows (above; a.k.a. Bourgoin’s Plate 67), but then repeating a somewhat random subset of it. In other words, incorrect tessellation.

It would appear that the manufacturer of these pre-cast concrete blocks selected a piece out of the overall pattern that was not the all-important basic square, but rather a rectangle.

Hence each of the concrete blocks looks like this:

When you put them together, lines match up, but the effect of the original design is lost.

The wallpaper group of this pattern would be*2222 (pmm).

Elsewhere in the Old City, there were pre-cast patterns that did tessellate pleasingly, again with octagons.

But back at the Taza Pir mosque, I spotted this on an adjacent building, which I believe is Baku Islamic University:

The grill pattern is octagons packed together, with squares in between; and eight radial lines emanating from the centre of each octagon. It’s basically the central column of this pattern:

But look at what they did in the point of the arch. It’s beyond my knowledge to know whether this is best practice or not, but it is definitely creative.

# A window in the Divankhana, Baku

Eric Broug, from the School of Islamic Geometric Design, writes that sometimes while travelling he sees a piece of contemporary Islamic geometric design and recognizes it as, well, let’s say less than best practice. (He sometimes posts images of this sort of thing on Instagram under the hashtag #cpigd, which stands for Common Problems in Islamic Geometric Design.)

What sort of mistakes are they? He explains on his page on best practices, but to me the most common one is incorrect tessellation, where a block of pattern is repeated in ways that cause lines to abruptly stop instead of continuing on.

I thought it would be interesting to take a look at the designs I found in Azerbaijan, in the cases where they were identifiably Islamic, and ask the same question: are they examples of good, traditional design.

So, bearing that in mind, let’s look at a grill window I found in the Divankhana of the Palace of the Shirvanshahs in Baku, Azerbaijan.

The Palace of the Shirvanshahs is the premier piece of historical architecture in the Old City of Baku, or, as it’s called in Azerbaijani, İçərişəhər or Icheri Sheher. The original buildings have been deduced to date from the 15th century, but most of the palace was heavily renovated/restored in the 20th century so it’s not immediately clear whether the details one sees are original or the work of a restorer.

The Divankhana (which is also variously called the Divan-Khane or Divanhane) is a structure in its own courtyard just off the outer courtyard of the palace. It holds a pleasing octagonal pavilion. whose original function is unknown (there are many theories). The pavilion is domed and consequently two stories in height, so it stands above the courtyard wall and can be seen from the palace courtyard. In one corner of the Divankhana, there is a staircase leading up to a locked door on the upper storey of the pavilion.

This window is at the top of that staircase, and looks out into the palace’s outer courtyard. Here it is seen from inside.

What I thought when I saw this is There’s no way this can be a best practices design. There were so many wacky elements that I had never seen in an Islamic geometric design before. For one, I couldn’t find a single axis of reflection in it, anywhere. For another, it contained a number of strange, three-way intersections.

Is it a bad design, perhaps a modern artist not working within traditional lines, or could this be authentic traditional design?

Let’s look at how the pattern works.

The whole pattern begins with a pair of adjacent large octagons.

These fill the window, as shown.

Centred on the vertices of each octagon are eight smaller octagons, sized so that when they overlap they bisect each other’s sides.

These circles of smaller octagons define an empty space in the centre of each of the larger octagons, a space which is an eight-pointed star.

So far, so good, and very symmetrical and, in fact, infinitely tile-able. The cleverness comes with what they did inside each 8-pointed star.

They divided this space with a four-armed pattern which has rotational symmetry but no mirror symmetry.

They ran it in opposite directions in the top and bottom halves of the window. So, looking from inside the window, there is a counter-clockwise star in the top half and a clockwise star in the bottom half.

This is fairly mind-boggling for traditional Islamic design — I think. (I’m no expert.) The little pattern of four “hammerhead” shapes that circles within each 8-pointed star looks more M. C. Escher than standard geometric design.

Mathematically, we might ask: does this pattern at least pass the test of being able to be  continued in all directions?

Well, yes. One would simply add more big octagons above, below and to the sides, and then add the smaller octagons, etc. This could go on forever.

Each big octagon hosts an eight-pointed star in its centre. If you alternate big octagons that host clockwise stars with big octagons that host counter-clockwise stars, following a checkerboard-like pattern, the centre of each star would be a four-fold centre of rotation. The corners where four octagon tiles come together are two-fold centres of rotation. And there are axes of reflection along the lines where adjacent big octagons touch.

Patterns with these axes of reflection and pattern of rotational centres belong (mathematically) to the “wallpaper group” known in orbifold notation as 4*2, and in IUC notation as p4g. (There are seventeen possible wallpaper groups.) Patterns that are 4*2 have a “twist” to them so that the basic square unit of the pattern is not the same as its mirror image.

In fact, the fundamental tile for this pattern, a tile from which the entire pattern can be generated through reflection and rotation, is triangular.

4*2 is an unusual wallpaper group for an Islamic geometric design, most of which are *442 (p4m) or *632 (p6m). But it’s not unheard of, and would not lead us to conclude that this pattern doesn’t use best practices.

And, then it gets a little more complicated.

At the Palace of the Shirvanshahs I never shot a picture of the outside of the window, so when I got home I found that, while there’s no Google Street view in Azerbaijan, someone had conveniently taken a photosphere in the main courtyard of the palace four months before I was there.

Here, in the photosphere, is the facade of the Divankhana as seen from the outer courtyard. The Divankhana is the building with the whitish dome.

Here’s a close-up of the three window openings on its second floor:

Only the central one is full-sized; the outer ones are reduced in height, and the left one is completely blocked. I was inside the rightmost window, and in this image we see it as the reverse of my original picture.

But look at the centre window.

The stars in this window turn the same way. They’re both counter-clockwise (as seen from outside).

This pattern group, incidentally, would be 442 (orbifold) or p4 (IUC); it has no axes of reflection and two kinds of centres of four-fold rotation. I have not seen an Islamic geometric design before in the 442 wallpaper group.

So the windows are not the same. Prompting some contemplation.

This reminded me of a #cpigd instagram posting by Eric Broug at https://www.instagram.com/p/Biq9_d3la-Y/?tagged=cpigd

Design problems in Cairo. *All* the restored, replaced geometrical windows on *all* the restored buildings in this street in Darb al-Ahmar are identical. Originally, there would have been a great diversity of design.

This post prompted some pretty hot discussion in the comments about whether window patterns in a single building were traditionally uniform or not. Broug was of the opinion that

conventionally in Islamic Architecture, different patterns would be next to each other. possibly to encourage contemplation and reflection by looking for differences and similarities

At the Divankhana, the blocked and un-restored left-hand window suggests that this second storey of the Divankhana has not been aggressively restored, and that we’re not looking at recently restored windows here.

And I wonder if this subtle difference between the centre and right windows was an example of encouraging contemplation and reflection (no pun intended). It certainly got me contemplating.

# A Little Geometric Creativity

There’s a nice old geometric pattern…

…that Eric Broug presents in his book Islamic Geometric Patterns as being from the Great Mosque of Herat.

Its construction is based on a hexagon, and the pattern repeats as additional hexagons are tiled around the first one. The underlying hexagons (which are not drawn in the finished pattern) are shown in red here:

This makes a fine star/wedge/triangle pattern that is quite satisfying to look at, and suitable for decorating things in your house.

The construction of one of the hexagon-based units of the pattern begins with drawing a circle, and then a hexagon within that circle. Once the hexagon is drawn, the lines of the pattern can be drawn within in it.

But in the real world, a medium on which we are drawing (or painting) always has edges, and those edges do not go along the edges of the hexagons. For example, let’s say we want to draw one of the pattern units, plus however much extra fits, on a square board or tile. Something like this:

Draw in the supporting hexagons and you will see that the one complete pattern is indeed surrounded by only parts of adjacent copies of the pattern.

It’s a bit of a construction conundrum. How will we fill in the portion of the pattern that lies outside the one central hexagon that we will have room to draw? How will we extend the pattern to the edges of the square when we cannot place a compass foot at the centre of any of the adjacent circles that would define the basic hexagons?

There is a way.

Let’s review how the pattern is made, and then examine how to extend it without being able to draw more circles. Follow along with your own piece of paper, pencil, compass and straightedge.

There is something magical in the fact that this pattern, with all of its exact proportions, can be constructed solely with a compass and a straightedge. You never need to measure an angle or a line.

To construct a hexagon, one begins by drawing a circle.

Within the circle inscribe a hexagon using the standard method of walking the legs of your compass (still set to the same radius) around the circle. We’ll call this Hexagon A. It is oriented so it is a point-up hexagon.

Draw radii through all the vertices of the hexagon. We’ll call these the ribs of hexagon A.

Now we need to construct a second hexagon of the same size, and on the same centre, but rotated 30°. To do this we will connect the midpoints of the six arcs from the hexagon already made.

If we had plenty of room, we would find those midpoints by first drawing six more circles (of the same radius) centred on the six vertices of hexagon A; and then draw lines from the centre of the original circle to the outer points of intersection of adjacent secondary circles. These ribs (purple, below) would show us the midpoints of the arcs.

But with the limited space we are working with, we need instead to bisect one side of the hexagon. This can be done with two compass arcs from adjacent vertices of hexagon A; we can then connect their intersection outside the circle with the circle’s centre.

This line bisects one of the arcs, and we can use that as a starting point for making a second hexagon. We’ll call this Hexagon B, and it is oriented so it is a side-up hexagon.

Draw ribs for hexagon B.

Note that the ribs of hexagon B (purple) go through the midpoints of the sides of hexagon A (red).

Connect the hexagon A midpoints to make a six-pointed star. We’ll call these star 1 lines (green in the next figure).

Similarly, we can note that the ribs of hexagon A pass through the midpoints of the sides of hexagon B. Connect the midpoints of hexagon B to make Star 2 — being sure, however, in this case to extend the star lines beyond hexagon B as far as the first edge of hexagon A you encounter. We’ll call these star 2 lines (blue in the next figure).

Finally, we ink certain portions of the Star 1 and Star 2 lines to make the final pattern. Note that the entire pattern lies within hexagon A.

The pattern that is inked consists only of portions of star 1 and star 2 lines. In fact we drew hexagons A and B, and their ribs, only to create the star 1 and 2 lines. And we drew the initial circle only to create hexagons A and B.

In order to extend the pattern we need to draw more star 1 and 2 lines outside our original hexagon. On a larger medium we could easily place the centres of new circles outside hexagon A, but on this limited surface where we are working we cannot.

It may help to look at what we need to find.

From our earlier construction of adjacent hexagonal cells, we know a few things about what should be happening. Star 1 lines leaving the hexagon (A in the diagram above) go a certain distance to a point Y (which we’re not really sure how to locate) and then turn through 120° to follow a line D.

Star 2 lines leaving the hexagon (B in the diagram above) go a certain distance to a point X (which we’re not really sure how to locate) where they cross a star 2 line (C) in the adjacent hexagons. At some farther point Z they also go through 120° turns.

As well, it’s valuable to make a few observations about what happens as these lines enter adjacent hexagons…

1. Hexagon A sides, when extended, become the ribs of other hexagons A, and vice versa.

2. Star 1 lines become some of the star 1 lines of adjacent hexagons.

3. Star 2 lines become some of the star 2 lines of adjacent hexagons.

This means that by extending these hexagon sides, ribs and star lines, we have much (but not all) of the construction information for the pattern outside the original hexagon.

What’s missing are lines C and D, and points X, Y and Z.

But now we can see that point X is the place where the extended hexagon A sides meet the extended star 2 lines. Drawing a line through them gives us line C.

Similarly, Point Y is the place where the extended hexagon A sides meet the extended star 1 lines. Drawing a line through pairs of points Y gives us line D.

At this point we have all the construction lines we need to ink the rest of the pattern outside the original hexagon…

…and then remove the construction lines.

It’s very satisfying to be able to construct this figure in a limited space, and to solve the problems associated. But now, as a bonus, it appear that the pattern presents us with a fascinating geometry problem!

As the pattern is extended, a new, larger hexagon has appeared, a hexagon that is similar to hexagon B, but is formed by star 1 lines that pass on the outside of the six small triangles. We’ll call it hexagon C. In the illustration below, hexagon B is purple, and hexagon C is blue.

It’s a bit of a puzzler, but I’ll just leave the problem here for the intrigued reader to solve. In terms of the side length of hexagon B, what is the side length of hexagon C?