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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