Bridges 2020 Supplemental Page
From Computer to Compass: Analysis and Reconstruction of a Self-Similar Islamic Geometric Pattern at Madrassa Madar-i-Shah
Welcome Bridges 2020 participants! And thank you for your deeper interest in my paper. As a reminder, here is the pattern in question:
Analyzing and reconstructing this pattern was a real adventure, and as I advocate in the paper, I think there is different-but-equal merit to doing computer-aided analysis and figuring out the construction using traditional tools.
One thing I had to omit from the paper due to length constraints was a pivotal construction to establish the rhombic framework for the whole design: the construction of a decagon in a given circle. This page will provide that construction, step by step, as well as explaining WHY it works (something many authors omit, but I think would be of interest to the Bridges community).
The “Yin Yang” Construction of a Decagon in a Circle
The construction outlined here was described to me as the “yin yang” decagon construction, because of the beautiful yin yang-like image that emerges early in the construction.
If you care to follow along — which I heartily encourage; doing a construction helps seal it in your memory far more than just reading it! — you will need the following equipment:
- A piece of paper, 8.5″ x 11″ or larger suggested
- A compass
- A straightedge
That’s it – which is kind of the whole point. 😉
Step 1: Create a circle with a horizontal diameter
NOTE: Almost every construction you’ll ever do will start this way. 🙂
- Arrange your paper in portrait orientation (longer side up and down).
- Find the approximate center of the paper, and draw a horizontal line through it. (You can measure if you’d like to, but anything reasonably close to the center and parallel to the bottom edge of the paper will do.)
- Place the point of your compass at the center of the page (point O), open it to a radius of 3 inches*, and draw the circle.
- Mark the points where the circle intersects the horizontal line A and B
* TIP: The exact radius doesn’t matter, of course. But I’m suggesting this particular radius for a reason that will become clear later. If you’re working on a different size paper, go for a radius that’s a tad less than 1/4 of the height of your paper.
Step 2: Construct a vertical line through the center
NOTE: This is a VERY common second step, and a good general skill to have.
- We need to create crossings above and below our horizontal line as guides for the vertical line. Normally we’d just widen our compass to some distance greater than the current radius to do this. However, I’m going to do this in a couple of steps using the current radius for two reasons:
- The fewer times we adjust our compass, the greater our accuracy will be. If we widen the compass to do our bisection, we’ll have to reset it in the next step, and it will likely be slightly off from the original setting.
- The extra marks we are about to make we have to make anyway – so why not use them for two things instead of just one?
- SO… keeping your compass at the same radius as the circle you just drew – place the point where the horizontal line meets the circle on the left (point A), and swing two small arcs marking points on the circle above and below (points C and E).
- Repeat this on the right side, centering your compass on point B and marking points D and F.
- Maintaining the same radius, place your compass at point C, and swing a short arc above the circle, about where you think the vertical will pass*. Repeat from point D to create a pair of crossing arcs above the circle at point G. This is our first reference point for the vertical.
- Repeat the previous steps below the circle from points E and F to create another pair of crossing arcs below the circle at point H.
- Place your straightedge so it lines up with points G, O, and H. (If these don’t line up, check your work – they should!). Use these three points to draw a vertical through the center of the circle, creating intersections J and K at the top and bottom of the circle.
* TIP: If you set your compass as suggested in step 1, there will be room for this on your paper. If you find that your crossing point will be off the paper, simply reduce your compass radius to any size that will place the crossing on the paper but above your circle. In this case, though, be sure you have marked all four points C, D, E, and F before adjusting your compass!
Step 3: Bisect horizontal radii
OK, that last step felt long, but it’s because we already did most of the work for this one. Remember those marks on the circle we just made? Time to use them for purpose number two – bisecting the horizontal radii.
- Locate points C and E on the left part of your circle. Place your straightedge in line with these two points, and mark where it intersects the horizontal line (I’ve shown the whole line as a dashed line in the illustration, but there’s no need to draw the whole line – it will just clutter up your drawing). This will divide the left radius in half at point L**.
- Repeat this procedure on the right side to bisect the right radius at point M.
** WHY DOES IT WORK?
Since we kept our radius the same as the original circle, the marks at point C and E are a pair of crossing points just like the ones we made to bisect the entire circle when we drew our central vertical line. Since these arcs originated from point O (to draw the circle) and point A (to create the crossings) we are bisecting line segment AO, a.k.a. the left radius of the circle.
Step 4: Create the “yin yang”
- Now comes the step that gives this particular construction of the decagon its name. (Yes, there are other ways to do this, but I find this one to be the most elegant and accurate of the ones I’ve tried.)
- We are finally ready to change our compass radius to 1/2 of its original length. Place your compass point on point M, and reduce the radius until the lead swings through points O and B*.
- On the right side, swing 1/2 of a circle above the original horizontal, running from point O to point B.
- Repeat this on the left side, using point L as your center, except swing the 1/2 circle below the original horizontal from point O to point A.
- These two 1/2-radius circles, along with our original circle, create the “yin yang” symbol from which this construction gets its name.
* TIP: At this point, before swinging your arcs, test out your new radius on both sides, checking both the inner and outer tangencies. If any error has crept into your construction, it will show up as things not quite lining up. If this is the case, make micro adjustments to your compass radius and placement to find the best overall compromise.
Step 5: Find decagon edge length
Believe it or not, we’re already incredibly close to our goal!
- Line your straightedge up along points J and M, and mark the point N where it intersects the top semicircle. (Again, I’ve shown the whole line as a dashed line in the illustration, but there’s no need to draw the whole line – it will just clutter up your drawing).
- Repeat with points K and L to find point P on the bottom semicircle.
- Place your compass at point J, and adjust it until it passes through point N. Check on your lower circle to make sure it works there, too (between points K and P), and make micro-adjustments if needed to account for any in accuracies.
- This radius is the desired edge length of our decagon!**
**WHY DOES THIS WORK?
For those who are interested, this construction is, in fact, theoretically accurate. By construction, the triangle OMJ has a height that is twice its base (that is, if we call the original circle radius OJ=2, then OM=1 because it bisects OB, another radius). By the Pythagorean theorem, this means that hypotenuse JM has length sqrt(5), and thus JN has length sqrt(5)-1. So the ratio of our edge length to the radius is sqrt(5)-1:2.
So what, you say? 🙂 WELL, in a regular decagon, each triangular “wedge” from the center to an edge is a 36-72-72 isosceles triangle. Half of that triangle is an 18-72-90 right triangle, with the short edge being 1/2 of the decagon’s edge length , and the hypotenuse being the circle radius. And the ratio of those two is, by definition, sin(18 degrees). And the sin of 18 degrees? You guessed it – (sqrt(5)-1)/4!
Step 6: Step edges around circle
- Starting from point J at the top*, swing an arc of this radius to the left and right, marking two points on the circle (which I’ve labeled 2 and 10).
- From each of those points, swing two successive arcs down each side, marking the circle as you go, to find points 3 and 9, then 4 and 8)
- Repeat all of this from the bottom, working upward to find 5 and 7, then 4 and 8, and so on. If your construction is perfectly accurate, your points coming down from the bottom will exactly match the ones coming up from the bottom. If this is not the case, adjust your compass slightly to find the best overall fit.
* NOTE: Depending on the orientation you need, you can also work from the left and right, where the horizontal meets the circle. This will put two of the edges of your decagon on the top and bottom, instead of at the sides. And if you really want to go crazy, you can mark both sets and make yourself a 20-gon! 🙂
Step 7: Draw the final decagon
- Once you have the points marked to your satisfaction, go ahead and use your straightedge to connect them one to the next all the way around to create your final decagon.
- NOTE: Be careful not to accidentally use the original four points C, D, E, and F that we used when constructing the vertical and bisecting the radii. They lie very close to points 10, 2, 7, and 5, respectively and can throw you off if you’re not paying attention. 🙂
Congratulations!
You now know a simple, elegant way to construct decagons and 20-gons. (And, did we mention, pentagons as well?)