**O**ne of the cool things about geometry is that the same principle can underlie a medieval church window and a design for a weird, Maker Faire-appropriate bicycle wheel. In this adaptation of material from our recent book, Make: Geometry, we’ll show you how the two are related, and how to draw either one at any scale.

Let’s start with medieval cathedral, or fortress, construction. One of the challenges of any structure is making windows and doors that will bear the weight around and above them. An early solution was the circular arch, which is an opening in a structure that is half a circle. The simplest way to lay one out is to pin one end of a rope and swing it around that pivot.

Then, someone hacked the design and came upon the Gothic arch. It is created by drawing two circles of the same radius. Each circle is centered at the base of the other arc. Gothic arches are much stronger than circular ones, since more force from material above the arch is transmitted downward through the base rather than pressing straight down on unsupported bricks. By the way, the term Gothic was used in a snarky way at the time by traditionalist architects, to imply association with the barbarians who destroyed Roman civilization. Improvements aren’t always appreciated when they are first introduced!

Now, what about that bicycle wheel? It, and the Gothic arch, both arise from the 2,000+ year-old construction of an equilateral triangle (one with all sides the same). Try it with us to see how!

1. Grab yourself a drawing compass.

Any kind will do, but the most common attaches or incorporates a pencil on one side. We’ll call the point that isn’t the pencil the “needle point.”

In a pinch (or to draw a really big version) you can use a loop of string instead of a compass. Put your pencil point inside the loop, hold down the other end of the loop as the center of the circle, and draw.

2. Set your compass points a convenient distance apart. This distance will be the length of the equilateral triangle’s sides, and you will leave them that way for the rest of the process.

Draw a circle. One corner of the triangle will be at the center of this circle.

3. Put the needle point at any point of the circle you just drew. This point will be the second corner of the triangle.

Draw another, intersecting circle of the same radius there.

4. Now put the needle point at one of the places where these circles intersect (below).

5. Draw a third intersecting circle (below).

6. You’ll see there is a shape like a rounded triangle in the middle.

This shape is called a Reuleaux triangle (grey shaded, above ), pronounced “roo-low.” Connecting all the

vertices of the Reuleaux triangle gives you an equilateral triangle.

7. Keeping one straight side and two rounded ones gives us our Gothic arch (below ).

What happens if you start making smaller arches within the arch? Here’s what we did (with a 3D-printed version) to create a tracery, the supports that would have held glass panes in a stained-glass window.

Try creating a window frame as shown in the gothic arch example above, then split into halves. Experiment with progressive repeating designs. The three intersecting circles we tucked into the remaining space in the big arch are called a trefoil — the same as the outside of the three intersecting circles. (If you want to find out how to 3D print the tracery shown below, or the other 3D prints in this article, check out our Make: Geometry book.)

Now, let’s return to the Reuleaux triangle that we constructed along the way. Draw one as we describe above on cardboard or something stiff, and cut it out. You will discover that the triangle will roll, just like a circular cutout would! That’s true because the width of a Reuleaux triangle is constant. (Why? Think about how it is constructed with intersecting circles, and remember that the distance from the center of a circle is the same everywhere on that circle.) However, the distance to its own center from all points on a Reuleaux triangle’s perimeter is not equal.

This means that you could in principle make a Reuleaux triangle wheel, but it would need a complicated mechanism to work since the center of rotation does not stay in the same place as the wheel rolls. People have made bicycles with Reuleaux triangle wheels, but they are complicated. Rather than rotating around a fixed hub, the centers of the wheels are mounted on a linkage that allows the center to move up and down, with rollers or another low-friction surface resting on top of the wheel to keep the frame’s height constant. If you cut a Reuleaux triangle out of cardboard, you can see that it rolls easily along a flat surface, but is not rotating around any one constant axis.

Finally, what happens if we confine a Reuleaux triangle in an appropriately sized square? It will turn freely. Actually, if you were to make a Reuleaux triangle drill bit, it could be used to drill out a square hole with slightly rounded corners.

CONCLUSION

We’ve seen here that a simple compass construction of an equilateral triangle can take us down a couple of very divergent paths. We’ve shown examples on paper (and with 3D printed examples we talk about more in Make: Geometry) which you could transfer to other, sturdier mediums if you wanted to make more substantial examples. Maybe you’ll come up with the next step in Gothic making — appreciated 1,000 years from now, even if too avant garde for the present!

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