How To Really Benefit From Curves Of Constant Width?

Curves of constant width represent one of the most elegant real-life applications of plane geometry. Imagine that you need to move a very massive and heavy object from one place to another. Conventional intuition would say that wheels would serve the purpose well. However, the heavier the object, the higher is the risk of wheel-axle-failure.

Historically, human beings have come up with more effective and elegant solutions for such a need. Often, you will see a setup that involves a flat platform with cylindrical rollers underneath.

Once the platform slides over and down the rollers, you move the rollers and place them in front of the platform again. Then, you repeat the whole process again until the object reaches the target destination.

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What are Curves of Constant Width?

If the underlying surface is a flat plane, perfectly cylindrical rollers would lead to zero bouncing (vertical movement) of the object you are transporting. This is because cylindrical rollers feature a circular cross-section. And a circle in turn is a curve of constant width.

Imagine two parallel lines touching the perimeter of a circle at two opposite points. No matter where you move this combination on a circle, you would get the same width (diameter) — hence a curve of constant width.

Now, here’s an interesting question:

Are circles the only possible curves of constant width?

The answer: of course not! Let us go back to the title image. The two different rollers below the wooden plank feature profiles (curves) of the same constant width. In other words, when we slide it, the wooden plank would roll without any vertical displacement (bobbing or bouncing). How? Let us find out.

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Non-Circular Curves of Constant Width — The Reuleaux Triangle

The Reuleaux triangle is the most well-known among the non-circular curves of constant width.

It was named after German engineer, Franz Reuleaux, who pioneered key work in this field. You can construct the Reuleaux triangle in several ways. Here are two prominent ones:

1. It is simply the area of symmetrical intersection of three identical circles as shown in the image below.

2. You consider an equilateral triangle, place the fixed end of a compass on one of the corners, and draw an arc connecting the other two corners. You then repeat the procedure for the other two corners to arrive at the Reuleaux triangle.

Now, you might be sceptical about the notion that the Reuleaux triangle features constant width. But think again; whenever two parallel lines are made to touch the perimeter of a Reuleaux triangle, they are necessarily tangential and feature the same width across all possible locations.

It is because of this property that a Reuleaux triangle can rotate inside a square whilst maintaining contact with all the sides of the square at all times.

If you look at the moving image above, you will notice that the path traced by the corners of the Reuleaux triangle almost forms a square (with rounded-off corners). All this geometrical significance is fine. But what could we use such properties for?

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Applications of the Reuleaux Triangle

One of the first well-documented uses of the Reuleaux triangle is to transform rotational motion into square motion. Engineer Harry James Watts invented a rotary drill in 1914 that could drill square holes!

It was able to do this because Watts used a profile based on the Reuleaux triangle for the drill. Similar drilling equipment was known even before Watts, but Watts just happened to popularize it better than before.

Another well-known application was to jerk-feed film into projectors. The Soviet Luch-2 8mm film projector, for instance, had to feed film in a jerky step-wise motion.

Each film frame stopped for a moment before the projector lens, after which an accelerated rotation enabled the next frame to be positioned. This was achieved by rotating a Reuleaux triangle profile within a square as you can see below:

Apart from this, window shapes based on the Reuleaux triangle were part of 13th- and 14th- century Gothic architecture. Leonardo da Vinci cut the spherical construct of the world map and mapped in onto eight octants, each featuring the shape of a Reuleaux triangle.

Curves of Constant Width Beyond Circles and Reuleaux Triangles

The Reuleaux triangle is the most famous non-circular curve of constant width because it features the lowest area (and sharpest angle) possible for a given width. It is in no way the only non-circular curve of constant width. There are, in fact, infinitely many such curves.

The Reuleaux triangle is based on a triangle. Similarly, it is possible to create curves of constant width based on polygons that feature an odd number of sides. Such curves are also known as Reuleaux polygons.

For instance, consider the irregular 7-pointed star shown above. First, we place the fixed end of the compass on one corner, and connect two other corners using a circular arc. Then, we repeat the process around each corner such that all the arcs connect together to form a curve of constant width.

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Final Thoughts

Curves of constant width not only offer a deeply satisfying geometrical experience but they enjoy a wealth of real-world applications as well. Curves of constant width also occur in nature in the form of Reuleaux polygons in two-dimensional soap bubbles.

Mathematicians, engineers, architects, and artists alike have shown their fascination towards this concept throughout human history. It appears that when it comes to curves of constant width, the shared fascination unites fields that are often at odds with one another.

This essay really just scratches the surface of the concept. There is so much more to curves of constant width. For anyone who wishes to dive deeper into this field, the possibilities seem endless!

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Reference and credit: Martin Gardner.

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Further reading that might interest you: How To Really Solve This Tricky Algebra Problem? and How To Casually Guess Numbers After Dice Throws?

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