The Strange Triangle We Encounter Everywhere

By Herm London
Originally written 2006

Most any elementary school pupil knows that a triangle is a simple geometric figure made up of three straight lines.  Most adults recall that a special case is the equilateral triangle in which each side is of exactly the same length. Few of us have heard of the Reuleaux triangle although every one of us encounters it almost daily.  This unique figure is nothing more than an equilateral triangle with each of its sides slightly bowed out in an arc instead of a straight line.

I first became aware of this artistic mathematical shape while on a tour of Lyndhurst, a National Trust Historic Site in Tarrytown, New York. High up upon a facade of one of the wings of the building, almost nestled near the peak of the structure, I spotted this little window.  I was determined to find out more about this shape.

Reuleaux triangle window at Lyndhurst Castle in Tarrytown, NY.

The Reuleaux triangle is easy enough to construct.  Merely draw an equilateral triangle on a piece of paper.  Then, adjust a drawing compass so that the distance between the pivot point and the pencil point is equal to the length of one side of your triangle.  You are ready to draw the curved sides. Place the pivot point at one vertex of the triangle and draw a sweeping arc between the two opposite vertices.  Repeat this process from the second vertex of the triangle and still one more time from the third vertex. You will have drawn a curved equilateral triangle “embracing” the conventional triangle.  With this simple ID picture, one can go in search of the Reuleaux shape everywhere around us.

Drawing an Reuleaux Triangle

In my own community of Chelsea, New York, right next to the post office (former school house) is St. Marks Episcopal Church.  The belfry facade is punctuated by one of these three-sided windows. Directly across the street, at the Chelsea Fire House, I saw men in uniform.  The shoulder patches on their sleeves, designating their affiliation, is a modified shield-shape or Reuleaux triangle with one vertex pointing down. Many police departments, fire departments, fire companies and ambulance corps use a similar shoulder patch.

St. Mark's Episcopal Church has an Reuleaux window in front.

Ambulance shoulder patch.

This geometric configuration was first mentioned by Swiss mathematician Leonard Euler in the 18th century.  However, it was Franz Reuleaux, German math teacher and engineer, who really studied the properties of the curved triangle in the late 19th Century, and after whom the triangle is named.

Collectors of odd coins may have noticed a Canadian coin, a British Twenty Pence piece, and a Fifty Pence piece that are all basically the shape of the Reuleaux triangle.  The coin that really attracted the attention of numismatists was the 1970 Bermuda silver (and gold commemorative) pieces. Complete with a representation of a ship on the Atlantic Ocean waters, it was clearly meant to depict the Bermuda Triangle!

Bermuda Silver pieces.

An old brain-teaser asks why manhole covers are circular.  The serious response is that the lid must have no diameter that is smaller than any other permitting the lid to accidentally drop into the sewer.  If the lid were square for example, it could be stood on edge and then dropped into the square manhole along the open diagonal which could accept it.  The MENSA members like to say that the circular lid has the same width in all directions. Then they ask you what other shape the manhole cover could be.  Yes...because all of its diameters are the same length, the Reuleaux triangle, or any Reuleaux polygon for that matter, would be suitable! There are communities in Minnesota that use triangular manhole covers.

Triangular manhole covers

Does your house of worship have a triangular window that is in disrepair?  The people of St. John the Baptist Catholic Church in Oceana County’s Claybanks Township, Michigan, knew that the quaint wooden structure needed a major restoration.  As long as they were spending the big bucks, they opted for the reinstallation of a Reuleaux Triangle stained glass window that was removed for a renovation project in the lat e1950s!  Taking a page from their book, your church can have a tourist site that may attract more regular worshippers in the long run.

St. John the Baptist Church in Michigan.

St. John the Baptist Church in Michigan.

A few years ago someone invented and produced a pistol that used rounds of ammunition that had triangular, rather than round, cross-section.  They were called “Trounds”. The shape and design enabled one to have a gun with virtually no recoil or kick-back. Although the companies stopped producing pistols and carbine rifles, the principle was sound and is still applied to larger firearms.  Wow! A Reuleaux triangular shaped bullet!

In the 1920s, Felix Wankel developed a unique engine whose inner rotor made it totally different from any reciprocating-piston gasoline engine.  Mazda produced a car with this rotary engine in the 1970s and has used it periodically ever since, though the technology of the standard piston engines had a seventy-five-year head start.  Rotary engines using the Reuleaux triangular shaped rotor are still used in lawnmowers and other small devices.

Mazda Rotary Engine

According to the old saying “You can’t put a round peg into a square hole.”  My question is “How do I drill square holes?” I can’t use an electric drill with a standard drill bit.  A typical drill bit is a cylinder-shaped like a pencil, but with sharp fluted indentations curled around it.  When this bit spins around in one spot it literally “carves out” a round hole.

Someone named Watt realized that if you constructed a drill bit with a Reuleaux triangle cross-section rather than a circular cross-section, and mounted it on an eccentric or off-center shaft, it would wobble from corner to corner as it spun around.  Sure enough this device can cut a pretty respectable square hole. You can go into a large hardware store and ask for a square-hole-cutting drill bit and they won’t look at you as if you asked for a left-handed screwdriver!    

If this spinning around has made you queasy and sick to your stomach, just take some Pepto-Bismol.  If you have a stuffy head-cold, take some Nyquil at bedtime. Take a good look at the shape of the bottoms of these bottles!  After you feel better and get a good night’s sleep, go out into the world prepared to notice the Reuleaux triangles everywhere around us.