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A Reuleaux triangle is a curved triangle of constant width, formed by drawing three circular arcs, each centered on one vertex of an equilateral triangle and spanning the other two. Because its width is the same in every direction, it can roll like a circle. For width w, its perimeter is πw and its area is ½(π − √3)w2.
Aside from a circle, what other shapes can a manhole be so that it won’t fall through the hole? One of the answers to this question is a Reuleaux triangle.
A Reuleaux triangle is a curved (or curvilinear) triangle made from three intersecting arcs, drawn when circles are centered on the vertices of an equilateral triangle. It is sometimes loosely called a spherical triangle in architecture, though that term properly refers to a triangle on the surface of a sphere. This shape has a constant width as it rotates, which makes it useful for many purposes. Sketched by Leonardo da Vinci and later studied formally by Leonhard Euler, among other mathematicians, it shows up in all sorts of applications, from rotary engines and architectural windows to guitar picks and graphic signage.
Construction Of Reuleaux Triangle
The Reuleaux Triangle is named after Franz Reuleaux, the 19th-century German engineer who pioneered the study of kinematics, the science of how machines translate one type of motion into another. Although the shape was known to earlier mathematicians, Reuleaux was the first to demonstrate its constant-width property, and he used it in many of his designs.
The construction of the Reuleaux Triangle is fairly straightforward:
- Make an equilateral triangle
- Using a compass, draw a circle, taking one of the vertices as the center and passing through the other two vertices.
- Repeat step 2 by taking the other points as the center, respectively.
Mathematical Properties Of Reuleaux Triangle
Curve Of Constant Width
The defining property of the Reuleaux Triangle is its constant width. If you trap the shape between two parallel supporting lines and then rotate it, the gap between those lines never changes, no matter which way the shape is turned. That width equals the side length of the equilateral triangle you started with, which is also the radius of each of the three arcs. The circle is the only other shape this simple that shares this trait, which is exactly why a Reuleaux triangle can roll smoothly under a flat board and why it makes a manhole cover that cannot fall through its own hole.
Perimeter And Area
Constant width leads to a surprisingly neat result for the perimeter. Barbier’s theorem, proved by Joseph-Émile Barbier in 1860, states that every curve of constant width w has the same perimeter as a circle of diameter w, namely πw. So a Reuleaux triangle of width w has a perimeter of exactly πw, identical to a circle that fits snugly through the same gap.
The area, however, is not the same. Each of the three arcs subtends an angle of 60° (π/3 radians) at its center, and adding up the three circular sectors and subtracting the equilateral triangle counted twice gives:
A = ½(π − √3)w2 ≈ 0.7048w2
That makes the Reuleaux triangle the smallest possible curve of constant width: among all shapes that share the perimeter πw, none enclose less area. (The circle, with area ¼πw2 ≈ 0.7854w2, encloses the most.)
Applications Of Reuleaux Triangle
Due to its compact geometry and equal proportions, the Reuleaux Triangle is used as a fundamental basis of structural and rotational strength. Some of the applications are:
Pencils
Modern pencil design has seen the adaptation of the Reuleaux Triangle as its predominant form. The triangle enables a stronger grip. As a person writes with three fingers, each curve of the triangle supports a finger, which provides better overall control of the pencil.
Guitar Picks
Similar to the pencil, guitar picks use the base geometry of the Reuleaux Triangle for better control and a tighter grip. It is easier to manipulate the strings with one of the edges of the curve while holding two fingers to hold the other two. Variations of the Reuleaux Triangle pick are even used for different genres of music to generate different sounds.
Carving Squares
The Reuleaux Triangle is the basis for a drill bit that cuts an almost-square hole. The Watts drill, patented by Henry Watts in 1914, uses a Reuleaux triangle reshaped with cutting edges. Held in a special floating chuck (so it has no fixed center of rotation), its constant width lets it sweep out a square with only slightly rounded corners.
Mechanism Design
Franz Reuleaux intensively studied the mechanical properties of a Reuleaux Triangle. The triangle can be used as a mechanical linkage that converts rotation around an axis into motion. A close relative of this shape shows up in the Wankel rotary engine used in some cars: the three-cornered rotor is essentially a fattened Reuleaux triangle (its sides bulge slightly outward rather than being true circular arcs), spinning inside a housing shaped like a curve called an epitrochoid.
Some early film projectors have also used the triangle to project images. The Luch-28mm film projector uses it as a key piece in the rotation of the mechanism.
Architecture
The Reuleaux Triangle has been extensively used in architectural forms. Its compact geometry and sturdy structural integrity make it highly versatile and aesthetically pleasing to the eye. In the late 13th or early 14th century, it was used as a decorative feature in many churches, mainly as a window design of the facade.
This shape is also used in staircase design. The 2D form of a triangle is used as the base plan for the staircase. The stairwell is then arranged on the three curves while the remaining center area is kept as a cavity.
In modern architecture, this triangle has been used as the floor plan of high-rise buildings. The KölnTriangle tower in Cologne, Germany, completed in 2006, has a ground plan in the shape of a Reuleaux triangle, and its office complex spans a gross floor area of 84,300 m2 (907,000 sq ft). The three curved sides give panoramic views in three directions while also varying the depth of the rooms arranged along each curve.
In Nature
The structure of soap bubbles has intrigued scientists for centuries, as it forms a fractal-like arrangement of varying sizes of bubbles.
Plateau’s laws state that soap films always meet in threes at 120°, which is also the interior angle at each corner of a Reuleaux triangle. Based on this law, it is possible to construct Reuleaux Triangles inside the intersecting bubbles.
Map Making
Leonardo da Vinci made maps of the world based on the geometry of the Reuleaux Triangle. He took 8 of these shapes and divided them into two parts to make the surfaces of the earth. Each Reuleaux Triangle represents 1/8th of the world. It is called Leonardo da Vinci’s Mappamundi.
In Closing
The Reuleaux Triangle has been used in many ways and in many things. Its three equal curved sides and constant width make it desirable for everything from pencils to projectors. That same blend of structural stability and roll-like-a-circle geometry has earned it a place in architecture and mechanical design. To this day, designers keep finding fresh uses for this deceptively simple curved triangle.
