Reuleaux triangle

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The Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.
Reuleaux triangle shaped manhole cover in San Francisco

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. It is a curve of constant width, the simplest and best known such curve other than the circle itself.[1] This means that the separation of two parallel supporting lines is independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"[2]

The term derives from Franz Reuleaux,[3] a 19th-century German engineer who did pioneering work using this shape on ways that machines translate one type of motion into another.[4] However, the concept was known before his time, for instance by Leonardo da Vinci, who used it for a map projection.[5] Other applications of the Reuleaux triangle include giving the shape to Gothic church windows, guitar picks, pencils, and drill bits for drilling square holes, as well as in graphic design in the shapes of some signs and corporate logos.

The Reuleaux triangle is the first of a sequence of Reuleaux polygons, curves of constant width formed from regular polygons with an odd number of sides. It can be generalized into three dimensions in multiple ways: the Reuleaux tetrahedron (the intersection of four spheres whose centers lie on a regular tetrahedron) does not have constant width, but can be modified to form the Meissner tetrahedron, which does. Alternatively, the surface of rotation of the Reuleaux triangle also has constant width.

Construction[edit]

To construct a Reuleaux triangle

The Reuleaux triangle may be constructed with a compass alone, not even needing a straightedge. By the Mohr–Mascheroni theorem the same is true more generally of any compass-and-straightedge construction, but the construction for the Reuleaux triangle is particularly simple. First, sweep an arc sufficient to enclose the desired figure. Then, with the radius unchanged, sweep a sufficient arc centred at a point on the first arc, so that the two arcs intersect each other. Finally, with the same radius and with the centre at the intersection point of the first two arcs, sweep a third arc to intersect both previous arcs. The result is a curve of constant width.

Alternatively, construct an equilateral triangle T. Then, draw three arcs of circles, each centered at one vertex of T and connecting the other two vertices.[1] Or, equivalently, intersect three disks centered at the vertices of T, with radius equal to the side length of T.[6]

Mathematical properties[edit]

Extremal measures[edit]

By many different measures, the Reuleaux triangle is one of the most extreme curves of constant width.

By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the smallest possible area of any curve of given constant width. This area is

\frac{1}{2}(\pi - \sqrt3)s^2 \approx 0.70477s^2,

where s is the constant width. One method for obtaining this area value is to partition the Reuleaux triangle into an inner equilateral triangle and three curvilinear regions between this inner triangle and the arcs forming the Reuleaux triangle, and then add the areas of these four sets. At the other extreme, the curve of constant width that has the maximum possible area is a circular disk, which has area \pi s^2 / 4\approx 0.78540s^2.[7]

The angle made by pairs of arcs at the corners of a Reuleaux triangle is 120°, the sharpest possible angle in any curve of constant width.[1]

Although the Reuleaux triangle has sixfold dihedral symmetry, the same as an equilateral triangle, it does not have central symmetry. The Reuleaux triangle is the least symmetric curve of constant width by two different measures of central asymmetry, the Kovner–Besicovitch measure (ratio of area to the largest centrally symmetric shape enclosed by the curve) and the Estermann measure (ratio of area to the smallest centrally symmetric shape enclosing the curve). For the Reuleaux triangle, the two centrally symmetric shapes that determine the measures of asymmetry are both regular hexagons.[8]

The shape of the largest possible constant width that avoids all points of an integer lattice is a Reuleaux triangle having one of its axes of symmetry parallel to the coordinate axes on a half-integer line. Its width, approximately 1.545, is the root of a degree-6 polynomial with integer coefficients.[8][9]

An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle

Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite that can be inscribed into a Reuleaux triangle.[10]

Other measures[edit]

By Barbier's theorem all curves of constant width including the Reuleaux triangle have equal perimeter, the perimeter of the circle with that width, which is \pi s.[11][12][1]

The radii of the largest inscribed circle of a Reuleaux triangle with width s, and of the circumscribed circle of the same triangle, are

\displaystyle\left(1-\frac{1}{\sqrt 3}\right)s\approx 0.42265s and \displaystyle\frac{s}{\sqrt 3}\approx 0.57735s

respectively; the sum of these radii equals the width of the Reuleaux triangle. More generally, for every curve of constant width, the largest inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve.[13]

Rotation within a square[edit]

Rotation of a Reuleaux triangle within a square, showing also the curve traced by the center of the triangle

Any curve of constant width can form a rotor within a square, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square; the Reuleaux triangle is the rotor with the minimum possible area.[1] As it rotates, its axis does not stay fixed, but instead follows a curve formed by the pieces of four ellipses.[14] Because of its 120° angles, the rotating Reuleaux triangle does not trace out the entire square, but rather covers a shape with slightly rounded corners, also in the shape of elliptical arcs.[1] At any point during this rotation, two of the sharp corners of the Reuleaux triangle touch two adjacent sides of the square, while the third corner of the triangle traces out a curve near the opposite vertex of the square. The shape traced out by the rotating Reuleaux triangle covers approximately 98.77% of the area of the square.[15]

As a counterexample[edit]

Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a planar object into a single position.[16]

The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.[17] Overlooking this fact may have played a role in the Space Shuttle Challenger disaster, as the roundness of sections of the rocket in that launch was tested only by measuring different diameters, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.[1]

Applications[edit]

Reaching into corners[edit]

Several types of machinery take the shape of the Reuleaux triangle, based on its property of being able to rotate within a square:

  • The Watts Brothers Tool Works square drill bit has the shape of a Reuleaux triangle, modified with concavities to form cutting surfaces. When mounted in a special chuck which allows for the bit not having a fixed centre of rotation, it can drill a hole that is nearly square.[18] The Harry Watt square is often used in mortising.[19] Although patented by Henry Watts in 1914, similar drills invented by others were used earlier.[1] Other Reuleaux polygons are used to drill pentagonal, hexagonal, and octagonal holes.[1][18]
  • Panasonic robotic vacuum cleaner RULO has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms.[20]
Comparison of a cylindrical and Reuleaux triangle roller

Rolling cylinders[edit]

Other objects take cylindrical shapes with a Reuleaux triangle cross section:

  • A Reuleaux triangle (along with all other curves of constant width) can roll but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers with cross-sections that were Reuleaux triangles would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels would bounce up and down three times per revolution.[1][21] This concept was used in a science fiction short story by Poul Anderson titled "Three-Cornered Wheel."[22]
  • Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels.[23] They are usually promoted as being more comfortable or encouraging proper grip, as well as being less likely to roll off tables (since the center of gravity moves up and down more than a rolling hexagon).
Reuleaux triangle based film advance mechanism in the Soviet Luch-2 8mm film projector

Mechanism design[edit]

Another class of applications of the Reuleaux triangles involve using it as a part of a mechanical linkage that can convert rotation around a fixed axis into reciprocating motion.[6] Reuleaux himself had models of several mechanisms involving the Reuleaux triangle, which he used to investigate their motion.[24] For instance, in a film projector, it is necessary to advance the film in a jerky, stepwise motion, in which each frame of film stops for a fraction of a second in front of the projector lens, and then much more quickly the film is moved to the next frame. This can be done using a mechanism in which the rotation of a Reuleaux triangle within a square is used to create a motion pattern for an actuator that pulls the film quickly to each new frame and then pauses the film's motion while the frame is projected.[25]

Other physical objects[edit]

Three Reuleaux triangle shaped guitar picks

Many guitar picks employ the Reuleaux triangle, as its unique shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Many players find the shape ergonomic, since it naturally tends to point in the proper direction. Its three equal tips also prevent wear and extend lifespan, as compared to the single tip of a pick shaped like an isosceles triangle.[26] The rotor of the Wankel engine is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle.[1] However, its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.[27]

Reuleaux triangle shaped window of the Church of Our Lady, Bruges in Belgium
The Kölntriangle building has a Reuleaux triangle cross-section

Architecture[edit]

In Gothic architecture, beginning in the late 13th century or early 14th century,[28] the Reuleaux triangle became one of several curvilinear forms frequently used for windows and architectural decorations.[3] In this context, the shape is more frequently called a spherical triangle,[28][29][30] but that term should be distinguished from the more usual mathematical meaning of the same phrase, a triangle on the surface of a sphere.

The Reuleaux triangle has also been used in other styles of architecture; for instance, a modern high-rise building, the Kölntriangle in Cologne, Germany, was built with a Reuleaux triangle cross-section. Together with the circular shape of its core, this gives varied depths to the rooms of the building.[31]

Mapmaking[edit]

Another early application of the Reuleaux triangle, by Leonardo da Vinci circa 1514 (or possibly by one of his followers at his direction), was a world map in which the spherical surface of the earth was divided into eight octants, each flattened into the shape of a Reuleaux triangle.[5][32][33]

Leonardo da Vinci's world map in eight Reuleaux-triangle quadrants

Similar maps also based on the Reuleaux triangle were published by Oronce Finé in 1551 and by John Dee in 1580.[33]

Signs and logos[edit]

The shape is used for signage for the National Trails System administered by the United States National Park Service, and for the logo of Colorado School of Mines.[34] The corporate logo of Petrofina (Fina), a Belgian oil company with major operations in Europe, North America and Africa, utilized a Reuleaux triangle with the Fina name from 1950 until Petrofina's merger with Total S.A. in 2000.[35] A rotated version of Fina's Reuleaux triangle is utilized by Alon USA, which acquired the American Petrofina operations spun off by Total in 2006.[36]

The Reuleaux triangle as the central bubble in a mathematical model of a four-bubble planar soap bubble cluster

In nature[edit]

By Plateau's laws, the 120° angle of the Reuleaux triangle's vertices is the same angle at which the circular arcs in two-dimensional soap bubble clusters meet, and it is possible to construct clusters in which some of the bubbles take the form of a Reuleaux triangle.[37]

The shape was first isolated in crystal form in 2014 as Reuleaux triangle disks.[38] Basic bismuth nitrate disks with the Reuleaux triangle shape were formed from the hydrolysis and precipitation of bismuth nitrate in an ethanol–water system in the presence of 2,3-bis(2-pyridyl)pyrazine.

Generalizations[edit]

Four spheres intersect to form a Reuleaux tetrahedron.

Three-dimensional version[edit]

The intersection of four spheres of radius s centered at the vertices of a regular tetrahedron with side length s is called the Reuleaux tetrahedron, but is not a surface of constant width.[39] It can, however, be made into a surface of constant width, called Meissner's tetrahedron, by replacing its edge arcs by curved surface patches. Alternatively, the surface of revolution of a Reuleaux triangle through one of its symmetry axes forms a surface of constant width, with minimum volume among all known surfaces of revolution of given constant width.[40]

Reuleaux polygons[edit]

Reuleaux polygons

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides, yielding a Reuleaux polygon. These are the only shapes of constant width whose boundaries are formed by finitely many circular arcs of equal length.[41]

The constant width of these shapes allows their use as coins that can be used in coin-operated machines.[1] The most commonly used of these, beginning in 1969 with the English 50-pence coin,[1] is the Reuleaux heptagon, which since then has been used as the approximate shape of several coins:

Similar methods can be used to enclose an arbitrary simple polygon within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge.[49] Although the regular-polygon based Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes based on irregular polygons that have an even number of sides.[50]

Yanmouti sets[edit]

The Yanmouti sets are defined as the convex hulls of an equilateral triangle together with three circular arcs, centered at the triangle vertices and spanning the same angle as the triangle, with equal radii that are at most equal to the side length of the triangle. Thus, when the radius is small enough, these sets degenerate to the equilateral triangle itself, but when the radius is as large as possible they equal the corresponding Reuleaux triangle. Every shape with width w, diameter d, and inradius r obeys the inequality

w - r \le \frac{d}{\sqrt 3},

and this inequality becomes an equality for the Yanmouti sets, showing that it cannot be improved.[51]

Related figures[edit]

Triquetra interlaced to form a trefoil knot

In the classical presentation of a three-set Venn diagram as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle.[3] The same three circles form one of the standard drawings of the Borromean rings, three mutually linked rings that cannot, however, be realized as geometric circles.[52] Parts of these same circles are used to form the triquetra, a figure of three overlapping semicircles (each two of which form a vesica piscis symbol) that again has a Reuleaux triangle at its center;[53] just as the three circles of the Venn diagram may be interlaced to form the Borromean rings, the three circular arcs of the triquetra may be interlaced to form a trefoil knot.[54]

The Deltoid curve is anothe type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius.[55] Similarly, depending on its parameters, a supercircle can be formed by replacing the sides of a square by non-circular curves that are either convex or concave.[56]

References[edit]

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