![ReuleauxTriangle](https://upload.wikimedia.org/wikipedia/commons/d/d2/ReuleauxTriangle.svg)
A Reuleaux triangle is a
curved triangle with
constant width
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific cons ...
, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three
circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel
supporting line In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely ...
s is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a
manhole cover
A manhole cover or maintenance hole cover is a removable plate forming the lid over the opening of a manhole, an opening large enough for a person to pass through that is used as an access point for an underground vault or pipe. It is designed to ...
be made so that it cannot fall down through the hole?"
Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.
They are named after
Franz Reuleaux
Franz Reuleaux (; ; 30 September 1829 – 20 August 1905), was a German mechanical engineer and a lecturer of the Berlin Royal Technical Academy, later appointed as the President of the Academy. He was often called the father of kinematics. He wa ...
,
[.] a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the designers of
Gothic
Gothic or Gothics may refer to:
People and languages
*Goths or Gothic people, the ethnonym of a group of East Germanic tribes
**Gothic language, an extinct East Germanic language spoken by the Goths
**Crimean Gothic, the Gothic language spoken b ...
church windows, by
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
, who used it for a
map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
, and by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in his study of constant-width shapes. Other applications of the Reuleaux triangle include giving the shape to
guitar pick
A guitar pick (American English) is a plectrum used for guitars. Picks are generally made of one uniform material—such as some kind of plastic (nylon, Delrin, celluloid), rubber, felt, tortoiseshell, wood, metal, glass, tagua, or stone. They ...
s,
fire hydrant
A fire hydrant, waterplug, or firecock (archaic) is a connection point by which firefighters can tap into a water supply. It is a component of active fire protection. Underground fire hydrants have been used in Europe and Asia since at least ...
nuts,
pencil
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
s, and
drill bit
Drill bits are cutting tools used in a drill to remove material to create holes, almost always of circular cross-section. Drill bits come in many sizes and shapes and can create different kinds of holes in many different materials. In order ...
s for drilling
filleted square holes, as well as in graphic design in the shapes of some signs and corporate logos.
Among constant-width shapes with a given width, the Reuleaux triangle has the minimum area and the sharpest (smallest) possible angle (120°) at its corners. By several numerical measures it is the farthest from being
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
. It provides the largest constant-width shape avoiding the points of an
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid l ...
, and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a complete rotation within a square while at all times touching all four sides of the square, and has the smallest possible area of shapes with this property. However, although it covers most of the square in this rotation process, it fails to cover a small fraction of the square's area, near its corners. Because of this property of rotating within a square, the Reuleaux triangle is also sometimes known as the Reuleaux rotor.
The Reuleaux triangle is the first of a sequence of
Reuleaux polygon
In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. These shapes are named after their prototypical example, the Reuleaux triangle, which in turn, is named after 19th-century German engineer ...
s whose boundaries are curves of constant width formed from
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s with an odd number of sides. Some of these curves have been used as the
shapes of coins. The Reuleaux triangle can also be generalized into three dimensions in multiple ways: the
Reuleaux tetrahedron
The Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three verti ...
(the intersection of four
balls whose centers lie on a regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
) does not have constant width, but can be modified by rounding its edges to form the
Meissner tetrahedron, which does. Alternatively, the
surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
of the Reuleaux triangle also has constant width.
Construction
![Construction triangle Reuleaux](https://upload.wikimedia.org/wikipedia/commons/2/27/Construction_triangle_Reuleaux.svg)
The Reuleaux triangle may be constructed either directly from three
circles
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, or by rounding the sides of an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
.
[.]
The three-circle construction may be performed with a
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
alone, not even needing a straightedge. By the
Mohr–Mascheroni theorem
In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.
It must be understood that by "any geometric construction", we are refer ...
the same is true more generally of any
compass-and-straightedge construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, but the construction for the Reuleaux triangle is particularly simple.
The first step is to mark two arbitrary points of the plane (which will eventually become vertices of the triangle), and use the compass to draw a circle centered at one of the marked points, through the other marked point. Next, one draws a second circle, of the same radius, centered at the other marked point and passing through the first marked point.
Finally, one draws a third circle, again of the same radius, with its center at one of the two crossing points of the two previous circles, passing through both marked points. The central region in the resulting arrangement of three circles will be a Reuleaux triangle.
Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle ''T'' by drawing three arcs of circles, each centered at one vertex of ''T'' and connecting the other two vertices.
Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of ''T'', with radius equal to the side length of ''T''.
Mathematical properties
![Reuleaux supporting lines](https://upload.wikimedia.org/wikipedia/commons/e/eb/Reuleaux_supporting_lines.svg)
The most basic property of the Reuleaux triangle is that it has constant width, meaning that for every pair of parallel
supporting line In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely ...
s (two lines of the same slope that both touch the shape without crossing through it) the two lines have the same
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
from each other, regardless of the orientation of these lines.
In any pair of parallel supporting lines, one of the two lines will necessarily touch the triangle at one of its vertices. The other supporting line may touch the triangle at any point on the opposite arc, and their distance (the width of the Reuleaux triangle) equals the radius of this arc.
[.]
The first mathematician to discover the existence of curves of constant width, and to observe that the Reuleaux triangle has constant width, may have been
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
.
[.] In a paper that he presented in 1771 and published in 1781 entitled ''De curvis triangularibus'', Euler studied
curvilinear
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
triangles as well as the curves of constant width, which he called orbiforms.
[. See in particular section 1.4, "Orbiforms, 1781"]
pp. 484–485
Extremal measures
By many different measures, the Reuleaux triangle is one of the most extreme curves of constant width.
By the
Blaschke–Lebesgue theorem
In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is ...
, the Reuleaux triangle has the smallest possible area of any curve of given constant width. This area is
:
where ''s'' is the constant width. One method for deriving this area formula 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
.
The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
of any curve of constant width.
Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles. The largest equilateral triangle inscribed in a Reuleaux triangle is the one connecting its three corners, and the smallest one is the one connecting the three
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimens ...
s of its sides. The subset of the Reuleaux triangle consisting of points belonging to three or more diameters is the interior of the larger of these two triangles; it has a larger area than the set of three-diameter points of any other curve of constant width.
![Symmetry measure of Reuleaux triangle](https://upload.wikimedia.org/wikipedia/commons/f/fb/Symmetry_measure_of_Reuleaux_triangle.svg)
Although the Reuleaux triangle has sixfold
dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
, the same as an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, it does not have
central symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
.
The Reuleaux triangle is the least symmetric curve of constant width according to two different measures of central asymmetry, the
Kovner–Besicovitch measure
In plane geometry the Kovner–Besicovitch measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the fraction of the area of the set that can be covered by its largest centrally symmet ...
(ratio of area to the largest
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
shape enclosed by the curve) and the
Estermann measure
In plane geometry the Estermann measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the ratio of areas between the given set and its smallest centrally symmetric convex superset. It ...
(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
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
al, although the inner one has curved sides.
[.] The Reuleaux triangle has diameters that split its area more unevenly than any other curve of constant width. That is, the maximum ratio of areas on either side of a diameter, another measure of asymmetry, is bigger for the Reuleaux triangle than for other curves of constant width.
Among all shapes of constant width that avoid all points of an
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid l ...
, the one with the largest width is a Reuleaux triangle. It has 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.
Just as it is possible for a circle to be surrounded by six congruent circles that touch it, it is also possible to arrange seven congruent Reuleaux triangles so that they all make contact with a central Reuleaux triangle of the same size. This is the maximum number possible for any curve of constant width.
![Reuleaux kite](https://upload.wikimedia.org/wikipedia/commons/7/72/Reuleaux_kite.svg)
Among all
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
s, the shape that has the greatest ratio of its
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
is an
equidiagonal kite
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
that can be inscribed into a Reuleaux triangle.
[; .]
Other measures
By
Barbier's theorem
In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860.
Examples
The most familiar examples of ...
all curves of the same constant width including the Reuleaux triangle have equal
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
s. In particular this perimeter equals the perimeter of the circle with the same width, which is
.
[.]
The radii of the largest
inscribed circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
of a Reuleaux triangle with width ''s'', and of the
circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of the same triangle, are
:
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.
The optimal
packing density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. I ...
of the Reuleaux triangle in the plane remains unproven, but is conjectured to be
:
which is the density of one possible
double lattice In mathematics, especially in geometry, a double lattice in is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of an ...
packing for these shapes. The best proven upper bound on the packing density is approximately 0.947275. It has also been conjectured, but not proven, that the Reuleaux triangles have the highest packing density of any curve of constant width.
Rotation within a square
![Rotation of Reuleaux triangle](https://upload.wikimedia.org/wikipedia/commons/2/22/Rotation_of_Reuleaux_triangle.gif)
Any curve of constant width can form a rotor within a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area.
As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s. Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs.
At any point during this rotation, two of the 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.
As a counterexample
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. The existence of Reuleaux triangles and other curves of constant width shows that diameter measurements alone cannot verify that an object has a circular cross-section.
In connection with the
inscribed square problem
The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is ...
, observed that the Reuleaux triangle provides an example of a constant-width shape in which no regular polygon with more than four sides can be inscribed, except the regular hexagon, and he described a small modification to this shape that preserves its constant width but also prevents regular hexagons from being inscribed in it. He generalized this result to three dimensions using a cylinder with the same shape as its
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
.
Applications
Reaching into corners
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
Drill bits are cutting tools used in a drill to remove material to create holes, almost always of circular cross-section. Drill bits come in many sizes and shapes and can create different kinds of holes in many different materials. In order ...
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.
[ (27 page brochure).] Although patented by Henry Watts in 1914, similar drills invented by others were used earlier.
Other Reuleaux polygons are used to drill pentagonal, hexagonal, and octagonal holes.
Panasonic
formerly between 1935 and 2008 and the first incarnation of between 2008 and 2022, is a major Japanese multinational corporation, multinational Conglomerate (company), conglomerate corporation, headquartered in Kadoma, Osaka, Kadoma, Osaka P ...
's RULO
robotic vacuum cleaner
A robotic vacuum cleaner, sometimes called a robovac or a roomba as a generic trademark, is an autonomous robotic vacuum cleaner which has a limited vacuum floor cleaning system combined with sensors and robotic drives with programmable controll ...
has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms.
Rolling cylinders
![Reuleaux triangle 54](https://upload.wikimedia.org/wikipedia/commons/5/58/Reuleaux_triangle_54.JPG)
Another class of applications of the Reuleaux triangle involves cylindrical objects with a Reuleaux triangle cross section. Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels.
[.] 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).
A Reuleaux triangle (along with all other
curves of constant width
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width ...
) can
roll
Roll or Rolls may refer to:
Movement about the longitudinal axis
* Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis
** Roll (aviation), ...
but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers that have Reuleaux triangle cross-sections would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels would bounce up and down three times per revolution.
This concept was used in a science fiction short story by
Poul Anderson
Poul William Anderson (November 25, 1926 – July 31, 2001) was an American fantasy and science fiction author who was active from the 1940s until the 21st century. Anderson wrote also historical novels. His awards include seven Hugo Awards and ...
titled "The Three-Cornered Wheel".
A bicycle with floating axles and a frame supported by the rim of its Reuleaux triangle shaped wheel was built and demonstrated in 2009 by Chinese inventor Guan Baihua, who was inspired by pencils with the same shape.
Mechanism design
![Luch2 greifer](https://upload.wikimedia.org/wikipedia/commons/2/2a/Luch2_greifer.gif)
Another class of applications of the Reuleaux triangle involves using it as a part of a
mechanical linkage
A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing i ...
that can convert
rotation around a fixed axis
Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
into
reciprocating motion
Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single r ...
.
[.] These mechanisms were studied by Franz Reuleaux. With the assistance of the Gustav Voigt company, Reuleaux built approximately 800 models of mechanisms, several of which involved the Reuleaux triangle.
Reuleaux used these models in his pioneering scientific investigations of their motion. Although most of the Reuleaux–Voigt models have been lost, 219 of them have been collected at
Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...
, including nine based on the Reuleaux triangle.
[.] However, the use of Reuleaux triangles in mechanism design predates the work of Reuleaux; for instance, some
steam engine
A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be trans ...
s from as early as 1830 had a
cam
Calmodulin (CaM) (an abbreviation for calcium-modulated protein) is a multifunctional intermediate calcium-binding messenger protein expressed in all eukaryotic cells. It is an intracellular target of the secondary messenger Ca2+, and the bin ...
in the shape of a Reuleaux triangle.
[. Reprinted in .]
One application of this principle arises in a
film projector
A movie projector is an opto-mechanical device for displaying motion picture film by projecting it onto a screen. Most of the optical and mechanical elements, except for the illumination and sound devices, are present in movie cameras. Moder ...
. In this application, 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.
The rotor of the
Wankel engine
The Wankel engine (, ) is a type of internal combustion engine using an Eccentric (mechanism), eccentric rotary combustion engine, rotary design to convert pressure into rotating motion. It was invented by German engineer Felix Wankel, and desi ...
is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle.
However, its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.
Architecture
![Reuleaux triangle shaped window of Onze-Lieve-Vrouwekerk, Bruges](https://upload.wikimedia.org/wikipedia/commons/4/4a/Reuleaux_triangle_shaped_window_of_Onze-Lieve-Vrouwekerk%2C_Bruges.jpg)
In
Gothic architecture
Gothic architecture (or pointed architecture) is an architectural style that was prevalent in Europe from the late 12th to the 16th century, during the High and Late Middle Ages, surviving into the 17th and 18th centuries in some areas. It e ...
, beginning in the late 13th century or early 14th century,
[.] the Reuleaux triangle became one of several curvilinear forms frequently used for windows, window
tracery
Tracery is an architecture, architectural device by which windows (or screens, panels, and vaults) are divided into sections of various proportions by stone ''bars'' or ''ribs'' of Molding (decorative), moulding. Most commonly, it refers to the s ...
, and other architectural decorations.
For instance, in
English Gothic architecture
English Gothic is an architectural style that flourished from the late 12th until the mid-17th century. The style was most prominently used in the construction of cathedrals and churches. Gothic architecture's defining features are pointed ar ...
, this shape was associated with the decorated period, both in its geometric style of 1250–1290 and continuing into its curvilinear style of 1290–1350.
It also appears in some of the windows of the
Milan Cathedral
Milan Cathedral ( it, Duomo di Milano ; lmo, Domm de Milan ), or Metropolitan Cathedral-Basilica of the Nativity of Saint Mary ( it, Basilica cattedrale metropolitana di Santa Maria Nascente, links=no), is the cathedral church of Milan, Lombard ...
. In this context, the shape is more frequently called a spherical triangle,
but the more usual mathematical meaning of a
spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
is a triangle on the surface of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(a shape also commonly used in architecture as a
pendentive
In architecture, a pendentive is a constructional device permitting the placing of a circular dome over a square room or of an elliptical dome over a rectangular room. The pendentives, which are triangular segments of a sphere, taper to points ...
). In its use in Gothic church architecture, the three-cornered shape of the Reuleaux triangle may be seen both as a symbol of the
Trinity
The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God the F ...
, and as "an act of opposition to the form of the circle".
The Reuleaux triangle has also been used in other styles of architecture. For instance,
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
sketched this shape as the plan for a fortification.
[.] Modern buildings that have been claimed to use a Reuleaux triangle shaped floorplan include the
MIT Kresge Auditorium, the
Kölntriangle, the
Donauturm
The Donauturm ( en, Danube Tower) is a tower in Vienna, the tallest structure in Austria at , and is the 68th tallest tower in the world. Opening in April 1964, the tower is located near the north bank of the Danube River in the district of ...
, the
Torre de Collserola
Torre de Collserola () is a uniquely designed tower located on the Tibidabo hill in the Serra de Collserola, in Barcelona, Catalonia, Spain. It was designed by the architect Sir Norman Foster and by the Spanish civil engineers Julio Martínez Calz ...
, and the
Mercedes-Benz Museum
The Mercedes-Benz Museum is an automobile museum in Stuttgart, Germany. It covers the history of the Mercedes-Benz brand and the brands associated with it. Stuttgart is home to the Mercedes-Benz brand and the international headquarters of the Merc ...
.
[ However in many cases these are merely rounded triangles, with different geometry than the Reuleaux triangle.
]
Mapmaking
Another early application of the Reuleaux triangle, da Vinci's world map from circa 1514, was a world map
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of map projection, projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensiona ...
in which the spherical surface of the earth was divided into eight octants, each flattened into the shape of a Reuleaux triangle.[.][.]
Similar maps also based on the Reuleaux triangle were published by Oronce Finé
Oronce Finé (or Fine; Latin: ''Orontius Finnaeus'' or ''Finaeus''; it, Oronzio Fineo; 20 December 1494 – 8 August 1555) was a French mathematician, cartographer, editor and book illustrator.
Life
Born in Briançon, the son and grandson of p ...
in 1551 and by John Dee
John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, teacher, occultist, and alchemist. He was the court astronomer for, and advisor to, Elizabeth I, and spent much of his time on alchemy, divinatio ...
in 1580.
Other objects
Many guitar pick
A guitar pick (American English) is a plectrum used for guitars. Picks are generally made of one uniform material—such as some kind of plastic (nylon, Delrin, celluloid), rubber, felt, tortoiseshell, wood, metal, glass, tagua, or stone. They ...
s employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Because all three points of the shape are usable, it is easier to orient and wears less quickly compared to a pick with a single tip.[.]
The Reuleaux triangle has been used as the shape for the cross section of a fire hydrant
A fire hydrant, waterplug, or firecock (archaic) is a connection point by which firefighters can tap into a water supply. It is a component of active fire protection. Underground fire hydrants have been used in Europe and Asia since at least ...
valve nut. The constant width of this shape makes it difficult to open the fire hydrant using standard parallel-jawed wrenches; instead, a wrench with a special shape is needed. This property allows the fire hydrants to be opened only by firefighters (who have the special wrench) and not by other people trying to use the hydrant as a source of water for other activities.
Following a suggestion of ,[.] the antennae of the Submillimeter Array, a radio-wave astronomical observatory on Mauna Kea
Mauna Kea ( or ; ; abbreviation for ''Mauna a Wākea''); is a dormant volcano on the island of Hawaii. Its peak is above sea level, making it the highest point in the state of Hawaii and second-highest peak of an island on Earth. The peak is ...
in Hawaii
Hawaii ( ; haw, Hawaii or ) is a state in the Western United States, located in the Pacific Ocean about from the U.S. mainland. It is the only U.S. state outside North America, the only state that is an archipelago, and the only stat ...
, are arranged on four nested Reuleaux triangles.[.][.] Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the signal from the array. The antennae may be moved from one Reuleaux triangle to another for different observations, according to the desired angular resolution of each observation. The precise placement of the antennae on these Reuleaux triangles was optimized using a neural network
A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
. In some places the constructed observatory departs from the preferred Reuleaux triangle shape because that shape was not possible within the given site.
Signs and logos
The shield shapes used for many signs and corporate logos feature rounded triangles. However, only some of these are Reuleaux triangles.
The corporate logo of Petrofina
Petrofina was a Belgian oil company. It merged with Total in 1999 to form TotalFina, which after subsequent mergers has changed its name back to Total. In the United States, Fina's former refining and marketing operations are now owned by De ...
(Fina), a Belgian oil company with major operations in Europe, North America and Africa, used a Reuleaux triangle with the Fina name from 1950 until Petrofina's merger with ''Total S.A.'' (today TotalEnergies
TotalEnergies SE is a French Multinational corporation, multinational integrated energy and List of oil exploration and production companies, petroleum company founded in 1924 and one of the seven Big Oil, supermajor oil companies. Its businesses ...
) in 2000.
Another corporate logo framed in the Reuleaux triangle, the south-pointing compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
of Bavaria Brewery, was part of a makeover by design company Total Identity that won the SAN 2010 Advertiser of the Year award. The Reuleaux triangle is also used in the logo of Colorado School of Mines
The Colorado School of Mines, informally called Mines, is a public research university in Golden, Colorado, founded in 1874. The school offers both undergraduate and graduate degrees in engineering, science, and mathematics, with a focus on ener ...
.
In the United States, the National Trails System
The National Trails System is a series of trails in the United States designated "to promote the preservation of, public access to, travel within, and enjoyment and appreciation of the open-air, outdoor areas and historic resources of the Nati ...
and United States Bicycle Route System
The United States Bicycle Route System (abbreviated USBRS) is the national cycling route network of the United States. It consists of interstate long-distance cycling routes that use multiple types of bicycling infrastructure, including off-road ...
both mark routes with Reuleaux triangles on signage.
In nature
According to Plateau's laws
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws.
Laws f ...
, the circular arcs in two-dimensional soap bubble
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact wi ...
clusters meet at 120° angles, the same angle found at the corners of a Reuleaux triangle. Based on this fact, it is possible to construct clusters in which some of the bubbles take the form of a Reuleaux triangle.[.]
The shape was first isolated in crystal form in 2014 as Reuleaux triangle disks. Basic bismuth nitrate
Bismuth(III) nitrate is a salt composed of bismuth in its cationic +3 oxidation state and nitrate anions. The most common solid form is the pentahydrate. It is used in the synthesis of other bismuth compounds. It is available commercially. It is th ...
disks with the Reuleaux triangle shape were formed from the hydrolysis
Hydrolysis (; ) is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution reaction, substitution, elimination reaction, elimination, and solvation reactions in which water ...
and precipitation
In meteorology, precipitation is any product of the condensation of atmospheric water vapor that falls under gravitational pull from clouds. The main forms of precipitation include drizzle, rain, sleet, snow, ice pellets, graupel and hail. ...
of bismuth nitrate in an ethanol–water system in the presence of 2,3-bis(2-pyridyl)pyrazine.
Generalizations
Triangular curves of constant width with smooth rather than sharp corners may be obtained as the locus of points at a fixed distance from the Reuleaux triangle. Other generalizations of the Reuleaux triangle include surfaces in three dimensions, curves of constant width with more than three sides, and the Yanmouti sets which provide extreme examples of an inequality between width, diameter, and inradius.
Three-dimensional version
The intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
with side length ''s'' is called the Reuleaux tetrahedron
The Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three verti ...
, but its surface is not a surface of constant width
In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the wi ...
.[ Weber also ha]
films of both types of Meissner body rotating
as well a
It can, however, be made into a surface of constant width, called Meissner's tetrahedron, by replacing three of its edge arcs by curved surfaces, the surfaces of rotation of a circular arc. Alternatively, the surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
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.[.]
Reuleaux polygons
The Reuleaux triangle can be generalized to regular or irregular polygons with an odd number of sides, yielding a Reuleaux polygon
In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. These shapes are named after their prototypical example, the Reuleaux triangle, which in turn, is named after 19th-century German engineer ...
, a curve of constant width formed from circular arcs of constant radius. The constant width of these shapes allows their use as coins that can be used in coin-operated machines. Although coins of this type in general circulation usually have more than three sides, a Reuleaux triangle has been used for a commemorative coin from Bermuda
)
, anthem = "God Save the King"
, song_type = National song
, song = " Hail to Bermuda"
, image_map =
, map_caption =
, image_map2 =
, mapsize2 =
, map_caption2 =
, subdivision_type = Sovereign state
, subdivision_name =
, e ...
.
Similar methods can be used to enclose an arbitrary simple polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise ...
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
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
, and can be drawn with compass and straightedge. Although the Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes with an even number of circular-arc sides of varying radii.
Yanmouti sets
The Yanmouti sets are defined as the convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
s 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'' (the radius of the largest possible circle contained in the shape) obeys the inequality
:
and this inequality becomes an equality for the Yanmouti sets, showing that it cannot be improved.
Related figures
In the classical presentation of a three-set Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between set (mathematics), sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple ...
as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle. The same three circles form one of the standard drawings of the Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
, three mutually linked rings that cannot, however, be realized as geometric circles. Parts of these same circles are used to form the triquetra
The triquetra ( ; from the Latin adjective ''triquetrus'' "three-cornered") is a triangular figure composed of three interlaced arcs, or (equivalently) three overlapping '' vesicae piscis'' lens shapes. It is used as an ornamental design in ar ...
, a figure of three overlapping semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of ...
s (each two of which form a vesica piscis
The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. In Latin, "vesica piscis" litera ...
symbol) that again has a Reuleaux triangle at its center; 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
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kno ...
.
Relatives of the Reuleaux triangle arise in the problem of finding the minimum perimeter shape that encloses a fixed amount of area and includes three specified points in the plane. For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii. In particular, when the three points are equidistant from each other and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure.
Circular triangle
In geometry, a circular triangle is a triangle with circular Arc (geometry), arc edge (geometry), edges.
Construction
A convex circular triangle may be constructed by three circles intersecting each other and represents the area of intersectio ...
s are triangles with circular-arc edges, including the Reuleaux triangle as well as other shapes.
The deltoid curve
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the insid ...
is another 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. Other planar shapes with three curved sides include the arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...
, which is formed from three semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of ...
s with collinear endpoints, and the Bézier triangle A Bézier triangle is a special type of Bézier surface that is created by (linear, quadratic, cubic or higher degree) interpolation of control points.
''n''th-order Bézier triangle
A general ''n''th-order Bézier triangle has (''n'' +1)('' ...
.
The Reuleaux triangle may also be interpreted as the conformal image of a spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
with 120° angles. This spherical triangle is one of the Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
s (with parameters 3/2, 3/2, 3/2), triangles bounded by great-circle arcs on the surface of a sphere that can tile the sphere by reflection.[.]
References
External links
*{{mathworld, id=ReuleauxTriangle, title=Reuleaux Triangle, mode=cs2
Piecewise-circular curves
Types of triangles
Constant width