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A Reuleaux triangle [ʁœlo] is a curved triangle with constant widththe simplest and paper presentation on bermuda triangle known curve of constant width other than the circle. Constant width means that the separation of every two paper presentation on bermuda triangle supporting lines is the same, independent of their orientation, paper presentation on bermuda triangle.
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? Reuleaux triangles have also been called spherical trianglespaper presentation on bermuda triangle, but that term more properly refers to triangles on the curved surface of a sphere.
They are named after Franz Reuleaux[3] 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.
Other applications of the Reuleaux triangle include giving the shape to guitar picksfire hydrant nuts, pencilsand drill bits 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 ° at its corners. By several numerical measures it is the farthest from being centrally symmetric.
It provides the largest constant-width shape avoiding the points of an integer latticeand 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 paper presentation on bermuda triangle Reuleaux rotor.
The Reuleaux triangle paper presentation on bermuda triangle the first of a sequence of Reuleaux polygons whose boundaries are curves of constant width formed from regular polygons 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 intersection of four balls whose centers lie on a regular tetrahedron does not have constant width, but can be modified by paper presentation on bermuda triangle its edges to form the Meissner tetrahedronwhich does.
Alternatively, the surface of revolution of the Reuleaux triangle also has constant width. The Reuleaux triangle may be constructed either directly from three circlesor by rounding the sides of an equilateral triangle. The three-circle construction may be performed 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[7] 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 triangleand 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.
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. The most basic property of the Reuleaux triangle is that it has constant width, meaning that for every pair of parallel supporting lines two lines of the same slope that both touch the shape without crossing through it the two lines have the same Euclidean distance from each other, paper presentation on bermuda triangle, regardless of the orientation of these lines.
The other supporting line paper presentation on bermuda triangle 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, paper presentation on bermuda triangle, and to observe that the Reuleaux triangle has constant width, may have been Leonhard Euler.
By many different measures, the Reuleaux triangle is one of the paper presentation on bermuda triangle extreme curves of constant width. By the Blaschke—Lebesgue theoremthe 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.
The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to °. This is the sharpest possible angle at any vertex of any curve of constant width. 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. Although the Reuleaux triangle has sixfold dihedral symmetrythe same as an equilateral triangleit does not have central symmetry.
The Reuleaux triangle is the least symmetric curve of constant width according to 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 hexagonalalthough the inner one has curved sides.
That is, the maximum ratio of areas on either side of a diameter, paper presentation on bermuda triangle, another measure of asymmetry, is bigger paper presentation on bermuda triangle the Paper presentation on bermuda triangle triangle than for other curves of constant width, paper presentation on bermuda triangle. Among all shapes of constant width that avoid all points of an integer latticethe 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. 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. Among all quadrilateralsthe shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite that can be inscribed into a Reuleaux triangle.
By Barbier's theorem all curves of the same constant width including the Reuleaux triangle have equal perimeters. The radii of the largest inscribed circle of a Reuleaux triangle with width spaper presentation on bermuda triangle, and of the circumscribed circle of the same triangle, are.
respectively; the sum of these radii equals the width of the Reuleaux triangle. More generally, paper presentation on bermuda triangle, 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 of the Reuleaux triangle in the plane remains unproven, but is conjectured to be. which is the density of one possible double lattice packing for these shapes. The best paper presentation on bermuda triangle upper bound on the packing density is approximately 0, paper presentation on bermuda triangle. Any curve of constant width can form a rotor within a squarea 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. 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 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, paper presentation on bermuda triangle.
In connection with the inscribed square problemEggleston 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.
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 paper presentation on bermuda triangle allows for paper presentation on bermuda triangle bit not having a fixed centre of rotation, it can drill a hole that is nearly square. Panasonic 's RULO robotic vacuum cleaner has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms, paper presentation on bermuda triangle.
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 paper presentation on bermuda triangle hexagonal barrels, paper presentation on bermuda triangle. 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 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.
Another class of applications of the Reuleaux triangle involves using it as a part of a mechanical linkage that can convert rotation around a fixed axis into reciprocating motion. With the assistance of the Gustav Voigt company, Reuleaux built approximately models of mechanisms, several of which involved the Reuleaux triangle.
One application of this principle arises in a film projector. 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 is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle. In Gothic architecturebeginning in the late 13th century or early 14th century, [47] the Reuleaux triangle became one of several curvilinear forms frequently used for windows, window traceryand other architectural decorations. 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[51] 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 sketched this shape as the plan for a fortification. Another early application of the Reuleaux triangle, da Vinci's world map from circawas 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.
Similar maps also based on the Reuleaux triangle were published by Oronce Finé in and by John Dee in Many guitar picks employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a paper presentation on bermuda triangle 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 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 by firefighters who have the special wrench but not by other people trying to use the hydrant as a source of water for other activities.
Following a suggestion of Keto[59] the antennae of the Submillimeter Arraya radio-wave astronomical observatory on Mauna Kea in Hawaiiare arranged on four nested Reuleaux triangles. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform of the signal from the array.
In some places the constructed observatory departs from the preferred Reuleaux triangle shape because that shape was not possible within the given site. 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 Finaa Belgian oil company with major operations in Europe, North America and Africa, used a Reuleaux triangle with the Fina name from until Petrofina's merger with Total S. in In the United States, the National Trails System and United States Bicycle Route System both mark routes with Reuleaux triangles on signage. According to Plateau's lawsthe circular arcs in two-dimensional soap bubble clusters meet at ° 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 as Reuleaux triangle disks. 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.
The intersection of four balls of radius s centered at the paper presentation on bermuda triangle of a regular tetrahedron with side length s is called the Reuleaux tetrahedronbut its surface is not a surface of constant width.
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. The Reuleaux triangle can be generalized to regular or irregular polygons with an odd number of sides, yielding a Reuleaux polygona 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, paper presentation on bermuda triangle.
The Bermuda Triangle Mystery Has Been Solved
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