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9. Contact - got a question about Reuleaux Triangle, or want to leave a comment then check out the sites contact page. Reputable companies have them and respond.
10. Payment - ready to pay for your Reuleaux Triangle, then use your credit card or PayPal! Be aware of companies that don't accept them, there may be genuine reasons but given the huge amount of choice you have when buying online there is no reason at all not to buy via credit card or PayPal.
A
Reuleaux polygon is a curve of constant width - that is, a curve in which all diameters are the same length. The best-known version is the
Reuleaux triangle. Both are named after
Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although it was known before his time.
The Reuleaux triangle is the simplest nontrivial example of a curve of constant width - a curve in which the distance between two opposite parallel tangent lines to its boundary is the same, regardless of the direction of those two parallel lines. (The trivial example would be a circle.)
To construct the Reuleaux triangle, start with an equilateral
triangle. Center a Compass (drafting) at one vertex (geometry) and sweep out the (minor) arc between the other two vertices. Do the same with the compass centered at each of the other vertices. Delete the original triangle. The result is a curve of constant width. Equivalently, given an equilateral triangle
T of side length
s, take the boundary of the
intersection (set theory) of the disks with
radius s centered at the vertices of
T.
By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is {1\over2}(\pi - \sqrt3)d^2, where
d is the constant diameter.
The Reuleaux triangle can be generalized to regular polygons with an odd number of sides. See also the United Kingdom
British Twenty Pence coin and British Fifty Pence coin coins.
Other uses
- Because all of its diameters are the same length, the Reuleaux triangle, along with all other Reuleaux polygons, is an answer to the question "Other than a circle, what shape can you make a manhole cover so that it cannot fall down through the hole?"
- The rotor of the Wankel engine is similar to a Reuleaux triangle. This engine has been used by Japanese car company Mazda.
- A drill bit in the shape of a Reuleaux triangle can drill a hole that is very nearly a perfect square.
- A Reuleaux triangle rolls smoothly and easily, but does not make a good wheel because it does not have a fixed center of rotation. While an object on top of rollers with cross-sections that were Reuleaux triangles would roll smoothly and flatly, an axle attached to wheels shaped like Reuleaux triangles would bounce up and down three times per revolution. This concept was used in a science fiction short story by Poul Anderson titled The Three-Cornered Wheel.
- The existence of Reuleaux polygons is a good demonstration of why you cannot use diameter measurements alone to verify that an object has a circular cross-section.
==Three-dimensional version==The intersection of the balls 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. 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 surfaces of revolution of given constant width.
External link
- Shapes of constant width at cut-the-knot
- Film about the Reuleaux triangle, also showing the Wankel engine and the mechanics of a famous sovietic film projector (self-explaining, with only few Russian words)
A
Reuleaux polygon is a curve of constant width - that is, a curve in which all diameters are the same length. The best-known version is the
Reuleaux triangle. Both are named after
Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although it was known before his time.
The Reuleaux triangle is the simplest nontrivial example of a
curve of constant width - a curve in which the distance between two opposite parallel tangent lines to its boundary is the same, regardless of the direction of those two parallel lines. (The trivial example would be a circle.)
To construct the Reuleaux triangle, start with an equilateral
triangle. Center a Compass (drafting) at one vertex (geometry) and sweep out the (minor) arc between the other two vertices. Do the same with the compass centered at each of the other vertices. Delete the original triangle. The result is a curve of constant width. Equivalently, given an equilateral triangle
T of side length
s, take the boundary of the
intersection (set theory) of the disks with
radius s centered at the vertices of
T.
By the
Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is {1\over2}(\pi - \sqrt3)d^2, where
d is the constant diameter.
The Reuleaux triangle can be generalized to
regular polygons with an odd number of sides. See also the United Kingdom
British Twenty Pence coin and
British Fifty Pence coin coins.
Other uses
- Because all of its diameters are the same length, the Reuleaux triangle, along with all other Reuleaux polygons, is an answer to the question "Other than a circle, what shape can you make a manhole cover so that it cannot fall down through the hole?"
- The rotor of the Wankel engine is similar to a Reuleaux triangle. This engine has been used by Japanese car company Mazda.
- A drill bit in the shape of a Reuleaux triangle can drill a hole that is very nearly a perfect square.
- A Reuleaux triangle rolls smoothly and easily, but does not make a good wheel because it does not have a fixed center of rotation. While an object on top of rollers with cross-sections that were Reuleaux triangles would roll smoothly and flatly, an axle attached to wheels shaped like Reuleaux triangles would bounce up and down three times per revolution. This concept was used in a science fiction short story by Poul Anderson titled The Three-Cornered Wheel.
- The existence of Reuleaux polygons is a good demonstration of why you cannot use diameter measurements alone to verify that an object has a circular cross-section.
==Three-dimensional version==The intersection of the balls 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. 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 surfaces of revolution of given constant width.
External link
- Shapes of constant width at cut-the-knot
- Film about the Reuleaux triangle, also showing the Wankel engine and the mechanics of a famous sovietic film projector (self-explaining, with only few Russian words)
Reuleaux triangle - Wikipedia, the free encyclopedia
A Reuleaux polygon is a curve of constant width - that is, a curve in which all diameters are the same length. The best-known version is the Reuleaux triangle.
Reuleaux Triangle -- from Wolfram MathWorld
A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest ...
K-MODDL > Tutorials > Reuleaux Triangle
If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels. Instead the object is placed on a flat platform that in turn ...
Learning module on Reuleaux triangle
Reuleaux triangle. By Daina Taimina and David W. Henderson. What is this triangle? If an enormously heavy object has to be moved from one spot to another, it may not be practical ...
Reuleaux Triangle
Reuleaux Triangle. This is an interesting geometric figure, one which I discovered in a roundabout way. In a newsgroup, we were having an idiotic discussion (the best kind).
Reuleaux Pentagon in a Hexagon
Reuleaux Pentagon in a Hexagon ... You may well know of the Reuleaux triangle, which is made by forming an arc centred on each corner of an equilateral triangle.
The Maths Site
This puzzle isn't too hard, but it will introduce you to a cool geometrical object -- the Reuleaux triangle (pronounced "Roo-low"). This "triangle" is pictured in the diagram at ...
A Rotating Reuleaux Triangle - Wolfram Demonstrations Project
A Reuleaux triangle is a curve of constant width formed from an equilateral triangle by joining each pair of vertices with a circular arc centered at the third vertex (each radius ...
A Rolling Reuleaux Triangle - Wolfram Demonstrations Project
A Reuleaux triangle is constructed from an equilateral triangle by joining each pair of vertices with a circular arc centered at the third vertex (each radius is equal to the side ...
Student Corner: Reuleaux Triangle
Engineers started to use the Reuleaux Triangle to solve technical problems in different areas of life. The Triangle is a constant width figure based on an equilateral triangle that ...
