What is the reuleaux triangle

Reuleaux triangle is the intersection area of three circles constructed from the vertices of a regular triangle. They have a radius equal to the side of the same triangle. It belongs to the category of simple shapes (like a circle) with constant width. That is, if it is to hold two parallel supporting straight, regardless of the chosen direction, the distance between them will be unchanged at any point regardless of their length.


According to historians, the name is complicated the simple shape gave the German engineer Franz reuleaux, who lived from 1829 to 1905. Many historians agree that he was the discoverer of the properties of this geometric figure. Because it is the first widely-used properties of the reuleaux triangle in its mechanisms.



Franz reuleaux was the first to give rigorous definitions of the terms "kinetic couple," "the kinetic chain". He first showed the possibility of a connection between the foundations of mechanics and design. That is linked the theory and the practical problems of design. That helped to create the mechanisms in the totality of their functionality with visual appeal/aesthetics. Here reuleaux is considered the poet of mechanics. Allowing followers to reconsider existing theories.

Other researchers discoverer of this figure recognize Leonhard Euler (18th century), which even then demonstrated the possibility of creating three circles.

And still others saw the reuleaux triangle in the manuscripts of the genius Leonardo Da Vinci. Manuscripts of this naturalist, with a picture of this "simple" shapes are stored in the Madrid Codex and the Institute of France.



But whoever was the discoverer of this "not simple" triangle is widespread in the modern world.


• Drill Wattsa. In 1914 Harry James Watts invented a unique tool for drilling square holes. This drill, made in the form of a reuleaux Triangle;

• The Wankel Engine. Since 1957, the reuleaux triangle, the German inventor of the Wankel F. created a unique mechanism. Where inside the chamber, a cylindrical shape, along a complex path moves the rotor-piston. Created in the shape of a reuleaux triangle. When it is constant motion, each of its faces in contact with the walls of the chamber, forming three chambers, later called "combustion chambers".

• Claw mechanism of the projectors. The reuleaux triangle inscribed in a square and the double of the parallelogram lie at its core. And he needed to evenly pull the film during the screening at a rate of 18 frames/s without deviations and delays;

• The basis of the Cam mechanism for zig-zag seam in the sewing machine, as well as in German watches of such famous brands as A. Lange & Söhne "Lange 31";

• A plectrum or pick is also nothing like the reuleaux triangle. They are necessary when playing stringed musical instruments.

• In the architecture. The design of the two arcs of the reuleaux triangle forms a pointed arch of the Gothic style. A window in the shape of reuleaux are in Bruges in the Church of the mother of God. As an ornament it is on the window bars of the Swiss municipality of Hauterive and the Cistercian Abbey.



Actually reuleaux is not the discoverer of this figure, while he explored her. In particular, he considered the question of how much contact (kinematic pairs) is necessary to prevent the movement of a plane figure, and the example of a curved triangle, inscribed in a square, showed that three contacts can be sufficient to ensure that the figure is not rotated.


Leonardo da Vinci, manuscript a, fragment of a leaf 15v

Some mathematicians think that first demonstrated the idea of a triangle an equal circular arcs Leonhard Euler in the XVIII century. Nevertheless, such a figure is found before, in the fifteenth century: it was used in his manuscripts, Leonardo da Vinci. The reuleaux triangle is in his manuscripts A and B stored in the Institute of France[10] and in the Madrid Codex.

Approximately in 1514 Leonardo da Vinci created one of the first of its kind map of the world. The surface of the globe it was separated by the equator and two meridians(the angle between the planes of these meridians is equal to 90°) into eight spherical triangles, which was shown on the map plane triangles reuleaux, gathered in four around the poles.

Even earlier, in the thirteenth century, the founders of the Church of our lady in Bruges used the reuleaux triangle as the shape for some Windows

Consequently, invented in the last century, the reuleaux triangle is widely used today. However, his study is not in place. Its properties as features of simple shape, is in constant theoretical and practical study.

It is the reuleaux triangle can help us in drilling square holes. Enough to move the center of this "triangle" on a certain trajectory, and its top will draw almost square, and the boundaries of the shapes, except for small portions at the corners, will be strictly straight! Such that, if continued cuts, thereby adding corners, you get exactly square.The area not swept up the corners is only about 2 percent of the area of the whole square!


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And here's another application :

A Chinese military officer Guan BAIHUA of Qingdao re-inventing the wheel. He has created an unusual bike: instead of round wheels had the rear triangle and a Pentagon on the front.

The inventor is confident that the new model will be popular because to be able to move on this bike requires more effort and, therefore, is to some extent can replace the sports load.

Volunteers tested a novelty, was surprised at how smoothly moves the bike with the new wheels. The fact that the angles of polygons are smoothed. This allows the bike not to jump up and down, as you would expect, says with reference to The Times InoPressa.ru.

In addition, the wheels on the form are the curves of constant length, otherwise known as the "reuleaux polygons" or "round polygons". The outline of these figures is a flat convex curve, the distance between any two parallel supporting straight which is constant and equals the "width" of the curve.

Despite the fact that the new bike did not enjoy commercial success, BAIHUA was not discouraged. Now he is busy creating new social network on the Internet.


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Source: masterok.livejournal.com/?skip=40


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