Circular Reuleaux triangle⁠

Here is Luch-2, the eight-millimeter film projector. It was in every Russian house where cine amateurs shot and looked films.

This cartoon presents geomet­rical notion often studied at math­e­mat­ical circles and its appli­ca­tions in our everyday life.

A wheel... A circle. One of the prop­er­ties of a circle is its constant width. Let's draw two parallel lines and fix the distance between them. Let’s start to rotate them. The curve(the circle in this case) perma­nently touches both lines. This is the defi­n­i­tion that closed curve have constant width.

Wether there are curves various from circles with constant width?

We consider a regular triangle. On each side of the triangle we shall draw an arc of a circle of radius equal to the length of the side. This curve is called "Reuleaux triangle". It turns out that this is a constant width curve. As well as in the case of a circle we shall draw two tangents, we shall fix the distance between them and we shall start to rotate them. Our curve contin­u­ally touches both lines. Indeed, one of the points of contact is always situ­ated at one of the "corners" of Reuleaux triangle, and the other on the oppo­site arc of the circle. There­fore the width is always equal to the radius of the circles, i.e. it is equal to the length of the initial triangle’s side.

In an everyday sense constant width of a curve means that if one make wheels with such profile then a book will roll over them without stir­ring.

However it is impos­sible to make a wheel with such profile since the center of this figure describes a compli­cated line while rolling.

Whether there are another constant width curves? It turns out that there are infi­nitely many of them.

On any regular polygon with odd numbers of vertices it is possible to plot constant width curve according to the scheme that Relo triangle has been constructed. It is neces­sary to draw an arc of a circle joining endpoints of the oppo­site side and of center in each vertex. In England 20-pence coin has the form of a constant width curve constructed on a septangle.

Consid­ered curves do not settle the whole class of curves of constant width. It appears that there exist non-symmetric curves of constant width. We regard arbi­trary collec­tion of inter­secting lines. Then we regard one of the sectors. We shall draw an arc of a circle of center in the point of inter­sec­tion of lines, defining this sector, and random radius. Then we shall consider the next sector and we shall draw a circle of center in the point of inter­sec­tion of lines, bounding this sector. Radius is chosen so that the part of curve, already drawn, can be contin­u­ously extended. We shall proceed further our construc­tion.It turns out that the curve will close and we will obtain a constant width curve. Prove it 

All curves of a given width have equal perime­ters. The circle and Reuleaux triangle stand out from all rest curves of a given width with its extreme prop­er­ties. The circle bounds the largest area and Reuleaux triangle — the least one.

Reuleaux triangle is often studied at math­e­mat­ical circles. It appears that this geomet­rical figure has inter­esting appli­ca­tions in mechanic.

Look, it is Mazda RX-7. Unlike the majority of serial cars it (and also model RX-8) is equipped with rotary engine of Vankel. How is it constructed? It is Reuleaux triangle that is used as a rotor! Rotor sepa­rates a chamber into three parts, each become combus­tion chamber by turns. At first a dark blue air-fuel mixture is injected, then because of move­ment of rotor it is compressed, fired and twists a rotor. A rotary engine is void of some lacks of free-piston engine. For instance, here rota­tion is trans­mitted directly to an axis and it is not neces­sary to use crank­shaft.

And this is claw mech­a­nism. It was used in film projec­tors. Engines give uniform rota­tion of an axis. But for a sharp image, it is neces­sary to pull a film for one frame, to stop it, and then again to pull quickly. And so 18 times per second. The claw mech­a­nism solves this problem. It is based on Reuleaux triangle inscribed into a square and two paral­lel­o­grams, which prevent his devi­a­tions. Indeed, since the lengths of the oppo­site sides are equal, then the middle section, the base, and the side of the square are always parallel to each other. The closer the axis of clamping to the vertex of Reuleaux triangle the closer the figure described by dentical of claw device to a square.

And that is that one would think purely math­e­mat­ical prob­lems find inter­esting appli­ca­tions.