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?

Reuleaux, Franz 1829—1905

Reuleaux, Franz — a french sci­en­tist. Was the first (1875) to for­mu­late accu­rately the main prob­lems of struc­ture and kine­mat­ics of mech­a­nisms; was devel­op­ping the prob­lems of design of tech­ni­cal mech­a­nisms.

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.