Drilling Square Hole

The film Cir­cu­lar Reuleaux tri­an­gle tells about the fig­ures of con­stant width. The Reuleuaux tri­an­gle —, the sim­plest fig­ure of con­stant width will help us to drill square holes. If one moves the cen­ter of the this «tri­an­gle» along some tra­jec­tory, its ver­tices will draw almost a square and itself will cover the area inside this fig­ure.

The bor­ders of the obtained fig­ure, except small angu­lar pieces, will be straight seg­ments! And if one extends the seg­ments adding the cor­ners, we get an exact square.

To achieve what we described above, the cen­ter of the Reuleaux tri­an­gle should move along the tra­jec­tory that con­sists of four equal patched arcs of ellipses. The cen­ters of the ellipses are placed in the ver­tices of the square, the semi-axes form­ing an angle of $45^\circ$ with the sides of the square and equal $k\cdot(1+1/\sqrt{3})/2$ and $k\cdot(1-1/\sqrt{3})/2$ where $k$ is the side of the square.

The curves round­ing the cor­ners are also arcs of ellipses cen­tered in the ver­tices of the square, semi-axes form­ing an angle of $45^\circ$ with the sides of the square and equal $k\cdot(\sqrt{3}+1)/2$ and $k\cdot(\sqrt{3}-1)/2$.

The area of the non-cov­ered cor­ners forms only around 2 per­cent of the area of the square!

Now if one makes a Reuleaux tri­an­gle shaped drill, one will be able to drill square holes with slightly rounded cor­ners but absolutely straight sides!

What is left is to make such a drill… Well, it's not hard to make the drill, its sec­tion should only remind of Reuleaux tri­an­gle and its cut­ting edges should coin­cide with its ver­tices!

The point is that its cen­ter's tra­jec­tory should con­sist of four arcs of ellipses, as we men­tioned before. Visu­ally this curve is very close to a cir­cle and is even math­e­mat­i­cally close to it, but it's still not a cir­cle. In addi­tion, all the eccentrics (a cir­cle placed on a cir­cle of dif­fer­ent radius with a shifted cen­ter) used in tech­nics pro­vide cir­cu­lar motion.

In 1914 an eng­lish engi­neer Garry James Watts invents how to orga­nize such a drilling. On a sur­face he places a direct­ing tem­plate with a square cut in which the drill, «put freely into a chuck», moves. A patent for such a chuck had been given to a com­pany that started man­u­fac­tur­ing Watts drills in 1916.

We'll use another known con­struc­tion. Fix the drill to the Reuleaux tri­an­gle placed into a square direct­ing frame. The frame is attached to the drill. It's left to trans­mit the drill chuck rota­tion to the Reuleaux tri­an­gle.

You should have already seen under the pass­ing trucks what helps us: the drive shaft. his trans­mis­sion is named after Gero­lamo Car­dano.

### Gero­lamo Car­dano 1501—1576

When in 1541 the emperor Charles V tri­umphally entered con­quered Milano, the rec­tor of the doc­tors col­lege Car­dano was walk­ing near the canopy. In response to this honor he sug­gested to sup­ply the royal car­riage with a sus­pen­sion with two shafts, the rolling of which would keep the car­riage hor­i­zon­tal […] To keep jus­tice we have to men­tion that the idea of such a sys­tem goes back to antique times, at least the «Atlantic Codex» by Leonardo da Vinci has a draw­ing of a ship com­pass with a gim­bal sus­pen­sion. Such com­passes became wide­spread in the first half of the XVI cen­tury, appar­ently, with­out Car­dano's influ­ence.

S. G. Gindikin. Sto­ries about physi­cists and math­e­mati­cians.

Now we have every­thing to start drilling. Take a sheet of veneer and… drill a square hole! As we've said before, the sides will be strictly straight and only the cor­ners will be sightly rounded. If needed, on may cor­rect them with a broach file.