Is it possible to drill a square hole?

macfel@realtime.net (Mac Ernest) wrote:

>The word drill implies circular motion.  If so, it is not possible to
> drill a square hole.
>If recprocating action ie. up and down motion is involved then a
>square hole is possible.

It is possible to drill a square hole with no reciprocating motion.
There are a set of curves known as curves of constant "diameter", of
which the circle is merely the most well known.  If, for instance,
wooden rollers are constructed whose cross-sections are curves of
constant diameter then a board placed on these rollers will roll
"smoothly" (assuming we made them all of equal diameter of course).

This is intuitively obvious when the cross-sections are circles (i.e.
cylindrical rollers) but seems impossible for any other cross-section.
Nevertheless such curves do exist.  The circle is the only such curve
with what we would call a "centre" and thus possessing a constant
"radius".  For this reason my description above cites rollers rather
than wheels as it is not possible to mount other curves of constant
diameter on an axle (without losing your smooth ride).

One such curve of constant diameter can be constructed by taking an
equilateral triangle and placing the point of a compass on each vertex
and inscribing the arc of a circle across the opposing side (you get an
equilateral triangle that looks "bloated").

In addition to the suprisingly smooth ride this latter shape gives as a
roller, if you rotate this shape around a point inside its boundary (I
am afraid I cannot remember exactly where the point is) the locus it
"covers" is a square (with ever so slightly rounded corners).  Thus if
an appropriate drill bit and an off-centre chuck is used you can drill
"square" holes.

I think this has actually had a commercial application in the "real"
world at some point but the exact details escape me.  I myself have
made rollers with cross-sections that are various such curves and it is
extremely weird to watch a board rolling smoothly on top of them.

-- Andrew McDonald <andrew@mitswa.com.au>

Take an equilateral triangle and replace each of the flat sides by an
arc of a circle passing through two vertices, with its center at the
third.  The resulting shape, a "puffed-up" triangle, has the property
that if you made some rollers having this shape for their
cross-section, you could roll a plank smoothly on the rollers, even
though each roller would have its center of mass bobbing up and down as
it moved.  The top surface of each roller would always remain at a
constant height off the ground.

If you place this shape inside a square box, it will rotate like the
rotor of a Wankel engine (this is not an accident -- it's precisely the
reason that a Wankel engine looks like that).  Although the center of
rotation will wobble a bit inside the box, the edges of the "cylinder"
always remain in contact with all four sides of the box.

To drill square holes, you use a drill bit with the Wankel-rotor shape,
except a little is cut away near the vertices in order to make sharp
edges.  The bit rotates inside a square casing, and cuts away nearly
all the wood inside the square, leaving just a tiny amount at the
corners.  By placing sharp edges at the corners of the square, it's
possible to drill perfectly square holes.

-- Dave Seaman <ags@seaman.cc.purdue.edu>, February 1996

[ Editor: see also geometry/manhole.cover for the Reuleaux triangle. ]