This is an interesting geometric figure, one which I discovered in a roundabout way. In a newsgroup, we were having an idiotic discussion (the best kind). The topic was, “Why are manhole covers round?” There were plenty of theories offered, my favorite being, “Because the hole is round.” I know a few things about manholes, having spent time crawling into them. I always knew that I would not have to worry about dropping the cover into the hole. Since it is circular, it needs an opening as wide as its diameter. But the hole, also circular, has a seat with a slightly smaller diameter. A regular polygon would not have that property. Someone mentioned this advantage in the newsgroup. In reply, another person described this figure:
Construct an equilateral triangle. On each vertex, center a compass, and draw an arc the short distance between the other two vertices. The perimeter will be three nonconcentric arcs. This is a reuleaux triangle. It is not a circle, but, like a circle, it has constant width, no matter how it is oriented. It is not difficult to see this property, but you should prove it.
I
made some sketches of the reuleaux and discovered some other interesting
properties. It can roll uphill, in a manner of speaking. Click on the image at left to see a Java simulation of a rolling reuleaux triangle. Notice that as it rolls, its height
is constant, but the height of its centroid changes. If it had mass, the
centroid would be the center of mass. Imagine that it is standing on one
vertex, so that the centroid is at its highest, and imagine that the surface
is very slightly inclined. If it moves forward one sixth turn, the centroid
will fall. So although the surface rises, the reuleaux is actually falling.
It
occurred to me that the figure had constant width and constant height.
It should be possible to inscribe one into a square and to turn it freely.
I did a sketch to simulate this. Click on the image. As it turned within the square, something
looked very familiar about it. It looked a lot like those diagrams for
the Wankel rotary engine. I did a Web search, and, sure enough, that is
the shape used for the rotor in the engine. That was when I found out what
to call it. A guy can learn a lot on the Web.
The square drill bit was not my idea either. Notice that as the reuleaux turns inside the square, its trace nearly fills the entire square. It was noticed that this shape might be used to drill square (actually squarish) holes, and in fact someone did manufacture a drill bit base on this concept. It is not just a simple matter of fitting a bit into a drill though. It required a more complex mechanism. Animate that last applet and watch the motion of the centroid. For every rotation of the reuleaux, the centroid makes three revolutions in the opposite direction, and its path is not circular. Your next assignment is to design a mechanism that would make it work.
The
reuleaux turns inside the square because the square is formed by two pairs
of parallel lines, equally spaced. However, I do not see why they should
have to intersect at right angles. Open it again and deform the square. It tilts over
into the shape of an oblique rhombus. It is still possible to inscribe
a reuleaux and to rotate it. You will see some interesting contortions
in the centroid locus as you change the rhombus.
As a reuleaux is inscribed between a pair of parallel lines and rotates between them, is there ever a time when no vertices touch the lines? One vertex? Two Vertices? All three vertices?
If a vertex of the reuleaux can touch a vertex of the rhombus, what can you say about the acute angle of the rhombus?
The shape of the centroid locus is not a single ellipse. It is comprised of parts from four ellipses. To understand this better, see the Sliding Triangle page. Even without understanding the shape of the centroid locus, you should be able to prove that it has four symmetries. Can you prove this and can you define the symmetries?
When I say that the reuleaux is inscribed in a rhombus, I mean that it touches every side, but crosses none of them. Is it possible to inscribe and rotate a reuleaux in any polygon other than a rhombus?
I
saved the toughest one for last. Is it possible to design a special road
for the reuleaux? I want it to be able to roll along without feeling any
bumps, so the centroid would have to trace a level line as the road rises
and falls. This one is not easy at all. Remember, it does not have a constant
radius, so if the rotational velocity is constant, the horizontal velocity
will vary. I tried parametrizing x and y as functions of
the rotation angle. What I got was an enormous integral which stumped Derive,
Mathematica, and me. Then I created an approximation in Sketchpad. Check
out reulroad.gsp below. The solution is outlined in the
sketch. I'll leave it to you to prove me right (or wrong). By the way,
this problem has nothing to do with the reuleaux's constant width. Sometimes
I just feel like being difficult.
Here are three Geometer's Sketchpad (version 3) files associated with this subject:
reulrhom.gsp
reulroad.gsp
reulroll.gsp
Geometry Junkyard Get more information on the reuleaux triangle and related figures at this site.
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