Why does a Reuleaux triangle appear when drilling a hole with a knife

Question

As the season commands, I was blowing out eggs for the kindergarten. In order to have large holes at both ends of the egg I first poke a small hole with a pin and then enlarge the said hole with a pointy knife (by rotating said knife in the hole in the hope of creating a round hole). To my surprise, enlarging the hole in such a fashion ended up giving me (in 7 out of 8 tries) a shape which rather looks like a Reuleaux triangle than a circle.

Now it's obvious that the resulting shape should have constant width. But my [first and vague] question is: are there reasons not to get a circle? (and is the phenomenon confirmed?)

In some sense a "Reuleaux triangle"-like shape is more probable since it has a much smaller symmetry group (so if the medium is not uniform, it might be a more "optimal" solution [to some optimisation problem]). So my [hopefully more precise] question is:

Question: What is a good model for this "drilling process" which could explain a "Reuleaux triangle"-like shape?

Just for details (but I doubt it matters) the knife was only sharp one side and both sides of the knife touch the edge of the hole at all times while the drilling occurs.

Answer 1

I wouldn't say this counts as a "good model for the drilling process", but...

The rouleax triangle has much smaller area than the circle (and although I didnt check, it's probably also smaller than the 5-gon etc). E.g. for diameter $1$:

$$Area(Rouleax) = \frac{\pi - \sqrt{3}}{2} \approx 0.705 < Area(Circle) = \frac{\pi}{4} \approx 0.785$$

This means it's a lesser amount of eggshell to be removed before your knife can make full rotations.

While this isn't a "mechanics" or "process" based explanation, maybe this counts as a "physics" style or "principle of least action / minimum energy" style explanation? :)

Just for Fun postscript: There must be another constraint at work, coz my explanation cannot explain why you don't get even smaller-area (non-convex, non-constant-width) Kakeya sets.

Answer 2

As an experiment, I took a sharp knife and stuck it through a piece of paper. Then I rotated then knife.

At first, the sharp edge of the knife stayed stuck in place while the dull edge carved a circular sector out of the piece of paper, or more precisely cut along a circular arc and pushed a sector-shaped flap of paper aside. When the knife had rotated about $60$ degrees, the sharp edge broke free and started to scrape along the previously intact edge of the original cut, although it curled the paper away slightly rather than cutting it.

After approximately another $60$ degrees, the sharp edge of the knife hit the end of the original cut and the dull edge broke free again and pushed past the sector-shaped flap.

I tried the experiment twice more. The second attempt resulted in something more like the Kakeya set mentioned here, but the third time was much like the first.

The holes I got the first and last time were not Reuleaux triangles. One side was an almost perfect circular arc, another was nearly straight, and the third was hard to identify as any shape because it still had a flap of paper attached. This shows that the experiment with paper was not a perfect model of what happens with an eggshell. But my hypothesis is that you get a roughly triangular shape because of the way the shell breaks under the force of the knife, that is, it tends to keep breaking away where it has already broken until the angle between the knife and shell at the other end of the cut has widened so much that the knife edge is able to break free. At that point you have a roughly triangular shape, and all that remains is for the knife to continue taking off bits of shell, mostly from the middle of each edge of the triangle, until it can turn $360$ degrees.

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Paper obviously is not a great model of the shell of an egg. It seems it would be a particularly poor model of the shell itself, but I was working on the assumption that the shell is still attached to the membrane underneath.

I still would not expect the paper to necessarily reproduce the properties of the combined membrane and shell, although I expect it would model the shell better than a sheet of peanut brittle or a sheet of glass would. And I did not want to sacrifice any actual eggs at that time.

The value of the experiment is not so much that it confirms the final shape, but rather the idea of closer observation of what happens as the knife turns.

I would not be surprised if different material properties produced a hole more like a higher-order Reuleaux polygon.

I suspect that in the case where I got the more Kakeya-set-like figure I had inadvertently applied a force to the blade (rather than just torque) causing the sharp end to slice further into the paper. Unfortunately I was not so observant that time.

If it turns out that the behavior of a knife in eggshell is similar to its behavior in paper, then the question is why the one edge continues to act as a fulcrum until the knife has made such a large angle with the original edge of the hole. I think this might be an engineering or materials science question, difficult to guess by mathematics alone.

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Update: Results for heavy-duty corrugated cardboard from a large box of vegetables.

The cardboard has a distinct bias. The knife is difficult to insert with the blade perpendicular to the corrugation. Rather than the material peeling away toward the side opposite from which the knife was inserted, the cardboard simply crushes in place. Both edges of the knife move at first, then one edge breaks free and sweeps across the triangular bit of cardboard that was obstructing it. The resulting hole is roughly triangular.

The knife is much easier to insert parallel to the corrugation but harder to turn. Again both edges push through the material at first. It seems I have to rotate the knife more than $90$ degrees before one edge breaks free. When it does, the hole ends up roughly square, loosely capturing the knife blade.