Take a sharpened pencil and look at the tip. Not a vague glance. Really look, under a magnifying glass if you have one. What you’ll find isn’t a perfect cone ending in a mathematical point. It curves. The sides sweep inward in a gentle arc before meeting at the apex, a shape mathematicians call a parabola. Now consider that the same rounded profile appears on a bee’s stinger, the thorn on a rose bush, the tip of an elephant’s tusk, and the stabbing mouthpart of a microscopic copepod. The similarity is, depending on your instinct, either deeply suspicious or deeply interesting.
For decades, the instinctive explanation was evolution. Natural selection, the story went, converged on the parabolic tip because it is mechanically superior (better at penetrating skin, less prone to snapping, distributing force more cleanly through tissue). A 2024 study in PNAS made precisely this argument, attributing the universal shape to evolutionary optimization pressure. It was a tidy hypothesis. The trouble is that a group of physicists at the Technical University of Denmark recently put it to the test with a bag of pencils and a vibrating plate, and found they could grow the exact same shape without evolution doing anything at all.
Kaare Hartvig Jensen, an associate professor at DTU Physics, had been bothered for a while by a nagging detail. “There is a general notion that almost everything in nature exists for a reason,” he says. “A previous study from 2024 argues that the rounded tips of teeth are due to evolution. But if you look at an unused tooth, it does not necessarily have that shape, and if you observe the shape later in the organism’s life, the parabola will emerge.” An unused tooth is sharp, conical, pointed. The parabola comes later. Which suggests, Jensen reasoned, that something else is going on.
That something else is wear. Random, repeated, indiscriminate wear.
Pencils as Proxies
To isolate the effect, Jensen and his colleague John Sebastian sharpened a handful of graphite pencils to conventional conical tips, placed them on a porcelain plate, and set the whole assembly vibrating at fifteen cycles per second. For four and a half hours, the pencils rattled around, colliding with each other at random angles, grinding away tiny flakes of graphite at every impact. The researchers also ran a parallel version of the experiment that required no lab equipment at all: they carried pencils around loose in a pocket and let the jostling of an ordinary day do the work. Photographed at regular intervals, the pencil tips told a consistent story. No matter how sharp they started, all of them converged toward the same rounded form. That form matched the profile found on bee stingers, rose thorns, and copepod mandibles to within the precision of the measurement.
The experiment also tested whether the starting shape mattered. Some pencils were sharpened normally; others were deliberately blunted to a flat truncated cone. After a few hours of collisions, both groups ended up at the same destination. “This points to something more fundamental,” Jensen says. “That random processes in and of themselves can lead to a universal form. The parabola is a stable shape across scales, from a thorn to an elephant’s tusk. It appears to be a coincidence that this shape is also the most effective for biting, stabbing, or tearing. The tips are thus not necessarily designed perfectly from the start; they become so through random wear.”
The mathematics isn’t especially exotic. Curvature-driven erosion means that sharp protrusions wear fastest, because they make contact most often. It’s the same principle that rounds off a cobblestone. Jensen’s team modelled this as a free boundary problem, and the stable attractor of that equation, the shape every tip inevitably tends toward, is a parabola. The same profile emerges from melting icicles, dissolving cylinders, and wind-eroded rock pinnacles via different physical routes entirely. That convergence across such different processes is the give-away: the shape may simply be what physics does to pointed things over time.
None of this means evolution plays no role. It almost certainly does, at least in setting the initial geometry of a stinger and the material properties of the tissue. But the study, published in PNAS, raises a pointed question about how much of what we take to be evolutionary optimization is really just physics operating on whatever shape an organism happened to start with. “We cannot rule out the role of evolution,” Jensen is careful to say, “but random processes are an equally good explanation.” For some biologists, that caveat will carry weight; for others, perhaps less so.
The paper itself offers a few telling counterexamples. Sharks, which depend on sharp teeth for feeding, have evolved polyphyodonty: they carry multiple rows of replacement teeth, fresh and conical beneath protective tissue, ready to roll forward when the front row blunts. The solution to wear, in that evolutionary context, was not to resist it but to keep producing new tips faster than wear could round them off. Copepod mandibles, likewise, begin sharp and become parabolic after use : the parabola is the used form, not the designed one.
Reading the Wear
There is also a practical upshot that has nothing to do with biology. Jensen points out that wear leaves a signature in the geometry of a pointed object, and if you have the right equations you can read that signature backward. “By analyzing wear patterns and the degree of rounding, we can begin to quantify the use… not just observe it,” he says. “We can say something about how much a tool has been used and what it has been used for.” For archaeologists trying to reconstruct how stone-age tools were actually deployed, the geometry of a worn tip could become something like a log file.
Jensen draws a broader lesson from the pine cone: when wet, it stays closed; when dry, it springs open and disperses its seeds. No genes are directing that movement, no proteins contracting. It’s just the physics of differential moisture response in fibrous tissue. “It shows that not everything in nature is controlled by genes and proteins,” he says. “Sometimes random and physical processes are both crucial and functional.” The parabolic stinger tip may be the same kind of thing: not a masterwork of selection but a side effect of usage, arriving at an efficient form by a route that had nothing to do with efficiency in mind.
Whether that distinction matters to the bee deploying its stinger, or to whoever it is being deployed against, is another question entirely.
Frequently Asked Questions
Why do teeth and thorns have a curved, rounded tip rather than a perfect point?
Most people assume that evolution shaped natural stingers into their familiar curved form because it’s the most efficient shape for piercing soft tissue. New research suggests a simpler explanation: random wear. Repeated collisions and abrasion over an organism’s lifetime gradually erode any initial sharp tip toward the same stable parabolic profile, regardless of what shape it started with.
Could a pencil really model how a shark tooth wears down?
Surprisingly well, according to DTU physicists who shook graphite pencils against each other on a vibrating plate for hours and found the tips converged on exactly the same power-law profile observed in biological stingers. They also tested a tip machined from actual bull horn material and got the same result. The key is that the underlying physics of curvature-driven erosion operates the same way regardless of what the material is made of.
Doesn’t this mean evolution had nothing to do with stinger shapes?
Not exactly. The study argues that wear is an equally plausible explanation for the observed shape, not that evolution is irrelevant. Evolution almost certainly influences the initial geometry and material properties of a stinger. What the pencil experiment shows is that you don’t need evolutionary optimization to arrive at the parabolic profile: physics will get you there anyway, given enough use.
What could this mean for archaeology?
Potentially quite a lot. Because the degree of rounding in a pointed tip reflects how much wear it has undergone, the equations developed in this study could allow researchers to infer how intensively a stone or bone tool was used, and possibly what kind of surface it was used against, turning the geometry of a worn tip into something closer to a usage record.
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