Aurora, my seven-year-old granddaughter, with a five-stick version of the stick bomb used in the experiments described here. Note the marks on the sticks. She made them in order to make the sticks distinguishable during the measurements. This is her first research project on dynamic systems!
Sometimes research begins with an expensive instrument, a large grant, and a carefully planned experimental program. This time, it began with six wooden sticks and a seven-year-old granddaughter who enjoyed participating in a research project.
It was a simple experiment designed to show how the “Hubbert” behavior, known as the basis of the “Peak Oil” concept, is common in complex systems. A mix of enthusiasm, curiosity, and patience, and the result could be turned into a full-fledged scientific paper, presently in the form of a preprint on Qeios.
Hubbert’s bell-shaped curve
The idea of the stick bomb experiment is about the Hubbert curve. It was developed in the 1950s to describe the production of crude oil from a finite resource. Oil production begins slowly, accelerates, reaches a maximum, and then declines. The resulting curve has the familiar bell shape. It was the origin of the term “Peak Oil,” coined by Colin Campbell, that became popular in the 2000s.
The explanation is straightforward. At the beginning, production is small because the industry is small. As oil extraction creates profits, some of the gains are reinvested in the exploitation of the wells and production grows. Eventually, however, depletion takes over. There may still be plenty of pumps and drilling rigs, but there is less accessible oil left to extract. Profits decline, and with them production. In between these two phases, there has to be a maximum. It is the peak.
This is a qualitative assessment of what happens. But there are rigorous ways to describe it. For simple systems, the Hubbert curve can be described by the Single-Cycle Lotka–Volterra, or SCLV model, developed by Alessandro Lavacchi, Ilaria Perissi, and I. We found it works in many more cases than just the oil industry. It describes very well, for instance, the overexploitaion of fisheries.
A few years ago, Ilaria Perissi and I tested the model with a mechanical experiment based on mousetraps. We were inspired by an idea created first by Sutton in the 1940s, and then shown to the world by Walt Disney in their movie Our Friend, the Atom (1956). The original idea of the mousetrap experiment was to simulate the mechanism of a nuclear explosion, but the mechanism is very general.
So, we built our experiment with 50 mousetraps storing elastic energy and throwing two wooden balls into the air when triggered. The falling balls activated other traps, and the feedback mechanism took over. The resulting pulse followed the expected Hubbert-like kinetics remarkably well.
Fine for mousetraps, but could the same behavior appear in a completely different mechanical device? And here comes the stick bomb.
A bomb made of six sticks
A stick bomb is a structure made by weaving flexible wooden sticks together so that they remain under tension. Once one element is released, the structure rapidly disassembles; it explodes. The sticks jump into the air, sometimes with remarkable enthusiasm. Above, you see a six-stick bomb being “detonated” using a screwdriver.
It is fun, but what can we say in quantitative terms about the kinetics of the explosion? Something that, apparently, nobody ever had tried to measure, at least in the form of the behavior of a single trap (the “cobra wave” of several connected traps has been studied). So, I set out to do that in a hot summer in Florence, with the help and the enthusiasm of a seven-year-old granddaughter who found that detonating stick bombs was a lot of fun.
Together, we set out to perform the experiments. Aurora acted as a competent laboratory assistant, rapidly becoming interested in playing with the apparatus, although sometimes requisitioning the sticks for purposes other than a mathematical analysis of an energy dissipation process. But, among other things, she was responsible for marking the sticks in different ways so that they would be recognizable during the explosion.
So, after several tests and many sticks lost in the nooks and crannies of the furniture of the living room, we managed to record in slow motion eight usable explosions with a Samsung phone.
A Hubbert curve lasting a few milliseconds
We aligned the eight experiments around the point of half depletion and combined the 48 measured release events. Despite the unavoidable experimental uncertainty—and despite the apparatus being assembled on the floor of the living room with the assistance of a child (note Aurora’s feet on the left of the images) —the result was quite clear.
The release rate rises, reaches a maximum, and declines. A logistic derivative gives a good empirical fit. More interestingly, the data can be fitted directly with the Single-Cycle Lotka–Volterra model (details in the article on Qeios).
The sticks still locked together represent the resource stock. In the oil analogy, they are the oil underground. The sticks released from the structure, measured as a function of time, represent the production flow. This is analogous to oil extraction.
The moving or flying sticks can be interpreted as the capital stock generated by the release of the resource. They remain active for a time and then hit the ground, where their energy is dissipated.
The fitted production pulse has a maximum of roughly 130 sticks per second and a width at half maximum of about 40 milliseconds.
Of course, nobody is actually producing 130 sticks every second. The entire stock contains only six sticks. The number describes the instantaneous rate during a very short event, much shorter than one second.
At first, the capital stock is too small to release much of the resource. Then the process becomes autocatalytic: the active collapse generates more active collapse. Eventually, resource depletion dominates. There are plenty of moving sticks, but almost no bound structure remains for them to release.
Production falls. That is the Hubbert mechanism in its purest form.
Oil wells, mousetraps, and wooden sticks
At first sight, oil extraction, mousetrap cascades, and exploding stick structures seem to have little in common.
But at a sufficiently abstract level, all three systems share the same structure:
a finite stock of potential;
an active agent capable of exploiting or releasing it;
positive feedback during the growth phase;
depletion during the declining phase;
eventual dissipation.
The “resource” need not be a material resource in the ordinary sense. It may be oil underground, compressed springs, chemical free energy, elastic strain, unstable connections, or any other form of stored thermodynamic potential.
The Hubbert curve may therefore be much more general than a model of petroleum production. It can be interpreted as a common kinetic pattern produced when stored potential is released through an autocatalytic mechanism.
This does not mean that every collapse follows a Hubbert curve. Systems can fragment, oscillate, stall, or fail asymmetrically. External forcing can alter the shape. Different timescales can generate multiple peaks.
Nevertheless, the humble stick bomb shows that the Hubbert mechanism does not require an economy, an oil industry, or even living predators and prey. Six pieces of wood are enough.
The Seneca Effect
The mechanism that I called the Seneca Effect is somewhat more complex than that of the Hubbert cycle described here. But the basic elements are the same: an energy potential that dissipates itself in steps. This small test is one more demonstration that the universe is fractal and repeats itself on widely different scales. Everything, including humankind, is subjected to the destiny that Seneca saw already 2,000 years ago and expressed as, “growth is slow, but decline is rapid.”
Acknowledgment: My thanks to Aurora Suleimanovic (*), age seven, for marking the sticks, helping assemble the stick structures, triggering the experiments, and periodically requisitioning the scientific equipment for alternative forms of research. Data extraction from the film clips and fitting of the experimental data were performed by ChatGPT 5.6 Sol.
(*) Note: I considered whether it could have been appropriate to list Aurora as a full-fledged co-author, but I thought I would have been stretching her role beyond the capabilities of a seven-year-old child. I don’t know what she will do when she grows up, but if she’ll become a scientist, this first step for her is an auspicious one.
Dear Ugo, another excellent story. Very good! Harald Sverdrup
Nicely done.