17 September 2026 · Nesin Mathematics Village · Şirince
And why can't even the most powerful AI fully solve it?
First number
I started two double pendulums one trillionth of a radian apart. Same masses, same lengths, same gravity, same equations — no randomness at all.
A difference smaller than one ten-billionth of a degree. The result: within ~20 seconds the two pendulums had diverged onto completely different trajectories.
Here is the interesting part — as I improved the precision 1000-fold at each step, I measured how much extra predictable time I gained:
Every 1000-fold gain in precision buys you only a few extra seconds — a constant amount, not an exponential one. You do not tear the wall down; for every 1000-fold effort you push it forward by just a few seconds.
The reason is not an engineering shortcoming: in chaotic systems uncertainty grows exponentially, while the benefit of precision is only logarithmic. And initial conditions can never be measured with infinite precision — no matter how much computing power you have.
This is not an engineering problem. It is a theorem.
What is a theorem is the exponential nature of the divergence (the Lyapunov exponent); the logarithmic payoff above follows directly from it.
One question, three acts
How do we step numbers forward in time — and what does that have to do with my antenna and electromagnetics research?
Does knowing the rules exactly mean knowing the future? (This is exactly where the pendulum experiment lands.)
The most powerful models (GraphCast, Pangu-Weather...) genuinely sped forecasting up. But they did not remove the wall — and we will see together why they cannot.
This is most likely the summary of a far more general lesson: some systems, however well they are understood, are inherently unpredictable.
Second number
The Feigenbaum constant. Systems whose physics have nothing in common slide into chaos at exactly this rate:
A water droplet, an electric current, liquid helium, population growth — none of them resemble one another. Mitchell Feigenbaum found this constant in 1975, not on a supercomputer, but on an HP-65 calculator he carried in his pocket.
So chaos is not an accident of particular systems — it is a structural feature of nature. The atmosphere is no exception.
Why I am giving this talk
Numerical modelling is not an abstract subject for me — I live inside it, in the middle of my research on antenna design and electromagnetic simulation (FDTD). In this talk I will show that the mathematics which carves a weather model onto a grid is exactly the same mathematics I use in my own lab.
Come, let's meet.
We start from scratch. No prior knowledge required — only curiosity.