A Lorenz section (cross-section through the folded surfaces of the Lorenz attractor) reveals a Cantor set, discovered by Georg Cantor in the 19th century, which is the prototype fractal.

A Cantor set is constructed by starting with a line segment, removing the middle third, then removing the middle third from each remaining segment, and repeating this process infinitely. What remains is the Cantor set.

The Lorenz equations, discovered by meteorologist Ed Lorenz in the early 1960s, describe a simple model of weather using three coupled non-linear differential equations, yet generate a geometry (now called fractal or attractor geometry) that would have been completely alien to Newton despite Newton's invention of differential calculus.

Fractal geometries never get boring when zoomed into at any scale—they retain structure at all magnifications, unlike Euclidean geometries (spheres, ellipses, hyperboloids) which become flat and featureless under sufficient zoom.

Fractal geometry provides a bridge between the classical mathematical and physical world of Newton and the greatest discoveries in 20th century mathematics by Gödel, Turing, and Wiles.

Chaos theory is one of the three great theories of 20th century physics, alongside quantum mechanics and relativity, because it describes how simple systems can exhibit extremely complex and unpredictable motion.

State space is an abstract multi-dimensional space in which each dimension represents a parameter or variable of a system, and a point in state space represents a complete state of that system.

The Cantor set has measure zero (no size) from outside measurement, but contains infinitely many points—as many points as the original full line—making it paradoxical like the Doctor Who TARDIS: small on the outside, palatial on the inside.

Syukuro Manabe won the 2021 Nobel Prize in Physics for developing one of the first climate models for studying climate change. Palmer expresses sadness that Lorenz died around 2008 without winning the Nobel Prize, which he 'certainly should have' won.

Cantor's contemporaries thought he was insane for proposing fractals, and his ideas were not recognized in his lifetime, sending Cantor into depression. However, later mathematicians realized how brilliant Cantor's work was.

For Hurricane Katrina, ensemble forecasts about a week before landfall showed multiple possible tracks with roughly equal probability; New Orleans was one possibility among several, so a disaster preparedness director would not have invested all resources into preparing New Orleans specifically.

Kurt Gödel proved in the 1930s that the set of mathematical truths is much larger than the set of things provable by algorithms, meaning mathematical truth exceeds what computers can prove.

In proving his result about the halting problem, Turing had to define what a computer was; he devised what is now called a universal Turing machine, which is the concept of a general-purpose computer.

Andrew Wiles proved Fermat's Last Theorem in the 1990s: the equation x^n + y^n = z^n has no solution in positive integers for any integer n greater than 2.

The Lorenz equations describe how the state of the system evolves over time—as if the state moves through state space, continually changing its position like trousers magically changing their waist size or length.

Andrew Wiles's proof of Fermat's Last Theorem relied on p-adic numbers, a number system unfamiliar to most physicists and even most mathematicians, which are designed to describe fractal geometry rather than Euclidean geometry.

Quantum mechanics and general relativity both affect our daily lives in practical ways: quantum mechanics underlies all modern electronics, and general relativity provides essential corrections to Newtonian gravity needed for GPS to function accurately.

Chaotic systems are often described as unpredictable, but this characterization is misleading; unpredictability is only about long-term behavior. Interestingly, the predictability of a chaotic system depends critically on where you start—some initial states are very predictable while others are very unpredictable.

Palmer and colleagues implemented ensemble forecast methods at the European Center for Medium-Range Weather Forecasts, applying the Lorenz model to real-world weather prediction for extreme weather events like tropical cyclones and hurricanes.

For cyclone Cider a week before landfall, ensemble forecasts predicted with high probability that it would hit Bangladesh; this high-confidence forecast allowed disaster preparedness agencies to give Bangladesh early warning and prepare effectively.