Chaos theory and geometry: can they predict our world? – with Tim Palmer
The Royal Institution・2 minutes read
Chaos theory explains complex motions in simple systems through irregular oscillations, linking fractal geometry to modern mathematics. Predictability in chaotic systems depends on initial conditions, with practical applications in daily life and technological advancements.
Insights
- Chaos theory explains how simple systems can display complex and unpredictable behavior, as seen in pendulum demonstrations and modeled by Lorenz's equations using fractal geometries.
- Fractal geometry, linked to mathematicians like Godel and Turing, bridges classical and modern mathematics, showcasing its importance in understanding chaotic systems and uncomputable problems.
- The predictability of chaotic systems depends on initial conditions, with some scenarios leading to explosive uncertainty while others are more predictable, highlighting the relevance of chaos theory in practical applications and technological advancements.
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Recent questions
What is chaos theory?
Chaos theory explains complex motions in simple systems.
How does fractal geometry differ from Euclidean geometry?
Fractal geometry is never boring when zoomed in.
How does chaos theory impact predictability in systems?
Predictability in chaotic systems depends on initial conditions.
What is the significance of fractal geometry in mathematics?
Fractal geometry is crucial in representing uncomputable problems.
How does chaos theory relate to real-world weather prediction?
Chaos theory aids in applying the Lorenz attractor to weather forecasts.
Related videos
Summary
00:00
"Chaos Theory: Unpredictable Systems in Physics"
- Chaos theory is a significant theory in 20th-century physics, explaining how simple systems can exhibit complex and unpredictable motions.
- Demonstrations with pendulums show the irregular and unpredictable oscillations that can occur in seemingly simple systems.
- Ed Lorenz's mathematical equations, based on differential calculus, were used to model weather and demonstrate chaotic behavior.
- Fractal geometries, unlike Euclidean geometries, never get boring when zoomed in, providing a bridge between classical and modern mathematics.
- Fractal geometry is linked to the work of 20th-century mathematicians like Kurt Godel, Alan Turing, and Andrew Wiles, showcasing its importance.
- Chaos theory is considered one of the three great theories of 20th-century physics, alongside quantum mechanics and relativity.
- The geometry of chaos may hold the key to unifying quantum mechanics and general relativity, a longstanding challenge in physics.
- Kurt Godel's work on mathematical truths, Alan Turing's proof on unsolvable problems, and Andrew Wiles' solution to Fermat's Last Theorem are significant in mathematics.
- State space, representing the state of a system, is crucial in understanding complex systems like those described by Lorenz's equations.
- Lorenz's equations describe the evolution of a system's state in a complex and unpredictable manner, showcasing the essence of chaos theory.
16:10
Fractal Geometry and Chaos Theory: Unpredictable Systems
- In the early 1960s, Lorenz discovered that the state of a system, when carried on indefinitely, did not fill the entire state space but remained within a specific subset with a butterfly-like structure.
- Lorenz realized this subset was not a solid or two-dimensional object but a fractal, consisting of an infinite complex of surfaces extremely close to each other.
- The fractal structure was formed by continually stretching and folding surfaces in state space, creating a Cantor Set, a geometric construct with no size but containing an infinite number of points.
- Cantor Set's size perception varied depending on whether it was viewed from the inside or outside, similar to the TARDIS concept from Doctor Who.
- The link between fractal geometry and mathematical problems like Hilbert's Decision Problem was explored, showing how uncomputable problems could be represented as properties of fractals.
- P-adic numbers, a type of number system, were crucial in understanding fractal geometry and played a role in Andrew Wiles' proof of Fermat's Last Theorem.
- Chaos theory, focusing on chaotic systems' unpredictability, revealed that predictability in chaotic systems depended on the initial conditions, with some states being highly predictable and others unpredictable.
- Different scenarios illustrated how initial conditions affected predictability in chaotic systems, with some leading to explosive uncertainty and others showing more predictable outcomes.
- To determine the predictability of a system, running multiple forecasts with slightly varied initial conditions could help identify whether the system was predictable or unpredictable.
- Understanding the predictability of chaotic systems was crucial for practical applications, as it could impact daily life and technological advancements, showcasing the relevance of chaos theory in modern society.
32:33
"Ensemble Forecasts: Revolutionizing Weather Prediction and Climate Change"
- The speaker worked with Roberta Wheat at the European Center for medium-range weather forecasts on applying the Lorenz attract to real-world weather prediction.
- Three examples are presented to illustrate different phases of the Lorenz attract in relation to extreme weather and tropical cyclones.
- The first example is a forecast a week ahead of a cyclone named Cider, showing a high probability of hitting Bangladesh.
- The second example is Hurricane Katrina, where a week before landfall, there was a possibility of hitting New Orleans, but it wasn't the most likely scenario.
- The third example is Hurricane Nadine, where the path was uncertain, leading to the development of Anticipatory Action in disaster preparedness.
- Anticipatory Action involves sending supplies ahead of time based on objective estimates of probability from ensemble forecasts.
- Ensemble forecasts have revolutionized humanitarian agencies' ability to provide emergency supplies efficiently.
- Climate change is explained using a magnetic pendulum analogy, showing how emissions of carbon dioxide alter weather statistics predictably.
- The speaker discusses the Nobel Prize awarded to Suki Manabe for climate modeling and the importance of ensemble prediction models in various fields.
- The talk concludes with a discussion on Bell experiments in quantum mechanics, challenging Einstein's views on determinism and showcasing the concept of entangled particles.
49:58
"Correlations, Counterfactuals, and Fractal Attractors"
- Bell's equation for correlation coefficients shows that the sum and difference of correlations exceeding a certain value indicates the presence of spooky action at a distance.
- Traditional deterministic models cannot satisfy this inequality without the occurrence of spooky action at a distance, where Alice's orientation affects Bob's measurement outcome.
- Clauser, Aspect, and Zeilinger's experimental results confirm the robustness of correlations exceeding two.
- Interpretation of these results leads to the conclusion that the world may not be deterministic, hinting at fuzzy, stochastic, or noisy realities.
- Counterfactual questions about measuring particles with different settings on subsequent days are crucial in understanding Bell's experiment.
- The inability to measure particles on different days due to their destruction raises questions about counterfactual worlds and their consistency with the laws of physics.
- Physicists often use counterfactuals to predict outcomes, as seen in examples like projectile motion calculations.
- Chaos in gravitationally bound systems, as demonstrated by Poincaré, supports the idea of the universe being a chaotic system evolving on a fractal attractor.
- The concept of fractal attractors challenges traditional reductionism in physics, suggesting a need to understand the laws of physics on a larger scale rather than focusing solely on smaller structures.
- The interpretation of Bell's experiment through fractal geometry implies a shift towards a top-down approach in understanding the laws of physics on a larger scale, potentially altering perspectives on unifying quantum mechanics and gravity.
01:07:21
Quantum Gravity: Deterministic yet Non-Computable Theory
- Mentioned Nobel Prize winners include Suki Manabe in 2021 and Roger Penrose in 2020, who emphasized the need for a radical new theory to comprehend quantum non-locality, akin to the revolutionary shift from Newtonian gravity to General Relativity.
- Penrose suggests that the correct theory of quantum gravity could be deterministic yet non-computable, aligning with the concept of fractal attractors, hinting at a potential breakthrough in understanding uncertainty and systems.
