The Geometry of Chaos
Like Duolingo, but for The Geometry of Chaos. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
For the part of you with thirty open tabs that never became anything.
A short one: 24 levels across 3 sections, about 48 minutes end to end, roughly 10 days at five minutes a day. It moves through Finding Order in the Noise, Measuring the Infinite, and The Mechanics of Complexity.
Free forever · No credit card · iPhone & Android
Key ideas in The Geometry of Chaos
- Deterministic systems follow fixed rules but can still be unpredictable
- A positive Lyapunov exponent indicates chaotic divergence
- Small initial differences grow exponentially in chaotic systems
- The distance between trajectories grows exponentially over time
- The distinction between deterministic rules and long-term predictability
- How the Lyapunov exponent quantifies the rate of divergence
- The exponent is derived from the ratio of final to initial separation
- Logarithmic scales are used to linearize exponential growth rates
- The process of calculating the Lyapunov exponent from a sequence of states
- A single time series contains 'echoes' of all other variables it interacted with
- Lagged versions of the data act as proxies for the missing dimensions
- Time lag must be large enough to let the system evolve but small enough to maintain a relationship
- Embedding dimension must be high enough to 'unfold' the geometry
- The practical consequences of choosing reconstruction parameters
- Reconstruction starts with a 1D stream and ends with a geometric object
- The conceptual mechanism of Takens' Embedding Theorem
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives The Geometry of Chaos the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
What does a positive Lyapunov exponent tell us about a system's behavior?
Get it right to open this lesson and 23 more in the app.
Where The Geometry of Chaos takes you
Master the hidden order within seemingly random systems. Learn to predict the unpredictable by exploring the deep mathematical structures of fractals, strange attractors, and nonlinear dynamics.
- 1
Finding Order in the Noise
- The Butterfly Effect and Sensitive Dependence
- Phase Space Reconstruction
- Iterative Maps and the Logistic Equation
- Bifurcation Diagrams and Period Doubling
- Identifying Strange Attractors
- 2
Measuring the Infinite
- Hausdorff Dimension and Self-Similarity
- The Mandelbrot Set and Complex Dynamics
- L-Systems and Algorithmic Botany
- 3
The Mechanics of Complexity
- Lyapunov Exponents and Predictability Horizons
- Ergodic Theory and Statistical Stability
- Hamiltonian Chaos in Celestial Mechanics
- Renormalization Group Methods
3 sections · 12 units · 24 levels. Built to play, not to enroll.
You pick the voice
The Geometry of Chaos is taught in the The Bestie style: your friend who just gets it. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
More Math on Tomo
The Architecture of Mathematics
A comprehensive journey through the abstract structures and rigorous foundations of modern mathematics, from category theory to non-standard analysis.
The Math of Moving Things
Discover how simple numbers and shapes predict exactly how objects fly, fall, and collide in the real world. Turn abstract math into a tool for mastering the physical environment.
Waves, Heat, and Hidden Patterns
Master the art of breaking complex signals into simple waves and solving the equations that describe how heat spreads and strings vibrate. Move beyond basic calculus to see the mathematical harmony in physical systems.
The Hidden Geometry of Your World
Move beyond basic shapes to see the invisible patterns that govern everything from city maps to nature's growth. Learn to predict outcomes and optimize your surroundings using the logic of space and connection.
Algebraic Fluency: The Art of Balance
Move beyond simple arithmetic to master the logic of equations. Learn to manipulate variables, scale relationships, and find the exact point where two paths cross.
Algebra: From Equations to Functions
Bridge the gap between basic arithmetic and advanced mathematical modeling. Learn to manipulate complex expressions, solve systems of constraints, and visualize the hidden curves that define our world.
Start The Geometry of Chaos today.
Download Tomo, search The Geometry of Chaos, and play your first lesson in under a minute.