In 1963, Edward Lorenz (1917-2008), studied convection in the Earth’s atmosphere. As the Navier-Stokes equations that describe fluid dynamics are very difficult to solve, he simplified them drastically. The model he obtained probably has little to do with what really happens in the atmosphere. Read More
Tag: fractals
Chaos II: Vector fields
At the end of the 17th century, Gottfried Wilhelm Leibniz (1646-1716) and Isaac Newton (1643-1727), independently one from the other, invented a brilliant mathematical tool: infinitesimal calculus or differential and integral calculus. Read More
The Science Behind the Butterfly Effect
The Lorenz Attractor in Processing
Evolution, Dynamical Systems and Markov Chains
In this post we present a high level introduction to evolution and to how we can use mathematical tools such as dynamical systems and Markov chains to model it. Questions about evolution then translate to questions about dynamical systems and Markov chains – some are easy to answer while others point to gaping holes in current techniques in algorithms and optimization.
Dissecting Hypercubes with Pascal’s Triangle
What does the inside of a tesseract look like? Pascal’s Triangle can tell us.
Splitting Rent with Triangles
You can find out how to fairly divide rent between three different people even when you don’t know the third person’s preferences! Find out how with Sperner’s Lemma.
Times Tables, Mandelbrot and the Heart of Mathematics
The good old times tables lead a very exciting secret life involving the infamous Mandelbrot set, the ubiquitous cardioid and a myriad of hidden beautiful patterns. Time for the Mathologer to go on a serious fact-finding mission.
Lecture 14: Divide and Conquer Recurrences
Lecture 13: Sums and Asymptotics
Lecture 3: Strong Induction
Lecture 2: Induction
A Computational Introduction to Number Theory and Algebra
The mathematical material covered includes the basics of number theory (including unique factorization, congruences, the distribution of primes, and quadratic reciprocity) and of abstract algebra (including groups, rings, fields, and vector spaces). It also includes an introduction to discrete probability theory—this material is needed to properly treat the topics of probabilistic algorithms and cryptographic applications.
The Nature of Code: Simulating Natural Systems with Processing
In this post, there is a playlist of video lectures that supplement the book.
How can we capture the unpredictable evolutionary and emergent properties of nature in software? How can understanding the mathematical principles behind our physical world help us to create digital worlds? This book focuses on a range of programming strategies and techniques behind computer simulations of natural systems, from elementary concepts in mathematics and physics to more advanced algorithms that enable sophisticated visual results.
Read The Book