A common problem when using Mathematica to derive expressions is similar to a big problem plaguing machine learning algorithms today: It is difficult or impossible to explain the result due to the internal complexity of the black-box which generates it.
Mathematica’s internal algorithms for performing various symbolic computation are built for speed, not simplicity, and in many cases the method Mathematica uses is nothing like the manual way humans would find the solution.

A small particle simulation was written in JS, utilizing a simplified (constant depth) quadtree structure. The model includes forces between nearby particles, so rather than invoke a O(n^2) operation to compute the net force for each particle, a quadtree is used so each particle may efficiently access its neighbors.
The forces used are tuned to provide some amount of clustering, but also to provide global homogeneity to prevent too many particles appearing in one quadtree section (which would decrease cpu-time efficiency).

This is a simulation of 2 bobs, connected by massless, perfectly rigid rods to a central pivot under the force of uniform gravity. In addition to being the motivating example for chaotic systems (in addition to the Lorenz system, its fluid mechanics counterpart), the double pendulum represents some interesting challenges.
Draw circle bounds Draw cherry tracer Draw connections Pause Clear cherry tracer
Simulation parameters:
Angle 1
Angle 2