We came across these interesting works that are somewhat tangential to our interests, but fascinating all the same.

Physics in N dimensions

Marc ten Bosch developed a formulation for rigid body dynamics that is independent of the dimension of the space, which is described using geometric algebra. An interesting issue that was also solved was that of collision resolution, which is handled elegantly for convex polytopes which are sufficient to represent typical objects to a high level of accuracy. The paper was accepted to SIGGRAPH 2020 and has been implemented as the commercially available 4D toys as well as an upcoming computer game. Two Minute Papers also covered this game/research.

Reaction-diffusion dynamics

Reaction-diffusion dynamics generalize the diffusion-only dynamics readers may be familiar with from the field of networked systems, which can actually be thought of as a discretization of the diffusion/heat partial differential equation. When the new reaction term is added, rather interesting behavior emerges due to the nonlinearities in the dynamics.

This page shows Robert Munafo’s exploration of different regimes of the parameter space of the reaction-diffusion dynamics.

One particularly interesting setting is called the “U-skate world,” which exhibits emergent patterns somewhat similar to Conway’s Game of Life e.g. the appearance of “still-lifes,” small portions of the state-space that retain their structure over time and “gliders,” small portions of the state-space that both retain their structure over time and move across the state-space by delicate balancing of the forces over their bodies.

SmoothLife

Conway’s Game of Life was a seminal development in computer science that spurred thought about information theory. It was a simple set of rules meant to mimic conditions of population pressures and incentives in living creatures, known as a cellular automata. The game is played on a grid and each cell has a binary state which is updated at discrete time steps. Stephan Rafler generalized the mathematical formulation of this game to continuous state and time, leading to qualitatively similar behavior as Conway’s Game, but with much more organic and arguably richer phenomena, including self-propelled gliders and chain-like structures. Check out the paper, the code, and some YouTube videos. Lex Fridman also interviewed Stephen Wolfram, another giant of computing, wherein they discussed cellular automata.