# Visualising high-dimensional state spaces with "Tuple Plots"

## TL;DR

This paper introduces novel geometric projections called 'plan tuple plots' and 'side tuple plots' for visualising high-dimensional state spaces, enhancing understanding of complex system trajectories across various domains.

## Contribution

It generalises Inselberg's parallel coordinates technique, providing new visualization methods that link high-dimensional state spaces with ordinary space representations.

## Key findings

- Effective visualisation of high-dimensional trajectories demonstrated across multiple domains.
- New geometric projections improve interpretability of complex system dynamics.
- Approach bridges high-dimensional data and intuitive spatial understanding.

## Abstract

Complex systems are described with high-dimensional data that is hard to visualise. Inselberg's parallel coordinates are one representation technique for visualising high-dimensional data. Here we generalise Inselberg's approach, and use it for visualising trajectories through high dimensional state spaces. We introduce two geometric projections of parallel coordinate representations -- 'plan tuple plots' and 'side tuple plots' -- and demonstrate a link between state space and ordinary space representations. We provide examples from many domains to illustrate use of the approach, including Cellular Automata, Random Boolean Networks, coupled logistic maps, reservoir computing, search algorithms, Turing Machines, and flocking.