# CAT(0) geometry, robots, and society

arXiv: https://arxiv.org/abs/1912.10007 · HTML: https://mlsanity.com/paper/1912.10007/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2019.

## Built on by (most cited first, 1 of 1)

- [Horoballs and the subgradient method](https://mlsanity.com/paper/2403.15749/) (2024)

## Abstract (arXiv)

How do we move a robot efficiently from one position to another? To answer this question, we need to understand its configuration space, a 'map' where we can find every possible position of the robot. Unfortunately, these maps are very large, they live in high dimensions, and they are very difficult to visualize. Fortunately, for some discrete robots they are CAT(0) cubical complexes, a family of spaces with favorable properties. In this case, using ideas from combinatorics and geometric group theory, we can construct a 'remote control' to navigate these complicated maps, and move the robots optimally. Along the way, we face larger ethical questions that we cannot ignore.