# A bound for crystallographic arrangements

## TL;DR

This paper proves, without computer assistance, that only finitely many irreducible crystallographic arrangements exist in each rank greater than two, refining the previous classification that relied on extensive computer checks.

## Contribution

It provides a computer-free proof establishing finiteness of irreducible crystallographic arrangements in higher ranks, advancing the theoretical understanding of these structures.

## Key findings

- Finiteness of irreducible crystallographic arrangements in ranks > 2
- Elimination of computer-based proofs for classification
- Enhanced theoretical framework for crystallographic arrangements

## Abstract

A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by Heckenberger and the author. However, this classification is based on two computer proofs checking millions of cases. In the present paper, we prove without using a computer that, up to equivalence, there are only finitely many irreducible crystallographic arrangements in each rank greater than two.