# A Note on the Faces of the Dual Koch Arrangement

## TL;DR

This paper studies the geometric properties of the dual Koch arrangement, revealing it contains no polygons larger than pentagons and quantifying the number of pentagons based on the parameter s.

## Contribution

It provides a detailed analysis of the face structure of the dual Koch arrangement, including bounds on polygon sizes and exact counts of pentagons.

## Key findings

- No k-gons for k > 5 in the arrangement
- Number of pentagons is 3 * 2^{s-1} - 3
- Characterization of faces in the dual Koch arrangement

## Abstract

We analyze the faces of the dual Koch arrangement, which is the arrangement of $2^s + 1$ lines obtained by projective duality from the Koch chain $K_s$. In particular, we show that this line arrangement does not contain any $k$-gons for $k > 5$, and that the number of pentagons is $3 \cdot 2^{s-1} - 3$.