# Platonic configurations of points and lines

## TL;DR

This paper introduces methods for constructing symmetric spatial configurations of points and lines based on Platonic solids, focusing on balanced and unbalanced arrangements preserved by Euclidean symmetries.

## Contribution

It presents new construction techniques for spatial point-line configurations aligned with Platonic symmetries, expanding understanding of geometric configurations in three-dimensional space.

## Key findings

- Constructed various balanced configurations $(n_{3}), (n_{4}), (n_{5})$
- Developed methods for unbalanced configurations $(p_{3}, n_{4})$, $(p_{3}, n_{5})$, $(p_{4}, n_{5})$
- Configurations are preserved by Euclidean space symmetries of Platonic solids

## Abstract

We present some methods for constructing connected spatial geometric configurations $(p_{q}, n_{k})$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^{3}$ as the chosen Platonic solid. In this paper we are primarily interested in balanced configurations $(n_{3}), (n_{4})$ and $(n_{5})$, but also in unbalanced configurations $(p_{3},n_{4}), (p_{3}, n_{5})$ and $(p_{4}, n_{5})$.