# The Projective Line as a Meridian

## TL;DR

This paper explores the concept of a 'meridian', a reinterpretation of the projective line, examining its structure through involutions, permutation groups, operators, and cross ratios, with a focus on real number fields.

## Contribution

It introduces the notion of a meridian as a new perspective on the projective line, analyzing its properties and symmetries in detail.

## Key findings

- Meridians can be characterized by involutions and permutation groups.
- The cross ratio plays a key role in defining equivalence classes.
- Existence of a real-valued exponential characterizes real number meridians.

## Abstract

The purpose of the present article is to examine the essence of what has commonlybeen described as a "projective line", but which is here named a "meridian". This shall be done in several papers: this first paper devoted to the meridian itself, the second to the character and form of the family of projective isomorphisms of one meridian onto another and the third to some connections between meridians and higher dimensional projective space. Here we view the meridian from various aspects: (1) as a set acted upon by a family of involutions; (2) as a set acted upon by a 3-transitive group of permutations; (3) as a set with a quinary operator; (4) as an equivalence class of quadruples, relating to the cross ratio. In the final section of this first paper we shall investigate how the existence of a certain single-valued exponential on a meridian is characteristic of the meridian corresponding to the field of real numbers.