Regular finite planar maps with equal edges

TL;DR

This paper proves that finite planar maps cannot have all edges equal in length while each vertex is connected to exactly five edges, highlighting a geometric restriction in such structures.

Contribution

The paper establishes a non-existence result for finite regular planar maps with uniform edge lengths and degree five vertices.

Findings

No finite planar map with all edges equal and vertices of degree five exists.

The result constrains possible configurations of regular planar graphs.

Provides a theoretical proof of non-existence.

Abstract

There doesn't exists a finite planar map with all edges having the same length, and each vertex on exactly 5 edges.