Rhombic embeddings of planar graphs with faces of degree 4

TL;DR

This work characterizes unit-length rhombic embeddings of planar graphs with all bounded faces of degree $4$, proving a simple necessary-and-sufficient condition for existence and showing the embedding space is convex when parametrized by rhombus angles. It provides a complete description of extreme points, extends to periodic graphs where the embedding space forms a convex polyhedron, and identifies a unique max-area canonical embedding on the torus. The study connects rhombic embeddings to isoradial embeddings via the diamond graph and frames the problem in terms of train-tracks, asymptotic directions, and a circle-homeomorphism action, yielding a coherent theory for both finite and periodic cases. The parallelogram embedding generalization shows how rhombic and parallelogram structures interrelate, enabling broader geometric control over graph embeddings in the plane.

Abstract

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.

Figures (3)

• Figure 1: Moving a strand past an intersection.
• Figure 2: Moving $\gamma$ at a triple intersection.
• Figure 3: An embedding of ${\mathbb Z}^2$ with image a strip.

Theorems & Definitions (37)

• Definition 1.1
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof